Optimal. Leaf size=127 \[ \frac {a^3 (5 B+i A)}{6 c^8 f (\tan (e+f x)+i)^6}+\frac {4 a^3 (A-2 i B)}{7 c^8 f (\tan (e+f x)+i)^7}-\frac {a^3 (B+i A)}{2 c^8 f (\tan (e+f x)+i)^8}+\frac {i a^3 B}{5 c^8 f (\tan (e+f x)+i)^5} \]
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Rubi [A] time = 0.18, antiderivative size = 127, 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} \[ \frac {a^3 (5 B+i A)}{6 c^8 f (\tan (e+f x)+i)^6}+\frac {4 a^3 (A-2 i B)}{7 c^8 f (\tan (e+f x)+i)^7}-\frac {a^3 (B+i A)}{2 c^8 f (\tan (e+f x)+i)^8}+\frac {i a^3 B}{5 c^8 f (\tan (e+f x)+i)^5} \]
Antiderivative was successfully verified.
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Rule 77
Rule 3588
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x)^2 (A+B x)}{(c-i c x)^9} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {4 a^2 (i A+B)}{c^9 (i+x)^9}-\frac {4 a^2 (A-2 i B)}{c^9 (i+x)^8}-\frac {i a^2 (A-5 i B)}{c^9 (i+x)^7}-\frac {i a^2 B}{c^9 (i+x)^6}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a^3 (i A+B)}{2 c^8 f (i+\tan (e+f x))^8}+\frac {4 a^3 (A-2 i B)}{7 c^8 f (i+\tan (e+f x))^7}+\frac {a^3 (i A+5 B)}{6 c^8 f (i+\tan (e+f x))^6}+\frac {i a^3 B}{5 c^8 f (i+\tan (e+f x))^5}\\ \end {align*}
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Mathematica [A] time = 10.68, size = 182, normalized size = 1.43 \[ -\frac {i a^3 (\cos (11 e+14 f x)+i \sin (11 e+14 f x)) (56 (55 A+i B) \cos (e+f x)+30 (55 A+9 i B) \cos (3 (e+f x))-280 i A \sin (e+f x)-450 i A \sin (3 (e+f x))-175 i A \sin (5 (e+f x))+385 A \cos (5 (e+f x))+616 B \sin (e+f x)+990 B \sin (3 (e+f x))+385 B \sin (5 (e+f x))+175 i B \cos (5 (e+f x)))}{53760 c^8 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.67, size = 129, normalized size = 1.02 \[ \frac {{\left (-105 i \, A - 105 \, B\right )} a^{3} e^{\left (16 i \, f x + 16 i \, e\right )} + {\left (-600 i \, A - 360 \, B\right )} a^{3} e^{\left (14 i \, f x + 14 i \, e\right )} + {\left (-1400 i \, A - 280 \, B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} + {\left (-1680 i \, A + 336 \, B\right )} a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} + {\left (-1050 i \, A + 630 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-280 i \, A + 280 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{53760 \, c^{8} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 9.60, size = 525, normalized size = 4.13 \[ -\frac {2 \, {\left (105 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{15} + 525 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 2975 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 140 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 8750 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} + 1190 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} + 22365 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 1596 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 39235 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 4711 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 58075 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 4600 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 63300 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 7380 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 58075 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 4600 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 39235 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 4711 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 22365 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 1596 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 8750 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1190 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2975 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 140 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 525 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 105 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{105 \, c^{8} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 90, normalized size = 0.71 \[ \frac {a^{3} \left (-\frac {4 i A +4 B}{8 \left (\tan \left (f x +e \right )+i\right )^{8}}-\frac {8 i B -4 A}{7 \left (\tan \left (f x +e \right )+i\right )^{7}}-\frac {-i A -5 B}{6 \left (\tan \left (f x +e \right )+i\right )^{6}}+\frac {i B}{5 \left (\tan \left (f x +e \right )+i\right )^{5}}\right )}{f \,c^{8}} \]
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 = 9.47, size = 160, normalized size = 1.26 \[ \frac {-\frac {a^3\,\left (-2\,B+A\,20{}\mathrm {i}\right )}{210}+\frac {a^3\,\mathrm {tan}\left (e+f\,x\right )\,\left (50\,A-B\,16{}\mathrm {i}\right )}{210}+\frac {B\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{5}+\frac {a^3\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (49\,B+A\,35{}\mathrm {i}\right )}{210}}{c^8\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+{\mathrm {tan}\left (e+f\,x\right )}^7\,8{}\mathrm {i}-28\,{\mathrm {tan}\left (e+f\,x\right )}^6-{\mathrm {tan}\left (e+f\,x\right )}^5\,56{}\mathrm {i}+70\,{\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,56{}\mathrm {i}-28\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,8{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.72, size = 498, normalized size = 3.92 \[ \begin {cases} \frac {\left (- 1803886264320 i A a^{3} c^{40} f^{5} e^{6 i e} + 1803886264320 B a^{3} c^{40} f^{5} e^{6 i e}\right ) e^{6 i f x} + \left (- 6764573491200 i A a^{3} c^{40} f^{5} e^{8 i e} + 4058744094720 B a^{3} c^{40} f^{5} e^{8 i e}\right ) e^{8 i f x} + \left (- 10823317585920 i A a^{3} c^{40} f^{5} e^{10 i e} + 2164663517184 B a^{3} c^{40} f^{5} e^{10 i e}\right ) e^{10 i f x} + \left (- 9019431321600 i A a^{3} c^{40} f^{5} e^{12 i e} - 1803886264320 B a^{3} c^{40} f^{5} e^{12 i e}\right ) e^{12 i f x} + \left (- 3865470566400 i A a^{3} c^{40} f^{5} e^{14 i e} - 2319282339840 B a^{3} c^{40} f^{5} e^{14 i e}\right ) e^{14 i f x} + \left (- 676457349120 i A a^{3} c^{40} f^{5} e^{16 i e} - 676457349120 B a^{3} c^{40} f^{5} e^{16 i e}\right ) e^{16 i f x}}{346346162749440 c^{48} f^{6}} & \text {for}\: 346346162749440 c^{48} f^{6} \neq 0 \\\frac {x \left (A a^{3} e^{16 i e} + 5 A a^{3} e^{14 i e} + 10 A a^{3} e^{12 i e} + 10 A a^{3} e^{10 i e} + 5 A a^{3} e^{8 i e} + A a^{3} e^{6 i e} - i B a^{3} e^{16 i e} - 3 i B a^{3} e^{14 i e} - 2 i B a^{3} e^{12 i e} + 2 i B a^{3} e^{10 i e} + 3 i B a^{3} e^{8 i e} + i B a^{3} e^{6 i e}\right )}{32 c^{8}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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