Optimal. Leaf size=92 \[ -\frac{2 i a^3}{c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{8 i a^3}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{8 i a^3}{5 f (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.163778, antiderivative size = 92, 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} \[ -\frac{2 i a^3}{c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{8 i a^3}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{8 i a^3}{5 f (c-i c \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{(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)}{(c-i c \tan (e+f x))^{11/2}} \, dx\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{(c-x)^2}{(c+x)^{7/2}} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (\frac{4 c^2}{(c+x)^{7/2}}-\frac{4 c}{(c+x)^{5/2}}+\frac{1}{(c+x)^{3/2}}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=-\frac{8 i a^3}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac{8 i a^3}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{2 i a^3}{c^2 f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 8.25086, size = 98, normalized size = 1.07 \[ \frac{2 a^3 \cos (e+f x) \sqrt{c-i c \tan (e+f x)} (-5 i \sin (2 (e+f x))+11 \cos (2 (e+f x))-4) (\sin (3 (e+2 f x))-i \cos (3 (e+2 f x)))}{15 c^3 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 66, normalized size = 0.7 \begin{align*}{\frac{2\,i{a}^{3}}{f{c}^{2}} \left ( -{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}+{\frac{4\,c}{3} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{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.
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Maxima [A] time = 1.07673, size = 88, normalized size = 0.96 \begin{align*} -\frac{2 i \,{\left (15 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{3} - 20 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{3} c + 12 \, a^{3} c^{2}\right )}}{15 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}} c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35031, size = 209, normalized size = 2.27 \begin{align*} \frac{\sqrt{2}{\left (-3 i \, a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 4 i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, a^{3}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \, c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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.
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