3.408 \(\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=612 \[ -\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \log \left (-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \log \left (\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{4 d}+\frac{i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d} \]

[Out]

(((15*I)/8)*a^3*(e*Sec[c + d*x])^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (((15*I)/8)*a^(7/2)*e^(3/2)*ArcTan[1
- (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a
- I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((15*I)/8)*a^(7/2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqr
t[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*S
qrt[a + I*a*Tan[c + d*x]]) + (((15*I)/16)*a^(7/2)*e^(3/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c
+ d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a - I*a*Tan
[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (((15*I)/16)*a^(7/2)*e^(3/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a
- I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt
[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((3*I)/4)*a^2*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c
+ d*x]])/d + ((I/3)*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/d

________________________________________________________________________________________

Rubi [A]  time = 0.689787, antiderivative size = 612, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3498, 3499, 3495, 297, 1162, 617, 204, 1165, 628} \[ -\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \log \left (-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \log \left (\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{4 d}+\frac{i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2),x]

[Out]

(((15*I)/8)*a^3*(e*Sec[c + d*x])^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (((15*I)/8)*a^(7/2)*e^(3/2)*ArcTan[1
- (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a
- I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((15*I)/8)*a^(7/2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqr
t[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*S
qrt[a + I*a*Tan[c + d*x]]) + (((15*I)/16)*a^(7/2)*e^(3/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c
+ d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a - I*a*Tan
[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (((15*I)/16)*a^(7/2)*e^(3/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a
- I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt
[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((3*I)/4)*a^2*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c
+ d*x]])/d + ((I/3)*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/d

Rule 3498

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] + Dist[(a*(m + 2*n - 2))/(m + n - 1), Int[(
d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] &&
GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 3499

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(3/2)/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(d*Sec
[e + f*x])/(Sqrt[a - b*Tan[e + f*x]]*Sqrt[a + b*Tan[e + f*x]]), Int[Sqrt[d*Sec[e + f*x]]*Sqrt[a - b*Tan[e + f*
x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]

Rule 3495

Int[Sqrt[(d_.)*sec[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(-4*b*d^
2)/f, Subst[Int[x^2/(a^2 + d^2*x^4), x], x, Sqrt[a + b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]]], x] /; FreeQ[{a, b,
d, e, f}, x] && EqQ[a^2 + b^2, 0]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx &=\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{1}{2} (3 a) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2} \, dx\\ &=\frac{3 i a^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{1}{8} \left (15 a^2\right ) \int (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)} \, dx\\ &=\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{1}{16} \left (15 a^3\right ) \int \frac{(e \sec (c+d x))^{3/2}}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{\left (15 a^3 e \sec (c+d x)\right ) \int \sqrt{e \sec (c+d x)} \sqrt{a-i a \tan (c+d x)} \, dx}{16 \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{\left (15 i a^4 e^3 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{4 d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac{\left (15 i a^4 e^2 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{8 d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (15 i a^4 e^2 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{8 d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{\left (15 i a^4 e \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{e}-\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}+x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{16 d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (15 i a^4 e \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{e}+\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}+x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{16 d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (15 i a^{7/2} e^{3/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{a}}{\sqrt{e}}+2 x}{-\frac{a}{e}-\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}-x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (15 i a^{7/2} e^{3/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{a}}{\sqrt{e}}-2 x}{-\frac{a}{e}+\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}-x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \log \left (a-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{15 i a^{7/2} e^{3/2} \log \left (a+\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{\left (15 i a^{7/2} e^{3/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{\left (15 i a^{7/2} e^{3/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}-\frac{15 i a^{7/2} e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \log \left (a-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{15 i a^{7/2} e^{3/2} \log \left (a+\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d}\\ \end{align*}

Mathematica [A]  time = 20.5643, size = 417, normalized size = 0.68 \[ \frac{i a^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2} \left (\sqrt{-\sin (c)+i \cos (c)+1} \left (\sqrt{-\sin (c)+i \cos (c)-1} \sqrt{-\tan \left (\frac{d x}{2}\right )+i} (-13 i \sin (c+d x)-45 i \sin (3 (c+d x))+239 \cos (c+d x)+45 \cos (3 (c+d x)))-180 \sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left (\frac{d x}{2}\right )+i} \cos ^3(c+d x) \tan ^{-1}\left (\frac{\sqrt{-\sin (c)+i \cos (c)-1} \sqrt{-\tan \left (\frac{d x}{2}\right )+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left (\frac{d x}{2}\right )+i}}\right )\right )-180 \sqrt{-\sin (c)-i \cos (c)+1} \sqrt{-\sin (c)+i \cos (c)-1} \sqrt{\tan \left (\frac{d x}{2}\right )+i} \cos ^3(c+d x) \tan ^{-1}\left (\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left (\frac{d x}{2}\right )+i}}{\sqrt{-\sin (c)-i \cos (c)+1} \sqrt{\tan \left (\frac{d x}{2}\right )+i}}\right )\right )}{96 d e \sqrt{-\sin (c)+i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left (\frac{d x}{2}\right )+i}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2),x]

[Out]

