Optimal. Leaf size=62 \[ \frac{2 a (B+i A) (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac{2 a B (c-i c \tan (e+f x))^{5/2}}{5 c f} \]
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Rubi [A] time = 0.106542, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {3588, 43} \[ \frac{2 a (B+i A) (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac{2 a B (c-i c \tan (e+f x))^{5/2}}{5 c f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 43
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int (A+B x) \sqrt{c-i c x} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left ((A-i B) \sqrt{c-i c x}+\frac{i B (c-i c x)^{3/2}}{c}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 a (i A+B) (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac{2 a B (c-i c \tan (e+f x))^{5/2}}{5 c f}\\ \end{align*}
Mathematica [A] time = 3.20934, size = 97, normalized size = 1.56 \[ \frac{2 a c (\cos (e)-i \sin (e)) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (5 i A+3 i B \tan (e+f x)+2 B) (A+B \tan (e+f x))}{15 f (A \cos (e+f x)+B \sin (e+f x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 55, normalized size = 0.9 \begin{align*}{\frac{2\,ia}{cf} \left ({\frac{i}{5}}B \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}+{\frac{-iBc+Ac}{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 = 1.18334, size = 66, normalized size = 1.06 \begin{align*} \frac{2 i \,{\left (3 i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}} B a +{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}{\left (5 \, A - 5 i \, B\right )} a c\right )}}{15 \, c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15164, size = 223, normalized size = 3.6 \begin{align*} \frac{\sqrt{2}{\left ({\left (20 i \, A + 20 \, B\right )} a c e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (20 i \, A - 4 \, B\right )} a c\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \,{\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, 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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int A c \sqrt{- i c \tan{\left (e + f x \right )} + c}\, dx + \int A c \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int B c \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan{\left (e + f x \right )}\, dx + \int B c \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\, dx\right ) \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{\left (B \tan \left (f x + e\right ) + A\right )}{\left (i \, a \tan \left (f x + e\right ) + a\right )}{\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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