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 = 0.14, antiderivative size = 101, normalized size of antiderivative = 1.00, 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 43
Rule 3487
Rule 3522
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*}
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Mathematica [B] time = 2.99, size = 374, normalized size = 3.70 \[ \frac {a^4 \sec (e) \sec (e+f x) (\cos (2 (e+3 f x))+i \sin (2 (e+3 f x))) \left (-6 i f x \sin (2 e+f x)+3 \sin (2 e+f x)-6 i f x \sin (2 e+3 f x)-\sin (2 e+3 f x)-6 i f x \sin (4 e+3 f x)+\sin (4 e+3 f x)+6 f x \cos (2 e+3 f x)-3 i \cos (2 e+3 f x)+6 f x \cos (4 e+3 f x)-i \cos (4 e+3 f x)-3 i \cos (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )+\cos (f x) \left (-3 i \log \left (\cos ^2(e+f x)\right )+6 f x+7 i\right )+\cos (2 e+f x) \left (-3 i \log \left (\cos ^2(e+f x)\right )+6 f x+9 i\right )-3 i \cos (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-3 \sin (f x) \log \left (\cos ^2(e+f x)\right )-3 \sin (2 e+f x) \log \left (\cos ^2(e+f x)\right )-3 \sin (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )-3 \sin (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-6 i f x \sin (f x)+\sin (f x)\right )}{4 c^2 f (\cos (f x)+i \sin (f x))^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 104, normalized size = 1.03 \[ \frac {-i \, a^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 3 i \, a^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 i \, a^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, a^{4} + {\left (-6 i \, a^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 6 i \, a^{4}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.59, size = 217, normalized size = 2.15 \[ -\frac {\frac {6 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c^{2}} - \frac {12 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{c^{2}} + \frac {6 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{c^{2}} - \frac {2 \, {\left (3 i \, a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 i \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} c^{2}} + \frac {25 i \, a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 108 \, a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 182 i \, a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 108 \, a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 25 i \, a^{4}}{c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{4}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 86, normalized size = 0.85 \[ -\frac {a^{4} \tan \left (f x +e \right )}{c^{2} f}-\frac {12 a^{4}}{f \,c^{2} \left (\tan \left (f x +e \right )+i\right )}+\frac {6 i a^{4} \ln \left (\tan \left (f x +e \right )+i\right )}{f \,c^{2}}+\frac {4 i a^{4}}{f \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 90, normalized size = 0.89 \[ -\frac {\frac {12\,a^4\,\mathrm {tan}\left (e+f\,x\right )}{c^2}+\frac {a^4\,8{}\mathrm {i}}{c^2}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}-1\right )}-\frac {a^4\,\mathrm {tan}\left (e+f\,x\right )}{c^2\,f}+\frac {a^4\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,6{}\mathrm {i}}{c^2\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 156, normalized size = 1.54 \[ \frac {2 i a^{4}}{- c^{2} f e^{2 i e} e^{2 i f x} - c^{2} f} - \frac {6 i a^{4} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{2} f} + \begin {cases} \frac {- i a^{4} c^{2} f e^{4 i e} e^{4 i f x} + 4 i a^{4} c^{2} f e^{2 i e} e^{2 i f x}}{c^{4} f^{2}} & \text {for}\: c^{4} f^{2} \neq 0 \\\frac {x \left (4 a^{4} e^{4 i e} - 8 a^{4} e^{2 i e}\right )}{c^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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