Optimal. Leaf size=94 \[ \frac{2 i a^3 (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.157665, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3522, 3487, 43} \[ \frac{2 i a^3 (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{\sqrt{c-i c \tan (e+f x)}} \, dx\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int (c-x)^2 (c+x)^{3/2} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (4 c^2 (c+x)^{3/2}-4 c (c+x)^{5/2}+(c+x)^{7/2}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac{8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{2 i a^3 (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}\\ \end{align*}
Mathematica [A] time = 4.47043, size = 100, normalized size = 1.06 \[ \frac{2 a^3 c^2 \sec ^4(e+f x) \sqrt{c-i c \tan (e+f x)} (\sin (2 e-f x)+i \cos (2 e-f x)) (55 i \sin (2 (e+f x))+71 \cos (2 (e+f x))+36)}{315 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 66, normalized size = 0.7 \begin{align*}{\frac{2\,i{a}^{3}}{f{c}^{2}} \left ({\frac{1}{9} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{9}{2}}}}-{\frac{4\,c}{7} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{4\,{c}^{2}}{5} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.34322, size = 90, normalized size = 0.96 \begin{align*} \frac{2 i \,{\left (35 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{9}{2}} a^{3} - 180 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{7}{2}} a^{3} c + 252 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}} a^{3} c^{2}\right )}}{315 \, c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5681, size = 331, normalized size = 3.52 \begin{align*} \frac{\sqrt{2}{\left (2016 i \, a^{3} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 1152 i \, a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 256 i \, a^{3} c^{2}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{315 \,{\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\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 (f x + e\right ) + a\right )}^{3}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]