Optimal. Leaf size=83 \[ -\frac{4 i c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )}+\frac{i c^3 \log (\cos (e+f x))}{a^2 f}+\frac{c^3 x}{a^2}+\frac{2 i c^3}{f (a+i a \tan (e+f x))^2} \]
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Rubi [A] time = 0.121946, antiderivative size = 83, 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} \[ -\frac{4 i c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )}+\frac{i c^3 \log (\cos (e+f x))}{a^2 f}+\frac{c^3 x}{a^2}+\frac{2 i c^3}{f (a+i a \tan (e+f x))^2} \]
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))^3}{(a+i a \tan (e+f x))^2} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(a+i a \tan (e+f x))^5} \, dx\\ &=-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{(a-x)^2}{(a+x)^3} \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \left (\frac{4 a^2}{(a+x)^3}-\frac{4 a}{(a+x)^2}+\frac{1}{a+x}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=\frac{c^3 x}{a^2}+\frac{i c^3 \log (\cos (e+f x))}{a^2 f}+\frac{2 i c^3}{f (a+i a \tan (e+f x))^2}-\frac{4 i c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 1.8576, size = 115, normalized size = 1.39 \[ -\frac{c^3 \sec ^2(e+f x) \left (\sin (2 (e+f x))+i \cos (2 (e+f x)) \left (\log \left (\cos ^2(e+f x)\right )+1\right )-\sin (2 (e+f x)) \log \left (\cos ^2(e+f x)\right )+2 \tan ^{-1}(\tan (f x)) (\cos (2 (e+f x))+i \sin (2 (e+f x)))-2 i\right )}{2 a^2 f (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 69, normalized size = 0.8 \begin{align*}{\frac{-2\,i{c}^{3}}{f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{i{c}^{3}\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{f{a}^{2}}}-4\,{\frac{{c}^{3}}{f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \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 = 1.45054, size = 220, normalized size = 2.65 \begin{align*} \frac{{\left (4 \, c^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 2 i \, c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 2 i \, c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c^{3}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{2 \, a^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.15936, size = 121, normalized size = 1.46 \begin{align*} \frac{i c^{3} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{2} f} + \frac{\left (\begin{cases} 2 c^{3} x e^{4 i e} - \frac{i c^{3} e^{2 i e} e^{- 2 i f x}}{f} + \frac{i c^{3} e^{- 4 i f x}}{2 f} & \text{for}\: f \neq 0 \\x \left (2 c^{3} e^{4 i e} - 2 c^{3} e^{2 i e} + 2 c^{3}\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 = 1.58941, size = 217, normalized size = 2.61 \begin{align*} -\frac{\frac{12 i \, c^{3} \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}{a^{2}} - \frac{6 i \, c^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac{6 i \, c^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} + \frac{-25 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 100 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 198 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 100 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 25 i \, c^{3}}{a^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{4}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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