Optimal. Leaf size=164 \[ -\frac {6 c^4 (A+3 i B)}{a^3 f (-\tan (e+f x)+i)}+\frac {2 c^4 (-5 B+3 i A)}{a^3 f (-\tan (e+f x)+i)^2}+\frac {8 c^4 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac {c^4 (-7 B+i A) \log (\cos (e+f x))}{a^3 f}-\frac {c^4 x (A+7 i B)}{a^3}+\frac {i B c^4 \tan (e+f x)}{a^3 f} \]
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Rubi [A] time = 0.21, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3588, 77} \[ -\frac {6 c^4 (A+3 i B)}{a^3 f (-\tan (e+f x)+i)}+\frac {2 c^4 (-5 B+3 i A)}{a^3 f (-\tan (e+f x)+i)^2}+\frac {8 c^4 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac {c^4 (-7 B+i A) \log (\cos (e+f x))}{a^3 f}-\frac {c^4 x (A+7 i B)}{a^3}+\frac {i B c^4 \tan (e+f x)}{a^3 f} \]
Antiderivative was successfully verified.
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Rule 77
Rule 3588
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(A+B x) (c-i c x)^3}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {i B c^3}{a^4}+\frac {8 (A+i B) c^3}{a^4 (-i+x)^4}+\frac {4 (-3 i A+5 B) c^3}{a^4 (-i+x)^3}-\frac {6 (A+3 i B) c^3}{a^4 (-i+x)^2}+\frac {i (A+7 i B) c^3}{a^4 (-i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(A+7 i B) c^4 x}{a^3}-\frac {(i A-7 B) c^4 \log (\cos (e+f x))}{a^3 f}+\frac {8 (A+i B) c^4}{3 a^3 f (i-\tan (e+f x))^3}+\frac {2 (3 i A-5 B) c^4}{a^3 f (i-\tan (e+f x))^2}-\frac {6 (A+3 i B) c^4}{a^3 f (i-\tan (e+f x))}+\frac {i B c^4 \tan (e+f x)}{a^3 f}\\ \end {align*}
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Mathematica [B] time = 9.44, size = 1239, normalized size = 7.55 \[ c^4 \left (\frac {\sec ^3(e+f x) \left (-\frac {1}{2} B \cos (3 e-f x)+\frac {1}{2} B \cos (3 e+f x)-\frac {1}{2} i B \sin (3 e-f x)+\frac {1}{2} i B \sin (3 e+f x)\right ) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {x \sec ^2(e+f x) \left (-\frac {1}{2} A \cos ^3(e)-\frac {7}{2} i B \cos ^3(e)-2 i A \sin (e) \cos ^2(e)+14 B \sin (e) \cos ^2(e)+3 A \sin ^2(e) \cos (e)+21 i B \sin ^2(e) \cos (e)+\frac {1}{2} A \cos (e)+\frac {7}{2} i B \cos (e)+2 i A \sin ^3(e)-14 B \sin ^3(e)+i A \sin (e)-7 B \sin (e)-\frac {1}{2} A \sin ^3(e) \tan (e)-\frac {7}{2} i B \sin ^3(e) \tan (e)-\frac {1}{2} A \sin (e) \tan (e)-\frac {7}{2} i B \sin (e) \tan (e)+i (A+7 i B) (\cos (3 e)+i \sin (3 e)) \tan (e)\right ) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{(A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {(A+5 i B) \cos (2 f x) \sec ^2(e+f x) (i \cos (e)-\sin (e)) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {(3 B-i A) \cos (4 f x) \sec ^2(e+f x) \left (\frac {\cos (e)}{2}-\frac {1}{2} i \sin (e)\right ) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {\sec ^2(e+f x) \left (-i A \cos \left (\frac {3 e}{2}\right )+7 B \cos \left (\frac {3 e}{2}\right )+A \sin \left (\frac {3 e}{2}\right )+7 i B \sin \left (\frac {3 e}{2}\right )\right ) \left (\cos \left (\frac {3 e}{2}\right ) \log (\cos (e+f x))+i \sin \left (\frac {3 e}{2}\right ) \log (\cos (e+f x))\right ) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {(A+i B) \cos (6 f x) \sec ^2(e+f x) \left (\frac {1}{3} i \cos (3 e)+\frac {1}{3} \sin (3 e)\right ) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {(A+7 i B) \sec ^2(e+f x) (-f x \cos (3 e)-i f x \sin (3 e)) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {(A+5 i B) \sec ^2(e+f x) (\cos (e)+i \sin (e)) \sin (2 f x) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {(A+3 i B) \sec ^2(e+f x) \left (\frac {1}{2} i \sin (e)-\frac {\cos (e)}{2}\right ) \sin (4 f x) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac {(A+i B) \sec ^2(e+f x) \left (\frac {1}{3} \cos (3 e)-\frac {1}{3} i \sin (3 e)\right ) \sin (6 f x) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 199, normalized size = 1.21 \[ -\frac {12 \, {\left (A + 7 i \, B\right )} c^{4} f x e^{\left (8 i \, f x + 8 i \, e\right )} - {\left (3 i \, A - 21 \, B\right )} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - {\left (-i \, A + 7 \, B\right )} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - {\left (2 i \, A - 2 \, B\right )} c^{4} + {\left (12 \, {\left (A + 7 i \, B\right )} c^{4} f x - {\left (6 i \, A - 42 \, B\right )} c^{4}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - {\left ({\left (-6 i \, A + 42 \, B\right )} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-6 i \, A + 42 \, B\right )} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{6 \, {\left (a^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} + a^{3} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.93, size = 429, normalized size = 2.62 \[ \frac {\frac {30 \, {\left (-i \, A c^{4} + 7 \, B c^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3}} + \frac {60 \, {\left (i \, A c^{4} - 7 \, B c^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}{a^{3}} - \frac {30 \, {\left (i \, A c^{4} - 7 \, B c^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{a^{3}} - \frac {30 \, {\left (-i \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 i \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i \, A c^{4} - 7 \, B c^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{3}} - \frac {147 i \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1029 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 1002 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6534 i \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2445 i \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 17115 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3820 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 23860 i \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2445 i \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 17115 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1002 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6534 i \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 147 i \, A c^{4} + 1029 \, B c^{4}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{6}}}{30 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 207, normalized size = 1.26 \[ \frac {i B \,c^{4} \tan \left (f x +e \right )}{a^{3} f}-\frac {8 c^{4} A}{3 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {8 i c^{4} B}{3 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {18 i c^{4} B}{f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {6 c^{4} A}{f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {i c^{4} A \ln \left (\tan \left (f x +e \right )-i\right )}{f \,a^{3}}-\frac {7 c^{4} B \ln \left (\tan \left (f x +e \right )-i\right )}{f \,a^{3}}+\frac {6 i c^{4} A}{f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {10 c^{4} B}{f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}} \]
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 = 11.01, size = 266, normalized size = 1.62 \[ -\frac {c^4\,\left (25\,B\,\mathrm {tan}\left (e+f\,x\right )-\frac {B\,32{}\mathrm {i}}{3}-A\,\mathrm {tan}\left (e+f\,x\right )\,6{}\mathrm {i}-\frac {8\,A}{3}-A\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-B\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,7{}\mathrm {i}+6\,A\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,{\mathrm {tan}\left (e+f\,x\right )}^2\,15{}\mathrm {i}+3\,B\,{\mathrm {tan}\left (e+f\,x\right )}^3+B\,{\mathrm {tan}\left (e+f\,x\right )}^4\,1{}\mathrm {i}+3\,A\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )+A\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}+B\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,21{}\mathrm {i}-7\,B\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-A\,\mathrm {tan}\left (e+f\,x\right )\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}+21\,B\,\mathrm {tan}\left (e+f\,x\right )\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{a^3\,f\,{\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.40, size = 408, normalized size = 2.49 \[ \frac {2 B c^{4}}{- a^{3} f e^{2 i e} e^{2 i f x} - a^{3} f} + \begin {cases} - \frac {\left (\left (- 2 i A a^{6} c^{4} f^{2} e^{6 i e} + 2 B a^{6} c^{4} f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (3 i A a^{6} c^{4} f^{2} e^{8 i e} - 9 B a^{6} c^{4} f^{2} e^{8 i e}\right ) e^{- 4 i f x} + \left (- 6 i A a^{6} c^{4} f^{2} e^{10 i e} + 30 B a^{6} c^{4} f^{2} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{6 a^{9} f^{3}} & \text {for}\: 6 a^{9} f^{3} e^{12 i e} \neq 0 \\x \left (- \frac {- 2 A c^{4} - 14 i B c^{4}}{a^{3}} + \frac {i \left (2 i A c^{4} e^{6 i e} - 2 i A c^{4} e^{4 i e} + 2 i A c^{4} e^{2 i e} - 2 i A c^{4} - 14 B c^{4} e^{6 i e} + 10 B c^{4} e^{4 i e} - 6 B c^{4} e^{2 i e} + 2 B c^{4}\right ) e^{- 6 i e}}{a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {i c^{4} \left (A + 7 i B\right ) \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{3} f} - \frac {x \left (2 A c^{4} + 14 i B c^{4}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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