\(\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx\) [213]

Optimal result

Integrand size = 28, antiderivative size = 215 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=-\frac {22 a^4 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^4 (e \sec (c+d x))^{3/2}}{9 d}+\frac {22 a^4 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {10 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )^2}{21 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{21 d} \]

[Out]

22/9*I*a^4*(e*sec(d*x+c))^(3/2)/d-22/3*a^4*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1
/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)+22/3*a^4*e*sin(d*x+c)*(e*sec(d*x+c))^(1/2)/d+2/
9*I*a*(e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^3/d+10/21*I*(e*sec(d*x+c))^(3/2)*(a^2+I*a^2*tan(d*x+c))^2/d+22/2
1*I*(e*sec(d*x+c))^(3/2)*(a^4+I*a^4*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3579, 3567, 3853, 3856, 2719} \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=-\frac {22 a^4 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^4 (e \sec (c+d x))^{3/2}}{9 d}+\frac {22 a^4 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{3 d}+\frac {22 i \left (a^4+i a^4 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{21 d}+\frac {10 i \left (a^2+i a^2 \tan (c+d x)\right )^2 (e \sec (c+d x))^{3/2}}{21 d}+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d} \]

[In]

Int[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^4,x]

[Out]

(-22*a^4*e^2*EllipticE[(c + d*x)/2, 2])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((22*I)/9)*a^4*(e*Sec
[c + d*x])^(3/2))/d + (22*a^4*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (((2*I)/9)*a*(e*Sec[c + d*x])^(3/2)
*(a + I*a*Tan[c + d*x])^3)/d + (((10*I)/21)*(e*Sec[c + d*x])^(3/2)*(a^2 + I*a^2*Tan[c + d*x])^2)/d + (((22*I)/
21)*(e*Sec[c + d*x])^(3/2)*(a^4 + I*a^4*Tan[c + d*x]))/d

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3567

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

Rule 3579

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 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = \frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {1}{3} (5 a) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx \\ & = \frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {10 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )^2}{21 d}+\frac {1}{21} \left (55 a^2\right ) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2 \, dx \\ & = \frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {10 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )^2}{21 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}+\frac {1}{3} \left (11 a^3\right ) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx \\ & = \frac {22 i a^4 (e \sec (c+d x))^{3/2}}{9 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {10 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )^2}{21 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}+\frac {1}{3} \left (11 a^4\right ) \int (e \sec (c+d x))^{3/2} \, dx \\ & = \frac {22 i a^4 (e \sec (c+d x))^{3/2}}{9 d}+\frac {22 a^4 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {10 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )^2}{21 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac {1}{3} \left (11 a^4 e^2\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx \\ & = \frac {22 i a^4 (e \sec (c+d x))^{3/2}}{9 d}+\frac {22 a^4 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {10 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )^2}{21 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac {\left (11 a^4 e^2\right ) \int \sqrt {\cos (c+d x)} \, dx}{3 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = -\frac {22 a^4 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^4 (e \sec (c+d x))^{3/2}}{9 d}+\frac {22 a^4 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {10 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )^2}{21 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{21 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 7.21 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.52 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\frac {(e \sec (c+d x))^{3/2} \left (\frac {22 i \sqrt {2} e^{-i (3 c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )}{-1+e^{2 i c}}+\frac {1}{56} \csc (c) \sec ^{\frac {9}{2}}(c+d x) (\cos (4 c)-i \sin (4 c)) (1260 \cos (d x)+1050 \cos (2 c+d x)+742 \cos (2 c+3 d x)+413 \cos (4 c+3 d x)+231 \cos (4 c+5 d x)-720 i \sin (d x)+720 i \sin (2 c+d x)-336 i \sin (2 c+3 d x)+336 i \sin (4 c+3 d x))\right ) (a+i a \tan (c+d x))^4}{9 d \sec ^{\frac {11}{2}}(c+d x) (\cos (d x)+i \sin (d x))^4} \]

[In]

Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^4,x]

[Out]

((e*Sec[c + d*x])^(3/2)*(((22*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c
+ d*x))]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4,
-E^((2*I)*(c + d*x))]))/(E^(I*(3*c + d*x))*(-1 + E^((2*I)*c))) + (Csc[c]*Sec[c + d*x]^(9/2)*(Cos[4*c] - I*Sin
[4*c])*(1260*Cos[d*x] + 1050*Cos[2*c + d*x] + 742*Cos[2*c + 3*d*x] + 413*Cos[4*c + 3*d*x] + 231*Cos[4*c + 5*d*
x] - (720*I)*Sin[d*x] + (720*I)*Sin[2*c + d*x] - (336*I)*Sin[2*c + 3*d*x] + (336*I)*Sin[4*c + 3*d*x]))/56)*(a
+ I*a*Tan[c + d*x])^4)/(9*d*Sec[c + d*x]^(11/2)*(Cos[d*x] + I*Sin[d*x])^4)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (208 ) = 416\).

