Optimal. Leaf size=101 \[ -\frac{a^4 \tan (e+f x)}{c^2 f}+\frac{12 i a^4}{f \left (c^2-i c^2 \tan (e+f x)\right )}-\frac{6 i a^4 \log (\cos (e+f x))}{c^2 f}+\frac{6 a^4 x}{c^2}-\frac{4 i a^4}{f (c-i c \tan (e+f x))^2} \]
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Rubi [A] time = 0.136019, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 43} \[ -\frac{a^4 \tan (e+f x)}{c^2 f}+\frac{12 i a^4}{f \left (c^2-i c^2 \tan (e+f x)\right )}-\frac{6 i a^4 \log (\cos (e+f x))}{c^2 f}+\frac{6 a^4 x}{c^2}-\frac{4 i a^4}{f (c-i c \tan (e+f x))^2} \]
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))^4}{(c-i c \tan (e+f x))^2} \, dx &=\left (a^4 c^4\right ) \int \frac{\sec ^8(e+f x)}{(c-i c \tan (e+f x))^6} \, dx\\ &=\frac{\left (i a^4\right ) \operatorname{Subst}\left (\int \frac{(c-x)^3}{(c+x)^3} \, dx,x,-i c \tan (e+f x)\right )}{c^3 f}\\ &=\frac{\left (i a^4\right ) \operatorname{Subst}\left (\int \left (-1+\frac{8 c^3}{(c+x)^3}-\frac{12 c^2}{(c+x)^2}+\frac{6 c}{c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^3 f}\\ &=\frac{6 a^4 x}{c^2}-\frac{6 i a^4 \log (\cos (e+f x))}{c^2 f}-\frac{a^4 \tan (e+f x)}{c^2 f}-\frac{4 i a^4}{f (c-i c \tan (e+f x))^2}+\frac{12 i a^4}{f \left (c^2-i c^2 \tan (e+f x)\right )}\\ \end{align*}
Mathematica [B] time = 2.68934, size = 374, normalized size = 3.7 \[ \frac{a^4 \sec (e) \sec (e+f x) (\cos (2 (e+3 f x))+i \sin (2 (e+3 f x))) \left (-6 i f x \sin (2 e+f x)+3 \sin (2 e+f x)-6 i f x \sin (2 e+3 f x)-\sin (2 e+3 f x)-6 i f x \sin (4 e+3 f x)+\sin (4 e+3 f x)+6 f x \cos (2 e+3 f x)-3 i \cos (2 e+3 f x)+6 f x \cos (4 e+3 f x)-i \cos (4 e+3 f x)-3 i \cos (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )+\cos (f x) \left (-3 i \log \left (\cos ^2(e+f x)\right )+6 f x+7 i\right )+\cos (2 e+f x) \left (-3 i \log \left (\cos ^2(e+f x)\right )+6 f x+9 i\right )-3 i \cos (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-3 \sin (f x) \log \left (\cos ^2(e+f x)\right )-3 \sin (2 e+f x) \log \left (\cos ^2(e+f x)\right )-3 \sin (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )-3 \sin (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-6 i f x \sin (f x)+\sin (f x)\right )}{4 c^2 f (\cos (f x)+i \sin (f x))^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 86, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}\tan \left ( fx+e \right ) }{{c}^{2}f}}-12\,{\frac{{a}^{4}}{{c}^{2}f \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{4\,i{a}^{4}}{{c}^{2}f \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}+{\frac{6\,i{a}^{4}\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{{c}^{2}f}} \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 = 1.3803, size = 281, normalized size = 2.78 \begin{align*} \frac{-i \, a^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 3 i \, a^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 i \, a^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, a^{4} +{\left (-6 i \, a^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 6 i \, a^{4}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.26356, size = 139, normalized size = 1.38 \begin{align*} - \frac{6 i a^{4} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{2} f} - \frac{2 i a^{4} e^{- 2 i e}}{c^{2} f \left (e^{2 i f x} + e^{- 2 i e}\right )} + \frac{\begin{cases} - \frac{i a^{4} e^{4 i e} e^{4 i f x}}{f} + \frac{4 i a^{4} e^{2 i e} e^{2 i f x}}{f} & \text{for}\: f \neq 0 \\x \left (4 a^{4} e^{4 i e} - 8 a^{4} e^{2 i e}\right ) & \text{otherwise} \end{cases}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.5818, size = 296, normalized size = 2.93 \begin{align*} -\frac{-\frac{12 i \, a^{4} \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}{c^{2}} + \frac{6 i \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{c^{2}} + \frac{6 i \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{c^{2}} - \frac{2 \,{\left (3 i \, a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 3 i \, a^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )} c^{2}} + \frac{25 i \, a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 108 \, a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 182 i \, a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 108 \, a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 25 i \, a^{4}}{c^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{4}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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