Optimal. Leaf size=101 \[ -\frac{c^4 \tan (e+f x)}{a^2 f}-\frac{12 i c^4}{f \left (a^2+i a^2 \tan (e+f x)\right )}+\frac{6 i c^4 \log (\cos (e+f x))}{a^2 f}+\frac{6 c^4 x}{a^2}+\frac{4 i c^4}{f (a+i a \tan (e+f x))^2} \]
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Rubi [A] time = 0.132704, 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} \[ -\frac{c^4 \tan (e+f x)}{a^2 f}-\frac{12 i c^4}{f \left (a^2+i a^2 \tan (e+f x)\right )}+\frac{6 i c^4 \log (\cos (e+f x))}{a^2 f}+\frac{6 c^4 x}{a^2}+\frac{4 i c^4}{f (a+i a \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx &=\left (a^4 c^4\right ) \int \frac{\sec ^8(e+f x)}{(a+i a \tan (e+f x))^6} \, dx\\ &=-\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \frac{(a-x)^3}{(a+x)^3} \, dx,x,i a \tan (e+f x)\right )}{a^3 f}\\ &=-\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \left (-1+\frac{8 a^3}{(a+x)^3}-\frac{12 a^2}{(a+x)^2}+\frac{6 a}{a+x}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^3 f}\\ &=\frac{6 c^4 x}{a^2}+\frac{6 i c^4 \log (\cos (e+f x))}{a^2 f}-\frac{c^4 \tan (e+f x)}{a^2 f}+\frac{4 i c^4}{f (a+i a \tan (e+f x))^2}-\frac{12 i c^4}{f \left (a^2+i a^2 \tan (e+f x)\right )}\\ \end{align*}
Mathematica [B] time = 2.37058, size = 279, normalized size = 2.76 \[ \frac{c^4 \sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left (-24 f x \sin ^2(e)-12 i f x \sin (2 e)+2 i \sin (2 e) \sin (4 f x)+12 i f x \tan (e)-2 \sin (2 e) \cos (4 f x)+i \sec (e) \cos (2 e-f x) \sec (e+f x)-i \sec (e) \cos (2 e+f x) \sec (e+f x)-\sec (e) \sin (2 e-f x) \sec (e+f x)+\sec (e) \sin (2 e+f x) \sec (e+f x)+6 \sin (2 e) \log \left (\cos ^2(e+f x)\right )-12 (\cos (2 e)+i \sin (2 e)) \tan ^{-1}(\tan (f x))+2 i \cos (2 e) \left (6 f x \tan (e)-3 \log \left (\cos ^2(e+f x)\right )+6 i f x+i \sin (4 f x)-\cos (4 f x)\right )+12 f x+8 \sin (2 f x)+8 i \cos (2 f x)\right )}{2 a^2 f (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 86, normalized size = 0.9 \begin{align*} -{\frac{{c}^{4}\tan \left ( fx+e \right ) }{{a}^{2}f}}-{\frac{6\,i{c}^{4}\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{{a}^{2}f}}-12\,{\frac{{c}^{4}}{{a}^{2}f \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{4\,i{c}^{4}}{{a}^{2}f \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}} \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 = 1.29552, size = 356, normalized size = 3.52 \begin{align*} \frac{12 \, c^{4} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c^{4} +{\left (12 \, c^{4} f x - 6 i \, c^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (6 i \, c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 6 i \, c^{4} e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.04238, size = 158, normalized size = 1.56 \begin{align*} \frac{6 i c^{4} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{2} f} - \frac{2 i c^{4} e^{- 2 i e}}{a^{2} f \left (e^{2 i f x} + e^{- 2 i e}\right )} + \frac{\left (\begin{cases} 12 c^{4} x e^{4 i e} - \frac{4 i c^{4} e^{2 i e} e^{- 2 i f x}}{f} + \frac{i c^{4} e^{- 4 i f x}}{f} & \text{for}\: f \neq 0 \\x \left (12 c^{4} e^{4 i e} - 8 c^{4} e^{2 i e} + 4 c^{4}\right ) & \text{otherwise} \end{cases}\right ) e^{- 4 i e}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.5997, size = 296, normalized size = 2.93 \begin{align*} -\frac{\frac{12 i \, c^{4} \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}{a^{2}} - \frac{6 i \, c^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac{6 i \, c^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac{2 \,{\left (-3 i \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 3 i \, c^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )} a^{2}} + \frac{-25 i \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 108 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 182 i \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 108 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 25 i \, c^{4}}{a^{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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