\(\int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx\) [261]

Optimal result

Integrand size = 28, antiderivative size = 163 \[ \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx=\frac {2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e^3 \sin (c+d x)}{117 a^4 d (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 a d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {4 i e^4}{117 d (e \sec (c+d x))^{5/2} \left (a^4+i a^4 \tan (c+d x)\right )} \]

[Out]

2/117*e^3*sin(d*x+c)/a^4/d/(e*sec(d*x+c))^(3/2)+2/39*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellip
ticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^4/d/cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)+4/13*I*e^2/a/d/(e*sec(d*x+c))^(1/
2)/(a+I*a*tan(d*x+c))^3+4/117*I*e^4/d/(e*sec(d*x+c))^(5/2)/(a^4+I*a^4*tan(d*x+c))

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3581, 3854, 3856, 2719} \[ \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx=\frac {4 i e^4}{117 d \left (a^4+i a^4 \tan (c+d x)\right ) (e \sec (c+d x))^{5/2}}+\frac {2 e^3 \sin (c+d x)}{117 a^4 d (e \sec (c+d x))^{3/2}}+\frac {2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{13 a d (a+i a \tan (c+d x))^3 \sqrt {e \sec (c+d x)}} \]

[In]

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

[Out]

(2*e^2*EllipticE[(c + d*x)/2, 2])/(39*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*e^3*Sin[c + d*x])/(1
17*a^4*d*(e*Sec[c + d*x])^(3/2)) + (((4*I)/13)*e^2)/(a*d*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^3) + (((4
*I)/117)*e^4)/(d*(e*Sec[c + d*x])^(5/2)*(a^4 + I*a^4*Tan[c + d*x]))

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 3581

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

Rule 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 {4 i e^2}{13 a d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {e^2 \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2} \, dx}{13 a^2} \\ & = \frac {4 i e^2}{13 a d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {4 i e^4}{117 d (e \sec (c+d x))^{5/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\left (5 e^4\right ) \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx}{117 a^4} \\ & = \frac {2 e^3 \sin (c+d x)}{117 a^4 d (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 a d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {4 i e^4}{117 d (e \sec (c+d x))^{5/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {e^2 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{39 a^4} \\ & = \frac {2 e^3 \sin (c+d x)}{117 a^4 d (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 a d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {4 i e^4}{117 d (e \sec (c+d x))^{5/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {e^2 \int \sqrt {\cos (c+d x)} \, dx}{39 a^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e^3 \sin (c+d x)}{117 a^4 d (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 a d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {4 i e^4}{117 d (e \sec (c+d x))^{5/2} \left (a^4+i a^4 \tan (c+d x)\right )} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 2.18 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.87 \[ \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx=\frac {i e^{-i d x} \sec ^2(c+d x) (e \sec (c+d x))^{3/2} (\cos (d x)+i \sin (d x)) \left (28+40 \cos (2 (c+d x))+\frac {24 e^{4 i (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+22 i \sin (2 (c+d x))\right )}{234 a^4 d (-i+\tan (c+d x))^4} \]

[In]

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

[Out]

((I/234)*Sec[c + d*x]^2*(e*Sec[c + d*x])^(3/2)*(Cos[d*x] + I*Sin[d*x])*(28 + 40*Cos[2*(c + d*x)] + (24*E^((4*I
)*(c + d*x))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + (22*I)*S
in[2*(c + d*x)]))/(a^4*d*E^(I*d*x)*(-I + Tan[c + 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. 540 vs. \(2 (167 ) = 334\).

Time = 8.18 (sec) , antiderivative size = 541, normalized size of antiderivative = 3.32

method	result	size
default	\(\frac {2 i \sqrt {e \sec \left (d x +c \right )}\, \left (-i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+72 \left (\cos ^{8}\left (d x +c \right )\right )-72 i \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )+72 \left (\cos ^{7}\left (d x +c \right )\right )+16 i \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-52 \left (\cos ^{6}\left (d x +c \right )\right )-72 i \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )-52 \left (\cos ^{5}\left (d x +c \right )\right )-i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \left (\cos ^{2}\left (d x +c \right )\right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 \left (\cos ^{2}\left (d x +c \right )\right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 i \cos \left (d x +c \right ) \sin \left (d x +c \right )+6 \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 ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-6 \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 ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+16 i \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 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}}-3 \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 ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\right ) e}{117 a^{4} d \left (\cos \left (d x +c \right )+1\right )}\)	\(541\)

[In]

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

[Out]

2/117*I/a^4/d*(e*sec(d*x+c))^(1/2)*(-I*cos(d*x+c)^2*sin(d*x+c)+72*cos(d*x+c)^8-72*I*cos(d*x+c)^7*sin(d*x+c)+72
*cos(d*x+c)^7+16*I*cos(d*x+c)^4*sin(d*x+c)-52*cos(d*x+c)^6-72*I*cos(d*x+c)^6*sin(d*x+c)-52*cos(d*x+c)^5-I*cos(
d*x+c)^3*sin(d*x+c)+3*cos(d*x+c)^2*EllipticF(I*(-csc(d*x+c)+cot(d*x+c)),I)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(
1/(cos(d*x+c)+1))^(1/2)-3*cos(d*x+c)^2*EllipticE(I*(-csc(d*x+c)+cot(d*x+c)),I)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/
2)*(1/(cos(d*x+c)+1))^(1/2)-3*I*cos(d*x+c)*sin(d*x+c)+6*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(-csc(d*
x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)-6*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(I*(-csc(
d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+16*I*cos(d*x+c)^5*sin(d*x+c)+3*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)-3*(cos(d*x+c)/(cos(d*x+c)+1))^(1
/2)*EllipticE(I*(-csc(d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2))*e/(cos(d*x+c)+1)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx=\frac {{\left (24 i \, \sqrt {2} e^{\frac {3}{2}} e^{\left (7 i \, d x + 7 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (24 i \, e e^{\left (8 i \, d x + 8 i \, c\right )} + 55 i \, e e^{\left (6 i \, d x + 6 i \, c\right )} + 59 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 37 i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 9 i \, e\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{468 \, a^{4} d} \]

[In]

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

[Out]

1/468*(24*I*sqrt(2)*e^(3/2)*e^(7*I*d*x + 7*I*c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I
*c))) + sqrt(2)*(24*I*e*e^(8*I*d*x + 8*I*c) + 55*I*e*e^(6*I*d*x + 6*I*c) + 59*I*e*e^(4*I*d*x + 4*I*c) + 37*I*e
*e^(2*I*d*x + 2*I*c) + 9*I*e)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c))*e^(-7*I*d*x - 7*I*c)/
(a^4*d)

Sympy [F]

\[ \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx=\frac {\int \frac {\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\tan ^{4}{\left (c + d x \right )} - 4 i \tan ^{3}{\left (c + d x \right )} - 6 \tan ^{2}{\left (c + d x \right )} + 4 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]

[In]

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

[Out]

Integral((e*sec(c + d*x))**(3/2)/(tan(c + d*x)**4 - 4*I*tan(c + d*x)**3 - 6*tan(c + d*x)**2 + 4*I*tan(c + d*x)
+ 1), x)/a**4

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

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

[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)