Optimal. Leaf size=90 \[ \frac{2 i a^3 \sqrt{c-i c \tan (e+f x)}}{c^2 f}+\frac{8 i a^3}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{8 i a^3}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.163147, antiderivative size = 90, 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} \[ \frac{2 i a^3 \sqrt{c-i c \tan (e+f x)}}{c^2 f}+\frac{8 i a^3}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{8 i a^3}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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))^{3/2}} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(c-i c \tan (e+f x))^{9/2}} \, dx\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{(c-x)^2}{(c+x)^{5/2}} \, 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)^{5/2}}-\frac{4 c}{(c+x)^{3/2}}+\frac{1}{\sqrt{c+x}}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=-\frac{8 i a^3}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac{8 i a^3}{c f \sqrt{c-i c \tan (e+f x)}}+\frac{2 i a^3 \sqrt{c-i c \tan (e+f x)}}{c^2 f}\\ \end{align*}
Mathematica [A] time = 4.82451, size = 94, normalized size = 1.04 \[ \frac{2 a^3 \sqrt{c-i c \tan (e+f x)} (9 \sin (2 (e+f x))+7 i \cos (2 (e+f x))+4 i) (\cos (2 e+5 f x)+i \sin (2 e+5 f x))}{3 c^2 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 64, normalized size = 0.7 \begin{align*}{\frac{2\,i{a}^{3}}{f{c}^{2}} \left ( \sqrt{c-ic\tan \left ( fx+e \right ) }+4\,{\frac{c}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}-{\frac{4\,{c}^{2}}{3} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26322, size = 92, normalized size = 1.02 \begin{align*} \frac{2 i \,{\left (\frac{3 \, \sqrt{-i \, c \tan \left (f x + e\right ) + c} a^{3}}{c} + \frac{4 \,{\left (3 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{3} - a^{3} c\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\right )}}{3 \, c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33281, size = 171, normalized size = 1.9 \begin{align*} \frac{\sqrt{2}{\left (-2 i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 8 i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{3}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3 \, c^{2} 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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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