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edited Nov 23, 2017 at 12:02

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integrand = q^2/(q^2 + ks^2) (1 - (1/\[Tau]^2τ^2 - c^2 q^2) \[Tau]^2τ^2)/(1 + 2 (\[Omega]^2ω^2 - (1/\[Tau]^2τ^2 - c^2 q^2)) \[Tau]^2τ^2 + (\[Omega]^2ω^2 + (1/\[Tau]^2τ^2 - c^2 q^2))^2 \[Tau]^4τ^4); fs = FullSimplify[ integrand, {ks > 0, \[Omega]ω > 0, \[Tau]τ > 0, c > 0}] (* (c^2 q^4)/((ks^2 + q^2) (4 \[Omega]^2ω^2 + \[Tau]^2τ^2 (-c^2 q^2 + \[Omega]^2ω^2)^2)) *) int = Integrate[fs, {q, 0, Infinity}, Assumptions -> {ks > 0, \[Omega]ω > 0, \[Tau]τ > 0, c > 0}] (* (\[Pi]π (4 c^3 ks^3 \[Tau]^τ^(3/2) Sqrt[-\[Tau]τ + ( 2 I)/\[Omega]]ω] + \[Omega]ω (4 + \[Tau]^2τ^2 \[Omega]^2ω^2) (2 + I \[Tau]τ \[Omega]ω + I Sqrt[-2 I - \[Tau]τ \[Omega]]ω] Sqrt[2 I - \[Tau]τ \[Omega]]ω]) + c^2 ks^2 \[Tau]τ (\[Tau]τ \[Omega]ω (4 + I \[Tau]τ \[Omega]ω + I Sqrt[-2 I - \[Tau]τ \[Omega]]ω] Sqrt[ 2 I - \[Tau]τ \[Omega]]ω]) - 2 (2 I + Sqrt[-2 I - \[Tau]τ \[Omega]]ω] Sqrt[ 2 I - \[Tau]τ \[Omega]]ω]))))/(8 c \[Tau]^τ^(3/2) Sqrt[-\[Tau]τ + ( 2 I)/\[Omega]]ω] (4 \[Omega]^2ω^2 + \[Tau]^2τ^2 (c^2 ks^2 + \ \[Omega]^2ω^2)^2)) *) ceRe = FullSimplify[ ComplexExpand[Re[int], TargetFunctions -> {Re, Im}], {ks > 0, \[Omega]ω > 0, \[Tau]τ > 0, c > 0}] (* (\[Pi]π (4 c^3 ks^3 \[Tau]^2τ^2 (\[Tau]^2τ^2 + 4/\[Omega]^2ω^2)^(1/4) + 2 Sqrt[\[Tau]Sqrt[τ/( 2 + (2 \[Tau]τ \[Omega]ω)/Sqrt[ 4 + \[Tau]^2τ^2 \[Omega]^2]ω^2])] (\[Omega]ω (4 + \[Tau]^2τ^2 \[Omega]^2ω^2) \ (\[Tau]τ \[Omega]ω + Sqrt[4 + \[Tau]^2τ^2 \[Omega]^2]ω^2]) + c^2 ks^2 \[Tau]τ (-4 + \[Tau]τ \[Omega]ω (\[Tau]τ \[Omega]ω + Sqrt[ 4 + \[Tau]^2τ^2 \[Omega]^2]ω^2])))))/(8 c \[Tau]^2τ^2 (\[Tau]^2τ^2 + 4/\[Omega]^2ω^2)^( 1/4) (4 \[Omega]^2ω^2 + \[Tau]^2τ^2 (c^2 ks^2 + \[Omega]^2ω^2)^2)) *)

