\(\int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx\) [404]

Optimal result

Integrand size = 30, antiderivative size = 38 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {3569} \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}} \]

[In]

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

[Out]

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

Rule 3569

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/(a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 9.75 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34

method	result	size
default	\(\frac {2 a \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (-i \cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}{3 d \sqrt {e \sec \left (d x +c \right )}\, e}\)	\(51\)
risch	\(-\frac {2 i a \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, {\mathrm e}^{i \left (d x +c \right )}}{3 e \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\)	\(72\)

[In]

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

[Out]

2/3/d*a*(a*(1+I*tan(d*x+c)))^(1/2)/(e*sec(d*x+c))^(1/2)/e*(-I*cos(d*x+c)+sin(d*x+c))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (28) = 56\).

Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.00 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (-i \, a e^{\left (3 i \, d x + 3 i \, c\right )} - i \, a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{3 \, d e^{2}} \]

[In]

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

[Out]

2/3*(-I*a*e^(3*I*d*x + 3*I*c) - I*a*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*
I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c)/(d*e^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (28) = 56\).

Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.00 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx=-\frac {2 i \, a^{\frac {3}{2}} {\left (-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac {3}{2}}}{3 \, d e^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

-2/3*I*a^(3/2)*(-2*I*sin(d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)^(3/2)/(d*e^(3/
2)*(-sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)^(3/2))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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