Optimal. Leaf size=101 \[ \frac{2 \tan (e+f x)}{3 a c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{\tan (e+f x)}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.129523, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {3523, 40, 39} \[ \frac{2 \tan (e+f x)}{3 a c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{\tan (e+f x)}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 40
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{5/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x)}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=\frac{\tan (e+f x)}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{2 \tan (e+f x)}{3 a c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.31066, size = 103, normalized size = 1.02 \[ \frac{(9 \sin (e+f x)+\sin (3 (e+f x))) \sec (e+f x) \sqrt{c-i c \tan (e+f x)} (\sin (2 (e+f x))-i \cos (2 (e+f x)))}{12 a c^2 f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 95, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \tan \left ( fx+e \right ) \left ( 2\, \left ( \tan \left ( fx+e \right ) \right ) ^{2}+3 \right ) }{3\,f{a}^{2}{c}^{2} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{3} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.88621, size = 61, normalized size = 0.6 \begin{align*} \frac{\sin \left (3 \, f x + 3 \, e\right ) + 9 \, \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right )}{12 \, a^{\frac{3}{2}} c^{\frac{3}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35622, size = 257, normalized size = 2.54 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (-i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 10 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 10 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-3 i \, f x - 3 i \, e\right )}}{24 \, a^{2} c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{3}{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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