Optimal. Leaf size=146 \[ \frac {a^5 x}{c^4}-\frac {i a^5 \log (\cos (e+f x))}{c^4 f}-\frac {4 i a^5}{f (c-i c \tan (e+f x))^4}-\frac {12 i a^5}{f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {32 i a^5 c^5}{3 f \left (c^3-i c^3 \tan (e+f x)\right )^3}+\frac {8 i a^5}{f \left (c^4-i c^4 \tan (e+f x)\right )} \]
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Rubi [A]
time = 0.10, antiderivative size = 146, 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 = {3603, 3568, 45}
\begin {gather*} \frac {8 i a^5}{f \left (c^4-i c^4 \tan (e+f x)\right )}-\frac {i a^5 \log (\cos (e+f x))}{c^4 f}+\frac {a^5 x}{c^4}-\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 {4 i a^5}{f (c-i c \tan (e+f x))^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rule 3603
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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 1.48, size = 151, normalized size = 1.03 \begin {gather*} \frac {a^5 \left (-6 i+16 i \cos (2 (e+f x))+3 \cos (4 (e+f x)) \left (-i+4 f x-2 i \log \left (\cos ^2(e+f x)\right )\right )+8 \sin (2 (e+f x))+3 \sin (4 (e+f x))-12 i f x \sin (4 (e+f x))-6 \log \left (\cos ^2(e+f x)\right ) \sin (4 (e+f x))\right ) (\cos (4 e+9 f x)+i \sin (4 e+9 f x))}{12 c^4 f (\cos (f x)+i \sin (f x))^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 79, normalized size = 0.54
method | result | size |
derivativedivides | \(\frac {a^{5} \left (-\frac {4 i}{\left (\tan \left (f x +e \right )+i\right )^{4}}-\frac {8}{\tan \left (f x +e \right )+i}+\frac {32}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}+i \ln \left (\tan \left (f x +e \right )+i\right )+\frac {12 i}{\left (\tan \left (f x +e \right )+i\right )^{2}}\right )}{f \,c^{4}}\) | \(79\) |
default | \(\frac {a^{5} \left (-\frac {4 i}{\left (\tan \left (f x +e \right )+i\right )^{4}}-\frac {8}{\tan \left (f x +e \right )+i}+\frac {32}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}+i \ln \left (\tan \left (f x +e \right )+i\right )+\frac {12 i}{\left (\tan \left (f x +e \right )+i\right )^{2}}\right )}{f \,c^{4}}\) | \(79\) |
risch | \(-\frac {i a^{5} {\mathrm e}^{8 i \left (f x +e \right )}}{4 c^{4} f}+\frac {i a^{5} {\mathrm e}^{6 i \left (f x +e \right )}}{3 c^{4} f}-\frac {i a^{5} {\mathrm e}^{4 i \left (f x +e \right )}}{2 c^{4} f}+\frac {i a^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{c^{4} f}-\frac {2 a^{5} e}{c^{4} f}-\frac {i a^{5} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{c^{4} f}\) | \(122\) |
norman | \(\frac {\frac {a^{5} x}{c}+\frac {a^{5} x \left (\tan ^{8}\left (f x +e \right )\right )}{c}+\frac {8 i a^{5}}{3 c f}+\frac {4 a^{5} x \left (\tan ^{2}\left (f x +e \right )\right )}{c}+\frac {6 a^{5} x \left (\tan ^{4}\left (f x +e \right )\right )}{c}+\frac {4 a^{5} x \left (\tan ^{6}\left (f x +e \right )\right )}{c}-\frac {40 a^{5} \left (\tan ^{3}\left (f x +e \right )\right )}{3 c f}+\frac {32 a^{5} \left (\tan ^{5}\left (f x +e \right )\right )}{3 c f}-\frac {8 a^{5} \left (\tan ^{7}\left (f x +e \right )\right )}{c f}+\frac {20 i a^{5} \left (\tan ^{6}\left (f x +e \right )\right )}{c f}+\frac {44 i a^{5} \left (\tan ^{2}\left (f x +e \right )\right )}{3 c f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} c^{3}}+\frac {i a^{5} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 c^{4} f}\) | \(226\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.61, size = 87, normalized size = 0.60 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.45, size = 209, normalized size = 1.43 \begin {gather*} - \frac {i a^{5} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{4} f} + \begin {cases} \frac {- 6 i a^{5} c^{12} f^{3} e^{8 i e} e^{8 i f x} + 8 i a^{5} c^{12} f^{3} e^{6 i e} e^{6 i f x} - 12 i a^{5} c^{12} f^{3} e^{4 i e} e^{4 i f x} + 24 i a^{5} c^{12} f^{3} e^{2 i e} e^{2 i f x}}{24 c^{16} f^{4}} & \text {for}\: c^{16} f^{4} \neq 0 \\\frac {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 )}{c^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 227, normalized size = 1.55 \begin {gather*} -\frac {\frac {420 i \, a^{5} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c^{4}} - \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 (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.07, size = 146, normalized size = 1.00 \begin {gather*} \frac {a^5\,\left (\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )-6\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2+\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^4-12\,{\mathrm {tan}\left (e+f\,x\right )}^2+\frac {8}{3}-\frac {\mathrm {tan}\left (e+f\,x\right )\,32{}\mathrm {i}}{3}-\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\mathrm {tan}\left (e+f\,x\right )\,4{}\mathrm {i}+\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3\,4{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^3\,8{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^4\,f\,{\left (-1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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