Optimal. Leaf size=83 \[ \frac{4 i a^3}{f \left (c^2-i c^2 \tan (e+f x)\right )}-\frac{i a^3 \log (\cos (e+f x))}{c^2 f}+\frac{a^3 x}{c^2}-\frac{2 i a^3}{f (c-i c \tan (e+f x))^2} \]
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Rubi [A] time = 0.122977, 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} \[ \frac{4 i a^3}{f \left (c^2-i c^2 \tan (e+f x)\right )}-\frac{i a^3 \log (\cos (e+f x))}{c^2 f}+\frac{a^3 x}{c^2}-\frac{2 i a^3}{f (c-i c \tan (e+f x))^2} \]
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))^3}{(c-i c \tan (e+f x))^2} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(c-i c \tan (e+f x))^5} \, dx\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{(c-x)^2}{(c+x)^3} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (\frac{4 c^2}{(c+x)^3}-\frac{4 c}{(c+x)^2}+\frac{1}{c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac{a^3 x}{c^2}-\frac{i a^3 \log (\cos (e+f x))}{c^2 f}-\frac{2 i a^3}{f (c-i c \tan (e+f x))^2}+\frac{4 i a^3}{f \left (c^2-i c^2 \tan (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 1.92661, size = 113, normalized size = 1.36 \[ \frac{a^3 (\cos (2 e+5 f x)+i \sin (2 e+5 f x)) \left (\cos (2 (e+f x)) \left (-i \log \left (\cos ^2(e+f x)\right )+2 f x-i\right )+\sin (2 (e+f x)) \left (-\log \left (\cos ^2(e+f x)\right )-2 i f x+1\right )+2 i\right )}{2 c^2 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 69, normalized size = 0.8 \begin{align*} -4\,{\frac{{a}^{3}}{f{c}^{2} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{2\,i{a}^{3}}{f{c}^{2} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}+{\frac{i{a}^{3}\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{f{c}^{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 = 1.38698, size = 150, normalized size = 1.81 \begin{align*} \frac{-i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, a^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{2 \, c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.09635, size = 102, normalized size = 1.23 \begin{align*} - \frac{i a^{3} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{2} f} + \frac{\begin{cases} - \frac{i a^{3} e^{4 i e} e^{4 i f x}}{2 f} + \frac{i a^{3} e^{2 i e} e^{2 i f x}}{f} & \text{for}\: f \neq 0 \\x \left (2 a^{3} e^{4 i e} - 2 a^{3} 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 = 1.41421, size = 217, normalized size = 2.61 \begin{align*} -\frac{-\frac{12 i \, a^{3} \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}{c^{2}} + \frac{6 i \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{c^{2}} + \frac{6 i \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{c^{2}} + \frac{25 i \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 100 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 198 i \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 100 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 25 i \, a^{3}}{c^{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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