Integrand size = 41, antiderivative size = 287 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {7 (4 A+i B) x}{128 a^3 c^5}+\frac {A+i B}{192 a^3 c^5 f (i-\tan (e+f x))^3}-\frac {3 i A-2 B}{128 a^3 c^5 f (i-\tan (e+f x))^2}-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (i-\tan (e+f x))}+\frac {A-i B}{80 a^3 c^5 f (i+\tan (e+f x))^5}-\frac {2 i A+B}{64 a^3 c^5 f (i+\tan (e+f x))^4}-\frac {5 A-i B}{96 a^3 c^5 f (i+\tan (e+f x))^3}+\frac {5 i A}{64 a^3 c^5 f (i+\tan (e+f x))^2}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (i+\tan (e+f x))} \]
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Time = 0.41 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3669, 78, 209} \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (-\tan (e+f x)+i)}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (\tan (e+f x)+i)}-\frac {-2 B+3 i A}{128 a^3 c^5 f (-\tan (e+f x)+i)^2}+\frac {A+i B}{192 a^3 c^5 f (-\tan (e+f x)+i)^3}-\frac {5 A-i B}{96 a^3 c^5 f (\tan (e+f x)+i)^3}-\frac {B+2 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^4}+\frac {A-i B}{80 a^3 c^5 f (\tan (e+f x)+i)^5}+\frac {7 x (4 A+i B)}{128 a^3 c^5}+\frac {5 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^2} \]
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Rule 78
Rule 209
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^4 (c-i c x)^6} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (\frac {A+i B}{64 a^4 c^6 (-i+x)^4}+\frac {i (3 A+2 i B)}{64 a^4 c^6 (-i+x)^3}-\frac {3 (7 A+3 i B)}{256 a^4 c^6 (-i+x)^2}+\frac {-A+i B}{16 a^4 c^6 (i+x)^6}+\frac {2 i A+B}{16 a^4 c^6 (i+x)^5}+\frac {5 A-i B}{32 a^4 c^6 (i+x)^4}-\frac {5 i A}{32 a^4 c^6 (i+x)^3}-\frac {5 (7 A+i B)}{256 a^4 c^6 (i+x)^2}+\frac {7 (4 A+i B)}{128 a^4 c^6 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {A+i B}{192 a^3 c^5 f (i-\tan (e+f x))^3}-\frac {3 i A-2 B}{128 a^3 c^5 f (i-\tan (e+f x))^2}-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (i-\tan (e+f x))}+\frac {A-i B}{80 a^3 c^5 f (i+\tan (e+f x))^5}-\frac {2 i A+B}{64 a^3 c^5 f (i+\tan (e+f x))^4}-\frac {5 A-i B}{96 a^3 c^5 f (i+\tan (e+f x))^3}+\frac {5 i A}{64 a^3 c^5 f (i+\tan (e+f x))^2}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (i+\tan (e+f x))}+\frac {(7 (4 A+i B)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{128 a^3 c^5 f} \\ & = \frac {7 (4 A+i B) x}{128 a^3 c^5}+\frac {A+i B}{192 a^3 c^5 f (i-\tan (e+f x))^3}-\frac {3 i A-2 B}{128 a^3 c^5 f (i-\tan (e+f x))^2}-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (i-\tan (e+f x))}+\frac {A-i B}{80 a^3 c^5 f (i+\tan (e+f x))^5}-\frac {2 i A+B}{64 a^3 c^5 f (i+\tan (e+f x))^4}-\frac {5 A-i B}{96 a^3 c^5 f (i+\tan (e+f x))^3}+\frac {5 i A}{64 a^3 c^5 f (i+\tan (e+f x))^2}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (i+\tan (e+f x))} \\ \end{align*}
Time = 6.79 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {\sec ^7(e+f x) (2100 i A \cos (e+f x)+63 (-8 i A+7 B) \cos (3 (e+f x))-56 i A \cos (5 (e+f x))+119 B \cos (5 (e+f x))-4 i A \cos (7 (e+f x))+16 B \cos (7 (e+f x))+872 A \sin (e+f x)+218 i B \sin (e+f x)-420 (4 A+i B) \arctan (\tan (e+f x)) \sec (e+f x) (\cos (2 (e+f x))-i \sin (2 (e+f x)))-956 A \sin (3 (e+f x))-239 i B \sin (3 (e+f x))-164 A \sin (5 (e+f x))-41 i B \sin (5 (e+f x))-16 A \sin (7 (e+f x))-4 i B \sin (7 (e+f x)))}{7680 a^3 c^5 f (-i+\tan (e+f x))^3 (i+\tan (e+f x))^5} \]
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Time = 0.38 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.17
method | result | size |
norman | \(\frac {\frac {7 \left (i B +4 A \right ) x}{128 a c}-\frac {8 i A +3 B}{40 a c f}+\frac {B \tan \left (f x +e \right )^{2}}{8 a c f}+\frac {79 \left (i B +4 A \right ) \tan \left (f x +e \right )^{3}}{192 a c f}+\frac {7 \left (i B +4 A \right ) \tan \left (f x +e \right )^{5}}{15 a c f}+\frac {49 \left (i B +4 A \right ) \tan \left (f x +e \right )^{7}}{192 a c f}+\frac {7 \left (i B +4 A \right ) \tan \left (f x +e \right )^{9}}{128 a c f}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{2}}{128 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{4}}{64 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{6}}{64 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{8}}{128 a c}+\frac {7 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{10}}{128 a c}+\frac {\left (-7 i B +100 A \right ) \tan \left (f x +e \right )}{128 a c f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{5} a^{2} c^{4}}\) | \(335\) |