Time = 24.52 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.23

method	result	size
default	\(-\frac {2 i e \,a^{4} \sqrt {e \sec \left (d x +c \right )}\, \left (231 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )-231 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+462 \cos \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )-462 F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+231 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )-231 F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-168+231 i \sin \left (d x +c \right )-168 \sec \left (d x +c \right )-91 i \tan \left (d x +c \right )+36 \left (\sec ^{2}\left (d x +c \right )\right )-91 i \tan \left (d x +c \right ) \sec \left (d x +c \right )+36 \left (\sec ^{3}\left (d x +c \right )\right )+7 i \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )+7 i \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )\right )}{63 d \left (\cos \left (d x +c \right )+1\right )}\)	\(480\)
parts	\(\text {Expression too large to display}\)	\(1322\)

[In]

int((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

-2/63*I*e*a^4/d*(e*sec(d*x+c))^(1/2)/(cos(d*x+c)+1)*(231*cos(d*x+c)^2*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(co
s(d*x+c)+1))^(1/2)*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I)-231*cos(d*x+c)^2*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)+462*cos(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*
x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I)-462*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)*(
1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+231*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I)-231*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)*(1/(co
s(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-168+231*I*sin(d*x+c)-168*sec(d*x+c)-91*I*tan(d*x+c)+36*se
c(d*x+c)^2-91*I*tan(d*x+c)*sec(d*x+c)+36*sec(d*x+c)^3+7*I*tan(d*x+c)*sec(d*x+c)^2+7*I*tan(d*x+c)*sec(d*x+c)^3)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.16 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (231 i \, a^{4} e e^{\left (9 i \, d x + 9 i \, c\right )} + 406 i \, a^{4} e e^{\left (7 i \, d x + 7 i \, c\right )} + 540 i \, a^{4} e e^{\left (5 i \, d x + 5 i \, c\right )} + 330 i \, a^{4} e e^{\left (3 i \, d x + 3 i \, c\right )} + 77 i \, a^{4} e e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 231 \, \sqrt {2} {\left (i \, a^{4} e e^{\left (8 i \, d x + 8 i \, c\right )} + 4 i \, a^{4} e e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, a^{4} e e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, a^{4} e e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{4} e\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{63 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-2/63*(sqrt(2)*(231*I*a^4*e*e^(9*I*d*x + 9*I*c) + 406*I*a^4*e*e^(7*I*d*x + 7*I*c) + 540*I*a^4*e*e^(5*I*d*x + 5
*I*c) + 330*I*a^4*e*e^(3*I*d*x + 3*I*c) + 77*I*a^4*e*e^(I*d*x + I*c))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2
*I*d*x + 1/2*I*c) + 231*sqrt(2)*(I*a^4*e*e^(8*I*d*x + 8*I*c) + 4*I*a^4*e*e^(6*I*d*x + 6*I*c) + 6*I*a^4*e*e^(4*
I*d*x + 4*I*c) + 4*I*a^4*e*e^(2*I*d*x + 2*I*c) + I*a^4*e)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-
4, 0, e^(I*d*x + I*c))))/(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2
*I*d*x + 2*I*c) + d)

Sympy [F]

\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=a^{4} \left (\int \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (- 6 \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (c + d x \right )}\right )\, dx + \int \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{4}{\left (c + d x \right )}\, dx + \int 4 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan {\left (c + d x \right )}\, dx + \int \left (- 4 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (c + d x \right )}\right )\, dx\right ) \]

[In]

integrate((e*sec(d*x+c))**(3/2)*(a+I*a*tan(d*x+c))**4,x)

[Out]

a**4*(Integral((e*sec(c + d*x))**(3/2), x) + Integral(-6*(e*sec(c + d*x))**(3/2)*tan(c + d*x)**2, x) + Integra
l((e*sec(c + d*x))**(3/2)*tan(c + d*x)**4, x) + Integral(4*I*(e*sec(c + d*x))**(3/2)*tan(c + d*x), x) + Integr
al(-4*I*(e*sec(c + d*x))**(3/2)*tan(c + d*x)**3, x))

Maxima [F]

\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

integrate((e*sec(d*x + c))^(3/2)*(I*a*tan(d*x + c) + a)^4, x)

Giac [F]

\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((e*sec(d*x + c))^(3/2)*(I*a*tan(d*x + c) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4 \,d x \]

[In]

int((e/cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

int((e/cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^4, x)