Optimal. Leaf size=251 \[ -\frac{3 A+2 i B}{64 a^2 c^5 f (-\tan (e+f x)+i)}+\frac{5 (3 A+i B)}{128 a^2 c^5 f (\tan (e+f x)+i)}-\frac{-B+i A}{128 a^2 c^5 f (-\tan (e+f x)+i)^2}+\frac{-B+5 i A}{64 a^2 c^5 f (\tan (e+f x)+i)^2}-\frac{B+3 i A}{64 a^2 c^5 f (\tan (e+f x)+i)^4}+\frac{A-i B}{40 a^2 c^5 f (\tan (e+f x)+i)^5}+\frac{3 x (7 A+3 i B)}{128 a^2 c^5}-\frac{A}{16 a^2 c^5 f (\tan (e+f x)+i)^3} \]
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Rubi [A] time = 0.304617, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073, Rules used = {3588, 77, 203} \[ -\frac{3 A+2 i B}{64 a^2 c^5 f (-\tan (e+f x)+i)}+\frac{5 (3 A+i B)}{128 a^2 c^5 f (\tan (e+f x)+i)}-\frac{-B+i A}{128 a^2 c^5 f (-\tan (e+f x)+i)^2}+\frac{-B+5 i A}{64 a^2 c^5 f (\tan (e+f x)+i)^2}-\frac{B+3 i A}{64 a^2 c^5 f (\tan (e+f x)+i)^4}+\frac{A-i B}{40 a^2 c^5 f (\tan (e+f x)+i)^5}+\frac{3 x (7 A+3 i B)}{128 a^2 c^5}-\frac{A}{16 a^2 c^5 f (\tan (e+f x)+i)^3} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 77
Rule 203
Rubi steps
\begin{align*} \int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^5} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^3 (c-i c x)^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{i (A+i B)}{64 a^3 c^6 (-i+x)^3}+\frac{-3 A-2 i B}{64 a^3 c^6 (-i+x)^2}+\frac{-A+i B}{8 a^3 c^6 (i+x)^6}+\frac{3 i A+B}{16 a^3 c^6 (i+x)^5}+\frac{3 A}{16 a^3 c^6 (i+x)^4}+\frac{-5 i A+B}{32 a^3 c^6 (i+x)^3}-\frac{5 (3 A+i B)}{128 a^3 c^6 (i+x)^2}+\frac{3 (7 A+3 i B)}{128 a^3 c^6 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A-B}{128 a^2 c^5 f (i-\tan (e+f x))^2}-\frac{3 A+2 i B}{64 a^2 c^5 f (i-\tan (e+f x))}+\frac{A-i B}{40 a^2 c^5 f (i+\tan (e+f x))^5}-\frac{3 i A+B}{64 a^2 c^5 f (i+\tan (e+f x))^4}-\frac{A}{16 a^2 c^5 f (i+\tan (e+f x))^3}+\frac{5 i A-B}{64 a^2 c^5 f (i+\tan (e+f x))^2}+\frac{5 (3 A+i B)}{128 a^2 c^5 f (i+\tan (e+f x))}+\frac{(3 (7 A+3 i B)) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{128 a^2 c^5 f}\\ &=\frac{3 (7 A+3 i B) x}{128 a^2 c^5}-\frac{i A-B}{128 a^2 c^5 f (i-\tan (e+f x))^2}-\frac{3 A+2 i B}{64 a^2 c^5 f (i-\tan (e+f x))}+\frac{A-i B}{40 a^2 c^5 f (i+\tan (e+f x))^5}-\frac{3 i A+B}{64 a^2 c^5 f (i+\tan (e+f x))^4}-\frac{A}{16 a^2 c^5 f (i+\tan (e+f x))^3}+\frac{5 i A-B}{64 a^2 c^5 f (i+\tan (e+f x))^2}+\frac{5 (3 A+i B)}{128 a^2 c^5 f (i+\tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 3.24652, size = 274, normalized size = 1.09 \[ \frac{\sec ^2(e+f x) (\cos (5 (e+f x))+i \sin (5 (e+f x))) (50 i (21 A+i B) \cos (e+f x)+20 (A (-42 f x+7 i)+3 B (1-6 i f x)) \cos (3 (e+f x))+350 A \sin (e+f x)-140 A \sin (3 (e+f x))+840 i A f x \sin (3 (e+f x))-175 A \sin (5 (e+f x))-14 A \sin (7 (e+f x))-105 i A \cos (5 (e+f x))-6 i A \cos (7 (e+f x))+150 i B \sin (e+f x)+60 i B \sin (3 (e+f x))-360 B f x \sin (3 (e+f x))-75 i B \sin (5 (e+f x))-6 i B \sin (7 (e+f x))+125 B \cos (5 (e+f x))+14 B \cos (7 (e+f x)))}{5120 a^2 c^5 f (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 397, normalized size = 1.6 \begin{align*}{\frac{{\frac{5\,i}{64}}A}{f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}+{\frac{3\,A}{64\,f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{21\,i}{256}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) A}{f{a}^{2}{c}^{5}}}+{\frac{9\,\ln \left ( \tan \left ( fx+e \right ) -i \right ) B}{256\,f{a}^{2}{c}^{5}}}+{\frac{B}{128\,f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{{\frac{5\,i}{128}}B}{f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{A}{40\,f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}}-{\frac{{\frac{3\,i}{64}}A}{f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}-{\frac{{\frac{i}{40}}B}{f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}}-{\frac{B}{64\,f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}+{\frac{15\,A}{128\,f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{{\frac{i}{32}}B}{f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{i}{128}}A}{f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{9\,\ln \left ( \tan \left ( fx+e \right ) +i \right ) B}{256\,f{a}^{2}{c}^{5}}}-{\frac{A}{16\,f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}-{\frac{{\frac{21\,i}{256}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) A}{f{a}^{2}{c}^{5}}}-{\frac{B}{64\,f{a}^{2}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}} \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 = 1.10827, size = 464, normalized size = 1.85 \begin{align*} \frac{{\left (120 \,{\left (7 \, A + 3 i \, B\right )} f x e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-4 i \, A - 4 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} +{\left (-35 i \, A - 25 \, B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-140 i \, A - 60 \, B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-350 i \, A - 50 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-700 i \, A + 100 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (140 i \, A - 100 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 10 i \, A - 10 \, B\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{5120 \, a^{2} c^{5} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.26768, size = 607, normalized size = 2.42 \begin{align*} \begin{cases} \frac{\left (\left (11258999068426240 i A a^{12} c^{30} f^{6} e^{2 i e} - 11258999068426240 B a^{12} c^{30} f^{6} e^{2 i e}\right ) e^{- 4 i f x} + \left (157625986957967360 i A a^{12} c^{30} f^{6} e^{4 i e} - 112589990684262400 B a^{12} c^{30} f^{6} e^{4 i e}\right ) e^{- 2 i f x} + \left (- 788129934789836800 i A a^{12} c^{30} f^{6} e^{8 i e} + 112589990684262400 B a^{12} c^{30} f^{6} e^{8 i e}\right ) e^{2 i f x} + \left (- 394064967394918400 i A a^{12} c^{30} f^{6} e^{10 i e} - 56294995342131200 B a^{12} c^{30} f^{6} e^{10 i e}\right ) e^{4 i f x} + \left (- 157625986957967360 i A a^{12} c^{30} f^{6} e^{12 i e} - 67553994410557440 B a^{12} c^{30} f^{6} e^{12 i e}\right ) e^{6 i f x} + \left (- 39406496739491840 i A a^{12} c^{30} f^{6} e^{14 i e} - 28147497671065600 B a^{12} c^{30} f^{6} e^{14 i e}\right ) e^{8 i f x} + \left (- 4503599627370496 i A a^{12} c^{30} f^{6} e^{16 i e} - 4503599627370496 B a^{12} c^{30} f^{6} e^{16 i e}\right ) e^{10 i f x}\right ) e^{- 6 i e}}{5764607523034234880 a^{14} c^{35} f^{7}} & \text{for}\: 5764607523034234880 a^{14} c^{35} f^{7} e^{6 i e} \neq 0 \\x \left (- \frac{21 A + 9 i B}{128 a^{2} c^{5}} + \frac{\left (A e^{14 i e} + 7 A e^{12 i e} + 21 A e^{10 i e} + 35 A e^{8 i e} + 35 A e^{6 i e} + 21 A e^{4 i e} + 7 A e^{2 i e} + A - i B e^{14 i e} - 5 i B e^{12 i e} - 9 i B e^{10 i e} - 5 i B e^{8 i e} + 5 i B e^{6 i e} + 9 i B e^{4 i e} + 5 i B e^{2 i e} + i B\right ) e^{- 4 i e}}{128 a^{2} c^{5}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (21 A + 9 i B\right )}{128 a^{2} c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31843, size = 363, normalized size = 1.45 \begin{align*} -\frac{\frac{20 \,{\left (-21 i \, A + 9 \, B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{2} c^{5}} + \frac{20 \,{\left (21 i \, A - 9 \, B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{2} c^{5}} + \frac{10 \,{\left (63 i \, A \tan \left (f x + e\right )^{2} - 27 \, B \tan \left (f x + e\right )^{2} + 150 \, A \tan \left (f x + e\right ) + 70 i \, B \tan \left (f x + e\right ) - 91 i \, A + 47 \, B\right )}}{a^{2} c^{5}{\left (-i \, \tan \left (f x + e\right ) - 1\right )}^{2}} + \frac{959 i \, A \tan \left (f x + e\right )^{5} - 411 \, B \tan \left (f x + e\right )^{5} - 5395 \, A \tan \left (f x + e\right )^{4} - 2255 i \, B \tan \left (f x + e\right )^{4} - 12390 i \, A \tan \left (f x + e\right )^{3} + 4990 \, B \tan \left (f x + e\right )^{3} + 14710 \, A \tan \left (f x + e\right )^{2} + 5550 i \, B \tan \left (f x + e\right )^{2} + 9275 i \, A \tan \left (f x + e\right ) - 3015 \, B \tan \left (f x + e\right ) - 2647 \, A - 483 i \, B}{a^{2} c^{5}{\left (\tan \left (f x + e\right ) + i\right )}^{5}}}{5120 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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