Optimal. Leaf size=55 \[ \frac{a (A-i B)}{5 c^5 f (\tan (e+f x)+i)^5}+\frac{a B}{4 c^5 f (\tan (e+f x)+i)^4} \]
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Rubi [A] time = 0.0889989, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {3588, 43} \[ \frac{a (A-i B)}{5 c^5 f (\tan (e+f x)+i)^5}+\frac{a B}{4 c^5 f (\tan (e+f x)+i)^4} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(c-i c x)^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{-A+i B}{c^6 (i+x)^6}-\frac{B}{c^6 (i+x)^5}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a (A-i B)}{5 c^5 f (i+\tan (e+f x))^5}+\frac{a B}{4 c^5 f (i+\tan (e+f x))^4}\\ \end{align*}
Mathematica [B] time = 2.84203, size = 124, normalized size = 2.25 \[ -\frac{i a (\cos (6 (e+f x))+i \sin (6 (e+f x))) (5 (6 A+i B) \cos (2 (e+f x))+4 (3 A+2 i B) \cos (4 (e+f x))-10 i A \sin (2 (e+f x))-8 i A \sin (4 (e+f x))+20 A+15 B \sin (2 (e+f x))+12 B \sin (4 (e+f x)))}{320 c^5 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 45, normalized size = 0.8 \begin{align*}{\frac{a}{f{c}^{5}} \left ({\frac{B}{4\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}-{\frac{-A+iB}{5\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}} \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 [B] time = 1.34209, size = 282, normalized size = 5.13 \begin{align*} \frac{{\left (-2 i \, A - 2 \, B\right )} a e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-10 i \, A - 5 \, B\right )} a e^{\left (8 i \, f x + 8 i \, e\right )} - 20 i \, A a e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-20 i \, A + 10 \, B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-10 i \, A + 10 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )}}{320 \, c^{5} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.1803, size = 350, normalized size = 6.36 \begin{align*} \begin{cases} \frac{- 10485760 i A a c^{20} f^{4} e^{6 i e} e^{6 i f x} + \left (- 5242880 i A a c^{20} f^{4} e^{2 i e} + 5242880 B a c^{20} f^{4} e^{2 i e}\right ) e^{2 i f x} + \left (- 10485760 i A a c^{20} f^{4} e^{4 i e} + 5242880 B a c^{20} f^{4} e^{4 i e}\right ) e^{4 i f x} + \left (- 5242880 i A a c^{20} f^{4} e^{8 i e} - 2621440 B a c^{20} f^{4} e^{8 i e}\right ) e^{8 i f x} + \left (- 1048576 i A a c^{20} f^{4} e^{10 i e} - 1048576 B a c^{20} f^{4} e^{10 i e}\right ) e^{10 i f x}}{167772160 c^{25} f^{5}} & \text{for}\: 167772160 c^{25} f^{5} \neq 0 \\\frac{x \left (A a e^{10 i e} + 4 A a e^{8 i e} + 6 A a e^{6 i e} + 4 A a e^{4 i e} + A a e^{2 i e} - i B a e^{10 i e} - 2 i B a e^{8 i e} + 2 i B a e^{4 i e} + i B a e^{2 i e}\right )}{16 c^{5}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.49759, size = 374, normalized size = 6.8 \begin{align*} -\frac{2 \,{\left (5 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 20 i \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 5 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 60 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 10 i \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 100 i \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 25 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 126 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 24 i \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 100 i \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 25 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 60 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 10 i \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 20 i \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 5 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 5 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{5 \, c^{5} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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