integrand = q^2/(q^2 + ks^2) (1 - (1/\[Tau]^2 - c^2 q^2) \[Tau]^2)/(1 + 2 (\[Omega]^2 - (1/\[Tau]^2 - c^2 q^2)) \[Tau]^2 + (\[Omega]^2 + (1/\[Tau]^2 - c^2 q^2))^2 \[Tau]^4); fs = FullSimplify[ integrand, {ks > 0, \[Omega] > 0, \[Tau] > 0, c > 0}] (* (c^2 q^4)/((ks^2 + q^2) (4 \[Omega]^2 + \[Tau]^2 (-c^2 q^2 + \[Omega]^2)^2)) *) int = Integrate[fs, {q, 0, Infinity}, Assumptions -> {ks > 0, \[Omega] > 0, \[Tau] > 0, c > 0}] (* (\[Pi] (4 c^3 ks^3 \[Tau]^(3/2) Sqrt[-\[Tau] + ( 2 I)/\[Omega]] + \[Omega] (4 + \[Tau]^2 \[Omega]^2) (2 + I \[Tau] \[Omega] + I Sqrt[-2 I - \[Tau] \[Omega]] Sqrt[2 I - \[Tau] \[Omega]]) + c^2 ks^2 \[Tau] (\[Tau] \[Omega] (4 + I \[Tau] \[Omega] + I Sqrt[-2 I - \[Tau] \[Omega]] Sqrt[ 2 I - \[Tau] \[Omega]]) - 2 (2 I + Sqrt[-2 I - \[Tau] \[Omega]] Sqrt[ 2 I - \[Tau] \[Omega]]))))/(8 c \[Tau]^(3/2) Sqrt[-\[Tau] + ( 2 I)/\[Omega]] (4 \[Omega]^2 + \[Tau]^2 (c^2 ks^2 + \ \[Omega]^2)^2)) *) ceRe = FullSimplify[ ComplexExpand[Re[int], TargetFunctions -> {Re, Im}], {ks > 0, \[Omega] > 0, \[Tau] > 0, c > 0}] (* (\[Pi] (4 c^3 ks^3 \[Tau]^2 (\[Tau]^2 + 4/\[Omega]^2)^(1/4) + 2 Sqrt[\[Tau]/( 2 + (2 \[Tau] \[Omega])/Sqrt[ 4 + \[Tau]^2 \[Omega]^2])] (\[Omega] (4 + \[Tau]^2 \[Omega]^2) \ (\[Tau] \[Omega] + Sqrt[4 + \[Tau]^2 \[Omega]^2]) + c^2 ks^2 \[Tau] (-4 + \[Tau] \[Omega] (\[Tau] \[Omega] + Sqrt[ 4 + \[Tau]^2 \[Omega]^2])))))/(8 c \[Tau]^2 (\[Tau]^2 + 4/\[Omega]^2)^( 1/4) (4 \[Omega]^2 + \[Tau]^2 (c^2 ks^2 + \[Omega]^2)^2)) *)

integrand = q^2/(q^2 + ks^2) (1 - (1/τ^2 - c^2 q^2) τ^2)/(1 + 2 (ω^2 - (1/τ^2 - c^2 q^2)) τ^2 + (ω^2 + (1/τ^2 - c^2 q^2))^2 τ^4); fs = FullSimplify[ integrand, {ks > 0, ω > 0, τ > 0, c > 0}] (* (c^2 q^4)/((ks^2 + q^2) (4 ω^2 + τ^2 (-c^2 q^2 + ω^2)^2)) *) int = Integrate[fs, {q, 0, Infinity}, Assumptions -> {ks > 0, ω > 0, τ > 0, c > 0}] (* (π (4 c^3 ks^3 τ^(3/2) Sqrt[-τ + ( 2 I)/ω] + ω (4 + τ^2 ω^2) (2 + I τ ω + I Sqrt[-2 I - τ ω] Sqrt[2 I - τ ω]) + c^2 ks^2 τ (τ ω (4 + I τ ω + I Sqrt[-2 I - τ ω] Sqrt[ 2 I - τ ω]) - 2 (2 I + Sqrt[-2 I - τ ω] Sqrt[ 2 I - τ ω]))))/(8 c τ^(3/2) Sqrt[-τ + ( 2 I)/ω] (4 ω^2 + τ^2 (c^2 ks^2 + \ ω^2)^2)) *) ceRe = FullSimplify[ ComplexExpand[Re[int], TargetFunctions -> {Re, Im}], {ks > 0, ω > 0, τ > 0, c > 0}] (* (π (4 c^3 ks^3 τ^2 (τ^2 + 4/ω^2)^(1/4) + 2 Sqrt[τ/( 2 + (2 τ ω)/Sqrt[ 4 + τ^2 ω^2])] (ω (4 + τ^2 ω^2) \ (τ ω + Sqrt[4 + τ^2 ω^2]) + c^2 ks^2 τ (-4 + τ ω (τ ω + Sqrt[ 4 + τ^2 ω^2])))))/(8 c τ^2 (τ^2 + 4/ω^2)^( 1/4) (4 ω^2 + τ^2 (c^2 ks^2 + ω^2)^2)) *)