risch | \(-\frac {i \sin \left (4 f x +4 e \right ) B}{128 a^{3} c^{5} f}+\frac {7 x A}{32 a^{3} c^{5}}-\frac {{\mathrm e}^{10 i \left (f x +e \right )} B}{2560 a^{3} c^{5} f}+\frac {7 i x B}{128 a^{3} c^{5}}-\frac {3 \,{\mathrm e}^{8 i \left (f x +e \right )} B}{1024 a^{3} c^{5} f}-\frac {13 i \sin \left (6 f x +6 e \right ) B}{1536 a^{3} c^{5} f}-\frac {5 \cos \left (6 f x +6 e \right ) B}{512 a^{3} c^{5} f}-\frac {21 i \cos \left (2 f x +2 e \right ) A}{256 a^{3} c^{5} f}-\frac {i {\mathrm e}^{8 i \left (f x +e \right )} A}{256 a^{3} c^{5} f}+\frac {29 \sin \left (6 f x +6 e \right ) A}{1536 a^{3} c^{5} f}-\frac {5 \cos \left (4 f x +4 e \right ) B}{256 a^{3} c^{5} f}+\frac {7 i \sin \left (2 f x +2 e \right ) B}{256 a^{3} c^{5} f}-\frac {3 i \cos \left (4 f x +4 e \right ) A}{64 a^{3} c^{5} f}+\frac {\sin \left (4 f x +4 e \right ) A}{16 a^{3} c^{5} f}-\frac {7 \cos \left (2 f x +2 e \right ) B}{256 a^{3} c^{5} f}-\frac {i {\mathrm e}^{10 i \left (f x +e \right )} A}{2560 a^{3} c^{5} f}-\frac {9 i \cos \left (6 f x +6 e \right ) A}{512 a^{3} c^{5} f}+\frac {49 \sin \left (2 f x +2 e \right ) A}{256 a^{3} c^{5} f}\) | \(367\) |
derivativedivides | \(-\frac {i A}{32 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {A}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {9 i B}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {5 A}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {5 i B}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}+\frac {35 A}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}-\frac {i B}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {B}{64 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{5}}-\frac {3 i A}{128 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {7 A \arctan \left (\tan \left (f x +e \right )\right )}{32 f \,a^{3} c^{5}}-\frac {A}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {5 i A}{64 a^{3} c^{5} f \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {i B}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {21 A}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {B}{64 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}\) | \(394\) |
default | \(-\frac {i A}{32 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {A}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {9 i B}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {5 A}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {5 i B}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}+\frac {35 A}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}-\frac {i B}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {B}{64 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{5}}-\frac {3 i A}{128 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {7 A \arctan \left (\tan \left (f x +e \right )\right )}{32 f \,a^{3} c^{5}}-\frac {A}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {5 i A}{64 a^{3} c^{5} f \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {i B}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {21 A}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {B}{64 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}\) | \(394\) |
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Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.55 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {{\left (840 \, {\left (4 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 6 \, {\left (i \, A + B\right )} e^{\left (16 i \, f x + 16 i \, e\right )} - 15 \, {\left (4 i \, A + 3 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} - 140 \, {\left (2 i \, A + B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} - 210 \, {\left (4 i \, A + B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - 2100 i \, A e^{\left (8 i \, f x + 8 i \, e\right )} - 420 \, {\left (-2 i \, A + B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 30 \, {\left (-4 i \, A + 3 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 10 i \, A - 10 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{15360 \, a^{3} c^{5} f} \]