((I/96)*a^2*(e*Sec[c + d*x])^(5/2)*(-180*ArcTan[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[1 -
I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Cos[c + d*x]^3*Sqrt[1 - I*Cos[c] - Sin[c]]*Sqrt[-1 + I*Cos[c] - S
in[c]]*Sqrt[I + Tan[(d*x)/2]] + Sqrt[1 + I*Cos[c] - Sin[c]]*(Sqrt[-1 + I*Cos[c] - Sin[c]]*(239*Cos[c + d*x] +
45*Cos[3*(c + d*x)] - (13*I)*Sin[c + d*x] - (45*I)*Sin[3*(c + d*x)])*Sqrt[I - Tan[(d*x)/2]] - 180*ArcTan[(Sqrt
[-1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Cos[c
+ d*x]^3*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]]))*Sqrt[a + I*a*Tan[c + d*x]])/(d*e*Sqrt[-1 + I*Co
s[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])

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Maple [A]  time = 0.294, size = 424, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x)

[Out]

-1/48/d*a^2*(cos(d*x+c)-1)^2*(45*I*arctanh(1/2*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1-sin(d*x+c)))*cos(d*x+c)^
3-45*I*arctanh(1/2*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))*cos(d*x+c)^3-90*I*(1/(cos(d*x+c)+1))^(1
/2)*cos(d*x+c)^2*sin(d*x+c)-45*cos(d*x+c)^3*arctanh(1/2*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1-sin(d*x+c)))-45
*cos(d*x+c)^3*arctanh(1/2*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))+68*I*(1/(cos(d*x+c)+1))^(1/2)*co
s(d*x+c)*sin(d*x+c)+90*cos(d*x+c)^3*(1/(cos(d*x+c)+1))^(1/2)+16*I*sin(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)+158*cos(
d*x+c)^2*(1/(cos(d*x+c)+1))^(1/2)+52*cos(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)-16*(1/(cos(d*x+c)+1))^(1/2))*(a*(I*si
n(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(e/cos(d*x+c))^(3/2)/(I*sin(d*x+c)+cos(d*x+c)-1)/(1/(cos(d*x+c)+1))^(3/
2)/cos(d*x+c)/sin(d*x+c)^3

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Maxima [B]  time = 3.39107, size = 4074, normalized size = 6.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

