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 = 0.182177, antiderivative size = 127, normalized size of antiderivative = 1., 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 3588
Rule 77
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*}
Mathematica [A] time = 9.59929, 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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Maple [A] time = 0.051, size = 90, normalized size = 0.7 \begin{align*}{\frac{{a}^{3}}{f{c}^{8}} \left ({\frac{{\frac{i}{5}}B}{ \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}}-{\frac{8\,iB-4\,A}{7\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{7}}}-{\frac{-5\,B-iA}{6\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{6}}}-{\frac{4\,iA+4\,B}{8\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{8}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42378, size = 402, normalized size = 3.17 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.14805, size = 498, normalized size = 3.92 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.66941, size = 709, normalized size = 5.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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