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Time = 0.80 (sec) , antiderivative size = 646, normalized size of antiderivative = 2.25 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\begin {cases} \frac {\left (- 7263405479023135948800 i A a^{21} c^{35} f^{7} e^{14 i e} e^{2 i f x} + \left (34587645138205409280 i A a^{21} c^{35} f^{7} e^{6 i e} - 34587645138205409280 B a^{21} c^{35} f^{7} e^{6 i e}\right ) e^{- 6 i f x} + \left (415051741658464911360 i A a^{21} c^{35} f^{7} e^{8 i e} - 311288806243848683520 B a^{21} c^{35} f^{7} e^{8 i e}\right ) e^{- 4 i f x} + \left (2905362191609254379520 i A a^{21} c^{35} f^{7} e^{10 i e} - 1452681095804627189760 B a^{21} c^{35} f^{7} e^{10 i e}\right ) e^{- 2 i f x} + \left (- 2905362191609254379520 i A a^{21} c^{35} f^{7} e^{16 i e} - 726340547902313594880 B a^{21} c^{35} f^{7} e^{16 i e}\right ) e^{4 i f x} + \left (- 968454063869751459840 i A a^{21} c^{35} f^{7} e^{18 i e} - 484227031934875729920 B a^{21} c^{35} f^{7} e^{18 i e}\right ) e^{6 i f x} + \left (- 207525870829232455680 i A a^{21} c^{35} f^{7} e^{20 i e} - 155644403121924341760 B a^{21} c^{35} f^{7} e^{20 i e}\right ) e^{8 i f x} + \left (- 20752587082923245568 i A a^{21} c^{35} f^{7} e^{22 i e} - 20752587082923245568 B a^{21} c^{35} f^{7} e^{22 i e}\right ) e^{10 i f x}\right ) e^{- 12 i e}}{53126622932283508654080 a^{24} c^{40} f^{8}} & \text {for}\: a^{24} c^{40} f^{8} e^{12 i e} \neq 0 \\x \left (- \frac {28 A + 7 i B}{128 a^{3} c^{5}} + \frac {\left (A e^{16 i e} + 8 A e^{14 i e} + 28 A e^{12 i e} + 56 A e^{10 i e} + 70 A e^{8 i e} + 56 A e^{6 i e} + 28 A e^{4 i e} + 8 A e^{2 i e} + A - i B e^{16 i e} - 6 i B e^{14 i e} - 14 i B e^{12 i e} - 14 i B e^{10 i e} + 14 i B e^{6 i e} + 14 i B e^{4 i e} + 6 i B e^{2 i e} + i B\right ) e^{- 6 i e}}{256 a^{3} c^{5}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (28 A + 7 i B\right )}{128 a^{3} c^{5}} \]
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Exception generated. \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.89 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=-\frac {\frac {420 \, {\left (-4 i \, A + B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{5}} + \frac {420 \, {\left (4 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{5}} + \frac {10 \, {\left (-308 i \, A \tan \left (f x + e\right )^{3} + 77 \, B \tan \left (f x + e\right )^{3} - 1050 \, A \tan \left (f x + e\right )^{2} - 285 i \, B \tan \left (f x + e\right )^{2} + 1212 i \, A \tan \left (f x + e\right ) - 363 \, B \tan \left (f x + e\right ) + 478 \, A + 163 i \, B\right )}}{a^{3} c^{5} {\left (\tan \left (f x + e\right ) - i\right )}^{3}} + \frac {3836 i \, A \tan \left (f x + e\right )^{5} - 959 \, B \tan \left (f x + e\right )^{5} - 21280 \, A \tan \left (f x + e\right )^{4} - 5095 i \, B \tan \left (f x + e\right )^{4} - 47960 i \, A \tan \left (f x + e\right )^{3} + 10790 \, B \tan \left (f x + e\right )^{3} + 55360 \, A \tan \left (f x + e\right )^{2} + 11230 i \, B \tan \left (f x + e\right )^{2} + 33260 i \, A \tan \left (f x + e\right ) - 5435 \, B \tan \left (f x + e\right ) - 8608 \, A - 667 i \, B}{a^{3} c^{5} {\left (\tan \left (f x + e\right ) + i\right )}^{5}}}{15360 \, f} \]
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Time = 10.41 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {\frac {3\,B}{40\,a^3\,c^5}+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (-\frac {7\,B}{24\,a^3\,c^5}+\frac {A\,7{}\mathrm {i}}{6\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (-\frac {7\,B}{64\,a^3\,c^5}+\frac {A\,7{}\mathrm {i}}{16\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (\frac {7\,A}{32\,a^3\,c^5}+\frac {B\,7{}\mathrm {i}}{128\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {35\,A}{96\,a^3\,c^5}+\frac {B\,35{}\mathrm {i}}{384\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (-\frac {77\,B}{320\,a^3\,c^5}+\frac {A\,77{}\mathrm {i}}{80\,a^3\,c^5}\right )-{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {49\,A}{480\,a^3\,c^5}+\frac {B\,49{}\mathrm {i}}{1920\,a^3\,c^5}\right )-\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {61\,A}{160\,a^3\,c^5}+\frac {B\,61{}\mathrm {i}}{640\,a^3\,c^5}\right )+\frac {A\,1{}\mathrm {i}}{5\,a^3\,c^5}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+{\mathrm {tan}\left (e+f\,x\right )}^7\,2{}\mathrm {i}+2\,{\mathrm {tan}\left (e+f\,x\right )}^6+{\mathrm {tan}\left (e+f\,x\right )}^5\,6{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^3\,6{}\mathrm {i}-2\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}-1\right )}+\frac {7\,x\,\left (4\,A+B\,1{}\mathrm {i}\right )}{128\,a^3\,c^5} \]
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