\(\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx\) [203]

Optimal result

Integrand size = 28, antiderivative size = 175 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx=-\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d} \]

[Out]

22/15*I*a^3*(e*sec(d*x+c))^(3/2)/d-22/5*a^3*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)+22/5*a^3*e*sin(d*x+c)*(e*sec(d*x+c))^(1/2)/d+2
/7*I*a*(e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^2/d+22/35*I*(e*sec(d*x+c))^(3/2)*(a^3+I*a^3*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3579, 3567, 3853, 3856, 2719} \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx=-\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 d}+\frac {22 i \left (a^3+i a^3 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{35 d}+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d} \]

[In]

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

[Out]

(-22*a^3*e^2*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((22*I)/15)*a^3*(e*Se
c[c + d*x])^(3/2))/d + (22*a^3*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (((2*I)/7)*a*(e*Sec[c + d*x])^(3/2
)*(a + I*a*Tan[c + d*x])^2)/d + (((22*I)/35)*(e*Sec[c + d*x])^(3/2)*(a^3 + I*a^3*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))^2}{7 d}+\frac {1}{7} (11 a) \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))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac {1}{5} \left (11 a^2\right ) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx \\ & = \frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac {1}{5} \left (11 a^3\right ) \int (e \sec (c+d x))^{3/2} \, dx \\ & = \frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}-\frac {1}{5} \left (11 a^3 e^2\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx \\ & = \frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}-\frac {\left (11 a^3 e^2\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = -\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 3.37 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.74 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx=\frac {a^3 (e \sec (c+d x))^{3/2} (1+i \tan (c+d x)) \left (-116 i-308 i \cos (2 (c+d x))+77 i e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+77 \sec (c+d x) \sin (3 (c+d x))+17 \tan (c+d x)\right )}{210 d} \]

[In]

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

[Out]

(a^3*(e*Sec[c + d*x])^(3/2)*(1 + I*Tan[c + d*x])*(-116*I - (308*I)*Cos[2*(c + d*x)] + ((77*I)*(1 + E^((2*I)*(c
+ d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 77*Sec[c + d*x]*
Sin[3*(c + d*x)] + 17*Tan[c + d*x]))/(210*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (174 ) = 348\).

Time = 16.21 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.58

method	result	size
default	\(-\frac {2 e \,a^{3} \sqrt {e \sec \left (d x +c \right )}\, \left (231 i \left (\cos ^{2}\left (d x +c \right )\right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-231 i \left (\cos ^{2}\left (d x +c \right )\right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+462 i \cos \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-462 i \cos \left (d x +c \right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+231 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-231 i E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-140 i-231 \sin \left (d x +c \right )-140 i \sec \left (d x +c \right )+63 \tan \left (d x +c \right )+15 i \left (\sec ^{2}\left (d x +c \right )\right )+63 \sec \left (d x +c \right ) \tan \left (d x +c \right )+15 i \left (\sec ^{3}\left (d x +c \right )\right )\right )}{105 d \left (\cos \left (d x +c \right )+1\right )}\)	\(452\)
parts	\(\text {Expression too large to display}\)	\(872\)

[In]

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

[Out]

-2/105*e*a^3/d*(e*sec(d*x+c))^(1/2)/(cos(d*x+c)+1)*(231*I*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)-231*I*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)+462*I*cos(d*x+c)*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)-462*I*cos(d*x+c)*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)+231*I*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)-231*I*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)-140*I-231*sin(d*x+c)-140*I*sec(d*x+c)+63
*tan(d*x+c)+15*I*sec(d*x+c)^2+63*sec(d*x+c)*tan(d*x+c)+15*I*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 = 208, normalized size of antiderivative = 1.19 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (231 i \, a^{3} e e^{\left (7 i \, d x + 7 i \, c\right )} + 287 i \, a^{3} e e^{\left (5 i \, d x + 5 i \, c\right )} + 253 i \, a^{3} e e^{\left (3 i \, d x + 3 i \, c\right )} + 77 i \, a^{3} e e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 231 \, \sqrt {2} {\left (i \, a^{3} e e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, a^{3} e e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, a^{3} e e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} e\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{105 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-2/105*(sqrt(2)*(231*I*a^3*e*e^(7*I*d*x + 7*I*c) + 287*I*a^3*e*e^(5*I*d*x + 5*I*c) + 253*I*a^3*e*e^(3*I*d*x +
3*I*c) + 77*I*a^3*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^3*e*e^(6*I*d*x + 6*I*c) + 3*I*a^3*e*e^(4*I*d*x + 4*I*c) + 3*I*a^3*e*e^(2*I*d*x + 2*I*c) + I*a^3*e)*sqrt(e)
*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c))))/(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x
+ 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F]

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

[In]

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

[Out]

-I*a**3*(Integral(I*(e*sec(c + d*x))**(3/2), x) + Integral(-3*(e*sec(c + d*x))**(3/2)*tan(c + d*x), x) + Integ
ral((e*sec(c + d*x))**(3/2)*tan(c + d*x)**3, x) + Integral(-3*I*(e*sec(c + d*x))**(3/2)*tan(c + d*x)**2, x))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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