Optimal. Leaf size=60 \[ \frac{2 i a^2}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.154045, antiderivative size = 60, 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} \[ \frac{2 i a^2}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx &=\left (a^2 c^2\right ) \int \frac{\sec ^4(e+f x)}{(c-i c \tan (e+f x))^{7/2}} \, dx\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{c-x}{(c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \left (\frac{2 c}{(c+x)^{5/2}}-\frac{1}{(c+x)^{3/2}}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=-\frac{4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac{2 i a^2}{c f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.45166, size = 93, normalized size = 1.55 \[ \frac{2 a^2 \cos (e+f x) \sqrt{c-i c \tan (e+f x)} (3 \sin (e+f x)+i \cos (e+f x)) (\cos (2 (e+2 f x))+i \sin (2 (e+2 f x)))}{3 c^2 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 47, normalized size = 0.8 \begin{align*}{\frac{-2\,i{a}^{2}}{cf} \left ( -{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}+{\frac{2\,c}{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 = 1.76929, size = 59, normalized size = 0.98 \begin{align*} \frac{2 i \,{\left (3 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{2} - 2 \, a^{2} c\right )}}{3 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}} c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45325, size = 165, normalized size = 2.75 \begin{align*} \frac{\sqrt{2}{\left (-i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, a^{2}\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 )}^{2}}{{\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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