Optimal. Leaf size=99 \[ \frac{c^2 (-3 B+i A)}{2 a^3 f (-\tan (e+f x)+i)^2}+\frac{2 c^2 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac{i B c^2}{a^3 f (-\tan (e+f x)+i)} \]
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Rubi [A] time = 0.156943, antiderivative size = 99, 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, 77} \[ \frac{c^2 (-3 B+i A)}{2 a^3 f (-\tan (e+f x)+i)^2}+\frac{2 c^2 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac{i B c^2}{a^3 f (-\tan (e+f x)+i)} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^3} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(A+B x) (c-i c x)}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{2 (A+i B) c}{a^4 (-i+x)^4}+\frac{(-i A+3 B) c}{a^4 (-i+x)^3}-\frac{i B c}{a^4 (-i+x)^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 (A+i B) c^2}{3 a^3 f (i-\tan (e+f x))^3}+\frac{(i A-3 B) c^2}{2 a^3 f (i-\tan (e+f x))^2}-\frac{i B c^2}{a^3 f (i-\tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 2.7558, size = 79, normalized size = 0.8 \[ -\frac{i c^2 \sec ^2(e+f x) (\cos (2 (e+f x))-i \sin (2 (e+f x))) ((A-5 i B) \tan (e+f x)-5 i A-B)}{24 a^3 f (\tan (e+f x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 69, normalized size = 0.7 \begin{align*}{\frac{{c}^{2}}{f{a}^{3}} \left ({\frac{iB}{\tan \left ( fx+e \right ) -i}}-{\frac{-iA+3\,B}{2\, \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{2\,iB+2\,A}{3\, \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.051, size = 128, normalized size = 1.29 \begin{align*} \frac{{\left ({\left (3 i \, A + 3 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (2 i \, A - 2 \, B\right )} c^{2}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{24 \, a^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.76566, size = 173, normalized size = 1.75 \begin{align*} \begin{cases} \frac{\left (\left (8 i A a^{3} c^{2} f e^{4 i e} - 8 B a^{3} c^{2} f e^{4 i e}\right ) e^{- 6 i f x} + \left (12 i A a^{3} c^{2} f e^{6 i e} + 12 B a^{3} c^{2} f e^{6 i e}\right ) e^{- 4 i f x}\right ) e^{- 10 i e}}{96 a^{6} f^{2}} & \text{for}\: 96 a^{6} f^{2} e^{10 i e} \neq 0 \\\frac{x \left (A c^{2} e^{2 i e} + A c^{2} - i B c^{2} e^{2 i e} + i B c^{2}\right ) e^{- 6 i e}}{2 a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.48839, size = 223, normalized size = 2.25 \begin{align*} -\frac{2 \,{\left (3 \, A c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 3 i \, A c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 3 \, B c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 8 \, A c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 2 i \, B c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 3 i \, A c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, B c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, A c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{3 \, a^{3} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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