Optimal. Leaf size=160 \[ -\frac {a^6 \tan (e+f x)}{c^4 f}+\frac {40 i a^6}{f \left (c^4-i c^4 \tan (e+f x)\right )}-\frac {10 i a^6 \log (\cos (e+f x))}{c^4 f}+\frac {10 a^6 x}{c^4}-\frac {40 i a^6}{f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {80 i a^6}{3 c f (c-i c \tan (e+f x))^3}-\frac {8 i a^6}{f (c-i c \tan (e+f x))^4} \]
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Rubi [A] time = 0.16, antiderivative size = 160, 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} \[ -\frac {a^6 \tan (e+f x)}{c^4 f}+\frac {40 i a^6}{f \left (c^4-i c^4 \tan (e+f x)\right )}-\frac {40 i a^6}{f \left (c^2-i c^2 \tan (e+f x)\right )^2}-\frac {10 i a^6 \log (\cos (e+f x))}{c^4 f}+\frac {10 a^6 x}{c^4}+\frac {80 i a^6}{3 c f (c-i c \tan (e+f x))^3}-\frac {8 i a^6}{f (c-i c \tan (e+f x))^4} \]
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))^6}{(c-i c \tan (e+f x))^4} \, dx &=\left (a^6 c^6\right ) \int \frac {\sec ^{12}(e+f x)}{(c-i c \tan (e+f x))^{10}} \, dx\\ &=\frac {\left (i a^6\right ) \operatorname {Subst}\left (\int \frac {(c-x)^5}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{c^5 f}\\ &=\frac {\left (i a^6\right ) \operatorname {Subst}\left (\int \left (-1+\frac {32 c^5}{(c+x)^5}-\frac {80 c^4}{(c+x)^4}+\frac {80 c^3}{(c+x)^3}-\frac {40 c^2}{(c+x)^2}+\frac {10 c}{c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^5 f}\\ &=\frac {10 a^6 x}{c^4}-\frac {10 i a^6 \log (\cos (e+f x))}{c^4 f}-\frac {a^6 \tan (e+f x)}{c^4 f}-\frac {8 i a^6}{f (c-i c \tan (e+f x))^4}+\frac {80 i a^6}{3 c f (c-i c \tan (e+f x))^3}-\frac {40 i a^6}{f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {40 i a^6}{f \left (c^4-i c^4 \tan (e+f x)\right )}\\ \end {align*}
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Mathematica [B] time = 6.45, size = 455, normalized size = 2.84 \[ \frac {a^6 \sec (e) \sec (e+f x) (\cos (4 (e+f x))+i \sin (4 (e+f x))) \left (40 \sin (2 e+f x)-60 i f x \sin (2 e+3 f x)+43 \sin (2 e+3 f x)-60 i f x \sin (4 e+3 f x)+55 \sin (4 e+3 f x)-60 i f x \sin (4 e+5 f x)-9 \sin (4 e+5 f x)-60 i f x \sin (6 e+5 f x)+3 \sin (6 e+5 f x)+20 i \cos (2 e+f x)+60 f x \cos (2 e+3 f x)+53 i \cos (2 e+3 f x)+60 f x \cos (4 e+3 f x)+65 i \cos (4 e+3 f x)+60 f x \cos (4 e+5 f x)-15 i \cos (4 e+5 f x)+60 f x \cos (6 e+5 f x)-3 i \cos (6 e+5 f x)-30 i \cos (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )-30 i \cos (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-30 i \cos (4 e+5 f x) \log \left (\cos ^2(e+f x)\right )-30 i \cos (6 e+5 f x) \log \left (\cos ^2(e+f x)\right )-30 \sin (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )-30 \sin (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-30 \sin (4 e+5 f x) \log \left (\cos ^2(e+f x)\right )-30 \sin (6 e+5 f x) \log \left (\cos ^2(e+f x)\right )+40 \sin (f x)+20 i \cos (f x)\right )}{24 c^4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 133, normalized size = 0.83 \[ \frac {-3 i \, a^{6} e^{\left (10 i \, f x + 10 i \, e\right )} + 5 i \, a^{6} e^{\left (8 i \, f x + 8 i \, e\right )} - 10 i \, a^{6} e^{\left (6 i \, f x + 6 i \, e\right )} + 30 i \, a^{6} e^{\left (4 i \, f x + 4 i \, e\right )} + 48 i \, a^{6} e^{\left (2 i \, f x + 2 i \, e\right )} - 12 i \, a^{6} + {\left (-60 i \, a^{6} e^{\left (2 i \, f x + 2 i \, e\right )} - 60 i \, a^{6}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{6 \, {\left (c^{4} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{4} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.64, size = 285, normalized size = 1.78 \[ -\frac {\frac {420 i \, a^{6} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c^{4}} - \frac {840 i \, a^{6} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{c^{4}} + \frac {420 i \, a^{6} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{c^{4}} - \frac {84 \, {\left (5 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 i \, a^{6}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} c^{4}} + \frac {2283 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 18936 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 69300 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 141512 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 183106 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 141512 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 69300 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 18936 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2283 i \, a^{6}}{c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{8}}}{42 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 131, normalized size = 0.82 \[ -\frac {a^{6} \tan \left (f x +e \right )}{c^{4} f}-\frac {40 a^{6}}{f \,c^{4} \left (\tan \left (f x +e \right )+i\right )}+\frac {10 i a^{6} \ln \left (\tan \left (f x +e \right )+i\right )}{f \,c^{4}}+\frac {80 a^{6}}{3 f \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{3}}+\frac {40 i a^{6}}{f \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {8 i a^{6}}{f \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{4}} \]
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 = 6.95, size = 170, normalized size = 1.06 \[ \frac {a^6\,\left (10\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )-60\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2+10\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^4-76\,{\mathrm {tan}\left (e+f\,x\right )}^2-4\,{\mathrm {tan}\left (e+f\,x\right )}^4+\frac {56}{3}-\frac {\mathrm {tan}\left (e+f\,x\right )\,197{}\mathrm {i}}{3}-\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\mathrm {tan}\left (e+f\,x\right )\,40{}\mathrm {i}+\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3\,40{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^3\,34{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^5\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^4\,f\,{\left (-1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.89, size = 246, normalized size = 1.54 \[ \frac {2 i a^{6}}{- c^{4} f e^{2 i e} e^{2 i f x} - c^{4} f} - \frac {10 i a^{6} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{4} f} + \begin {cases} \frac {- 3 i a^{6} c^{12} f^{3} e^{8 i e} e^{8 i f x} + 8 i a^{6} c^{12} f^{3} e^{6 i e} e^{6 i f x} - 18 i a^{6} c^{12} f^{3} e^{4 i e} e^{4 i f x} + 48 i a^{6} c^{12} f^{3} e^{2 i e} e^{2 i f x}}{6 c^{16} f^{4}} & \text {for}\: 6 c^{16} f^{4} \neq 0 \\\frac {x \left (4 a^{6} e^{8 i e} - 8 a^{6} e^{6 i e} + 12 a^{6} e^{4 i e} - 16 a^{6} e^{2 i e}\right )}{c^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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