(347136*a^2*e*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 387072*a^2*e*cos(5/4*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c))) + 138240*a^2*e*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 347136*I*a^2*e*
sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 387072*I*a^2*e*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*
d*x + 2*c))) + 138240*I*a^2*e*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (17280*sqrt(2)*a^2*e*cos(
6*d*x + 6*c) + 51840*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 51840*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + 17280*I*sqrt(2)*a
^2*e*sin(6*d*x + 6*c) + 51840*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 51840*I*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + 1728
0*sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, sqrt(2)*sin(1/4*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - (17280*sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 51840*sqrt(2)*a^2*e*c
os(4*d*x + 4*c) + 51840*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + 17280*I*sqrt(2)*a^2*e*sin(6*d*x + 6*c) + 51840*I*sqrt
(2)*a^2*e*sin(4*d*x + 4*c) + 51840*I*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + 17280*sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos
(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, -sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) + 1) - (17280*sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 51840*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 51840*sqrt(2)*a^2
*e*cos(2*d*x + 2*c) + 17280*I*sqrt(2)*a^2*e*sin(6*d*x + 6*c) + 51840*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 51840*
I*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + 17280*sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c))) - 1, sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - (17280*sqrt(2)*a^2*e*c
os(6*d*x + 6*c) + 51840*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 51840*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + 17280*I*sqrt(2
)*a^2*e*sin(6*d*x + 6*c) + 51840*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 51840*I*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + 1
7280*sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1, -sqrt(2)*sin(1/4
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - (17280*I*sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 51840*I*sqrt(2)
*a^2*e*cos(4*d*x + 4*c) + 51840*I*sqrt(2)*a^2*e*cos(2*d*x + 2*c) - 17280*sqrt(2)*a^2*e*sin(6*d*x + 6*c) - 5184
0*sqrt(2)*a^2*e*sin(4*d*x + 4*c) - 51840*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + 17280*I*sqrt(2)*a^2*e)*arctan2(sqrt(
2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))
, sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) + 1) - (-17280*I*sqrt(2)*a^2*e*cos(6*d*x + 6*c) - 51840*I*sqrt(2)*a^2*e*cos(4*d*x + 4*c) - 51840*I*sqr
t(2)*a^2*e*cos(2*d*x + 2*c) + 17280*sqrt(2)*a^2*e*sin(6*d*x + 6*c) + 51840*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 51
840*sqrt(2)*a^2*e*sin(2*d*x + 2*c) - 17280*I*sqrt(2)*a^2*e)*arctan2(-sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))), -sqrt(2)*cos(1/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - (8640*sqrt(2)*a^2*e*
cos(6*d*x + 6*c) + 25920*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 25920*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + 8640*I*sqrt(2
)*a^2*e*sin(6*d*x + 6*c) + 25920*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 25920*I*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + 8
640*sqrt(2)*a^2*e)*log(2*sqrt(2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))*sin(1/4*arctan2(sin(2*d*
x + 2*c), cos(2*d*x + 2*c))) + 2*(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1)*cos(1/2*ar
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*cos(1/
4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*
sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c))) + 1) + (8640*sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 25920*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 25920*sqrt(2)*a
^2*e*cos(2*d*x + 2*c) + 8640*I*sqrt(2)*a^2*e*sin(6*d*x + 6*c) + 25920*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 25920
*I*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + 8640*sqrt(2)*a^2*e)*log(-2*sqrt(2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2
*d*x + 2*c)))*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*
c), cos(2*d*x + 2*c))) - 1)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c)))^2 + 2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1
/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - (8640*I*sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 25920*I*sqrt(2
)*a^2*e*cos(4*d*x + 4*c) + 25920*I*sqrt(2)*a^2*e*cos(2*d*x + 2*c) - 8640*sqrt(2)*a^2*e*sin(6*d*x + 6*c) - 2592
0*sqrt(2)*a^2*e*sin(4*d*x + 4*c) - 25920*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + 8640*I*sqrt(2)*a^2*e)*log(2*cos(1/4*
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*
sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos
(2*d*x + 2*c))) + 2) - (-8640*I*sqrt(2)*a^2*e*cos(6*d*x + 6*c) - 25920*I*sqrt(2)*a^2*e*cos(4*d*x + 4*c) - 2592
0*I*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + 8640*sqrt(2)*a^2*e*sin(6*d*x + 6*c) + 25920*sqrt(2)*a^2*e*sin(4*d*x + 4*c
) + 25920*sqrt(2)*a^2*e*sin(2*d*x + 2*c) - 8640*I*sqrt(2)*a^2*e)*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2
*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) - (8640*I*
sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 25920*I*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 25920*I*sqrt(2)*a^2*e*cos(2*d*x + 2*
c) - 8640*sqrt(2)*a^2*e*sin(6*d*x + 6*c) - 25920*sqrt(2)*a^2*e*sin(4*d*x + 4*c) - 25920*sqrt(2)*a^2*e*sin(2*d*
x + 2*c) + 8640*I*sqrt(2)*a^2*e)*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arct
an2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) +
2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) - (-8640*I*sqrt(2)*a^2*e*cos(6*d*x + 6*c)
- 25920*I*sqrt(2)*a^2*e*cos(4*d*x + 4*c) - 25920*I*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + 8640*sqrt(2)*a^2*e*sin(6*d
*x + 6*c) + 25920*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 25920*sqrt(2)*a^2*e*sin(2*d*x + 2*c) - 8640*I*sqrt(2)*a^2*e
)*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c))) + 2))*sqrt(a)*sqrt(e)/(d*(-36864*I*cos(6*d*x + 6*c) - 110592*I*cos(4*d*x + 4*c
) - 110592*I*cos(2*d*x + 2*c) + 36864*sin(6*d*x + 6*c) + 110592*sin(4*d*x + 4*c) + 110592*sin(2*d*x + 2*c) - 3
6864*I))

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Fricas [A]  time = 2.29891, size = 2105, normalized size = 3.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/12*((113*I*a^2*e*e^(4*I*d*x + 4*I*c) + 126*I*a^2*e*e^(2*I*d*x + 2*I*c) + 45*I*a^2*e)*sqrt(a/(e^(2*I*d*x + 2*
I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(3/2*I*d*x + 3/2*I*c) - 6*sqrt(225/64*I*a^5*e^3/d^2)*(d*e^(5*I*
d*x + 5*I*c) + 2*d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*log(2/15*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*s
qrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(3/2*I*d*x + 3/2*I*c) + 8*sqrt(225/64*I*a
^5*e^3/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(a^2*e)) + 6*sqrt(225/64*I*a^5*e^3/d^2)*(d*e^(5*I*d*x
+ 5*I*c) + 2*d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*log(2/15*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(
a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(3/2*I*d*x + 3/2*I*c) - 8*sqrt(225/64*I*a^5*e
^3/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(a^2*e)) + 6*sqrt(-225/64*I*a^5*e^3/d^2)*(d*e^(5*I*d*x + 5
*I*c) + 2*d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*log(2/15*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(a/(
e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(3/2*I*d*x + 3/2*I*c) + 8*sqrt(-225/64*I*a^5*e^3
/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(a^2*e)) - 6*sqrt(-225/64*I*a^5*e^3/d^2)*(d*e^(5*I*d*x + 5*I
*c) + 2*d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*log(2/15*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(a/(e^
(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(3/2*I*d*x + 3/2*I*c) - 8*sqrt(-225/64*I*a^5*e^3/d
^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(a^2*e)))/(d*e^(5*I*d*x + 5*I*c) + 2*d*e^(3*I*d*x + 3*I*c) + d
*e^(I*d*x + I*c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))**(3/2)*(a+I*a*tan(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((e*sec(d*x + c))^(3/2)*(I*a*tan(d*x + c) + a)^(5/2), x)