Optimal. Leaf size=146 \[ \frac{8 i a^5}{f \left (c^4-i c^4 \tan (e+f x)\right )}-\frac{12 i a^5}{f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac{32 i a^5 c^5}{3 f \left (c^3-i c^3 \tan (e+f x)\right )^3}-\frac{i a^5 \log (\cos (e+f x))}{c^4 f}+\frac{a^5 x}{c^4}-\frac{4 i a^5}{f (c-i c \tan (e+f x))^4} \]
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Rubi [A] time = 0.14569, antiderivative size = 146, 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} \[ \frac{8 i a^5}{f \left (c^4-i c^4 \tan (e+f x)\right )}-\frac{12 i a^5}{f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac{32 i a^5 c^5}{3 f \left (c^3-i c^3 \tan (e+f x)\right )^3}-\frac{i a^5 \log (\cos (e+f x))}{c^4 f}+\frac{a^5 x}{c^4}-\frac{4 i a^5}{f (c-i c \tan (e+f x))^4} \]
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))^5}{(c-i c \tan (e+f x))^4} \, dx &=\left (a^5 c^5\right ) \int \frac{\sec ^{10}(e+f x)}{(c-i c \tan (e+f x))^9} \, dx\\ &=\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \frac{(c-x)^4}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{c^4 f}\\ &=\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \left (\frac{16 c^4}{(c+x)^5}-\frac{32 c^3}{(c+x)^4}+\frac{24 c^2}{(c+x)^3}-\frac{8 c}{(c+x)^2}+\frac{1}{c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^4 f}\\ &=\frac{a^5 x}{c^4}-\frac{i a^5 \log (\cos (e+f x))}{c^4 f}-\frac{4 i a^5}{f (c-i c \tan (e+f x))^4}+\frac{32 i a^5}{3 c f (c-i c \tan (e+f x))^3}-\frac{12 i a^5}{f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac{8 i a^5}{f \left (c^4-i c^4 \tan (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 2.64362, size = 151, normalized size = 1.03 \[ \frac{a^5 (\cos (4 e+9 f x)+i \sin (4 e+9 f x)) \left (8 \sin (2 (e+f x))-12 i f x \sin (4 (e+f x))+3 \sin (4 (e+f x))+16 i \cos (2 (e+f x))+3 \cos (4 (e+f x)) \left (-2 i \log \left (\cos ^2(e+f x)\right )+4 f x-i\right )-6 \sin (4 (e+f x)) \log \left (\cos ^2(e+f x)\right )-6 i\right )}{12 c^4 f (\cos (f x)+i \sin (f x))^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 114, normalized size = 0.8 \begin{align*}{\frac{-4\,i{a}^{5}}{f{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}-8\,{\frac{{a}^{5}}{f{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{12\,i{a}^{5}}{f{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}+{\frac{i{a}^{5}\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{f{c}^{4}}}+{\frac{32\,{a}^{5}}{3\,f{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}} \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 = 1.406, size = 238, normalized size = 1.63 \begin{align*} \frac{-3 i \, a^{5} e^{\left (8 i \, f x + 8 i \, e\right )} + 4 i \, a^{5} e^{\left (6 i \, f x + 6 i \, e\right )} - 6 i \, a^{5} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 i \, a^{5} e^{\left (2 i \, f x + 2 i \, e\right )} - 12 i \, a^{5} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{12 \, c^{4} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34123, size = 173, normalized size = 1.18 \begin{align*} - \frac{i a^{5} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{4} f} + \frac{\begin{cases} - \frac{i a^{5} e^{8 i e} e^{8 i f x}}{4 f} + \frac{i a^{5} e^{6 i e} e^{6 i f x}}{3 f} - \frac{i a^{5} e^{4 i e} e^{4 i f x}}{2 f} + \frac{i a^{5} e^{2 i e} e^{2 i f x}}{f} & \text{for}\: f \neq 0 \\x \left (2 a^{5} e^{8 i e} - 2 a^{5} e^{6 i e} + 2 a^{5} e^{4 i e} - 2 a^{5} e^{2 i e}\right ) & \text{otherwise} \end{cases}}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.68487, size = 309, normalized size = 2.12 \begin{align*} -\frac{-\frac{840 i \, a^{5} \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}{c^{4}} + \frac{420 i \, a^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{c^{4}} + \frac{420 i \, a^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{c^{4}} + \frac{2283 i \, a^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 18264 \, a^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 70644 i \, a^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 136808 \, a^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 191170 i \, a^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 136808 \, a^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 70644 i \, a^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 18264 \, a^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2283 i \, a^{5}}{c^{4}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{8}}}{420 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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