Optimal. Leaf size=60 \[ \frac{2 a B}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a (B+i A)}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.107126, antiderivative size = 60, 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} \[ \frac{2 a B}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a (B+i A)}{3 f (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 43
Rubi steps
\begin{align*} \int \frac{(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 \frac{A+B x}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{A-i B}{(c-i c x)^{5/2}}+\frac{i B}{c (c-i c x)^{3/2}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{2 a (i A+B)}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac{2 a B}{c f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.47424, size = 98, normalized size = 1.63 \[ \frac{2 a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\cos (2 e+3 f x)+i \sin (2 e+3 f x)) ((2 B-i A) \cos (e+f x)-3 i B \sin (e+f x))}{3 c^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 53, normalized size = 0.9 \begin{align*}{\frac{2\,ia}{cf} \left ({-iB{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}}-{\frac{c \left ( A-iB \right ) }{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 = 1.15275, size = 61, normalized size = 1.02 \begin{align*} -\frac{2 i \,{\left (3 i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )} B a +{\left (A - i \, B\right )} a 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 = 1.0852, size = 197, normalized size = 3.28 \begin{align*} \frac{\sqrt{2}{\left ({\left (-i \, A - B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-2 i \, A + 4 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-i \, A + 5 \, B\right )} a\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, 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 (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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