Optimal. Leaf size=59 \[ \frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{9/2}}{9 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0705784, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 43} \[ \frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{9/2}}{9 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x) (a+x)^{7/2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a+x)^{7/2}-(a+x)^{9/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{4 i (a+i a \tan (c+d x))^{9/2}}{9 a^2 d}+\frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.531602, size = 85, normalized size = 1.44 \[ -\frac{2 a^2 (9 \tan (c+d x)+13 i) \sec ^4(c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (4 c+6 d x)+i \sin (4 c+6 d x))}{99 d (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.327, size = 117, normalized size = 2. \begin{align*} -{\frac{2\,{a}^{2} \left ( 32\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}-32\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}-20\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -23\,i\cos \left ( dx+c \right ) +9\,\sin \left ( dx+c \right ) \right ) }{99\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.07692, size = 54, normalized size = 0.92 \begin{align*} \frac{2 i \,{\left (9 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{11}{2}} - 22 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a\right )}}{99 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.61219, size = 360, normalized size = 6.1 \begin{align*} \frac{\sqrt{2}{\left (-128 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 704 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{99 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]