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answered Nov 23, 2017 at 8:40

Akku14

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With Mathematica Version 8.0, I get good result, with no imaginary part.

integrand = q^2/(q^2 + ks^2) (1 - (1/\[Tau]^2 - c^2 q^2) \[Tau]^2)/(1 + 2 (\[Omega]^2 - (1/\[Tau]^2 - c^2 q^2)) \[Tau]^2 + (\[Omega]^2 + (1/\[Tau]^2 - c^2 q^2))^2 \[Tau]^4); fs = FullSimplify[ integrand, {ks > 0, \[Omega] > 0, \[Tau] > 0, c > 0}] (* (c^2 q^4)/((ks^2 + q^2) (4 \[Omega]^2 + \[Tau]^2 (-c^2 q^2 + \[Omega]^2)^2)) *) int = Integrate[fs, {q, 0, Infinity}, Assumptions -> {ks > 0, \[Omega] > 0, \[Tau] > 0, c > 0}] (* (\[Pi] (4 c^3 ks^3 \[Tau]^(3/2) Sqrt[-\[Tau] + ( 2 I)/\[Omega]] + \[Omega] (4 + \[Tau]^2 \[Omega]^2) (2 + I \[Tau] \[Omega] + I Sqrt[-2 I - \[Tau] \[Omega]] Sqrt[2 I - \[Tau] \[Omega]]) + c^2 ks^2 \[Tau] (\[Tau] \[Omega] (4 + I \[Tau] \[Omega] + I Sqrt[-2 I - \[Tau] \[Omega]] Sqrt[ 2 I - \[Tau] \[Omega]]) - 2 (2 I + Sqrt[-2 I - \[Tau] \[Omega]] Sqrt[ 2 I - \[Tau] \[Omega]]))))/(8 c \[Tau]^(3/2) Sqrt[-\[Tau] + ( 2 I)/\[Omega]] (4 \[Omega]^2 + \[Tau]^2 (c^2 ks^2 + \ \[Omega]^2)^2)) *) ceRe = FullSimplify[ ComplexExpand[Re[int], TargetFunctions -> {Re, Im}], {ks > 0, \[Omega] > 0, \[Tau] > 0, c > 0}] (* (\[Pi] (4 c^3 ks^3 \[Tau]^2 (\[Tau]^2 + 4/\[Omega]^2)^(1/4) + 2 Sqrt[\[Tau]/( 2 + (2 \[Tau] \[Omega])/Sqrt[ 4 + \[Tau]^2 \[Omega]^2])] (\[Omega] (4 + \[Tau]^2 \[Omega]^2) \ (\[Tau] \[Omega] + Sqrt[4 + \[Tau]^2 \[Omega]^2]) + c^2 ks^2 \[Tau] (-4 + \[Tau] \[Omega] (\[Tau] \[Omega] + Sqrt[ 4 + \[Tau]^2 \[Omega]^2])))))/(8 c \[Tau]^2 (\[Tau]^2 + 4/\[Omega]^2)^( 1/4) (4 \[Omega]^2 + \[Tau]^2 (c^2 ks^2 + \[Omega]^2)^2)) *)

Imaginary part is zero.