Optimal. Leaf size=55 \[ -\frac{a (A-i B)}{3 c^3 f (\tan (e+f x)+i)^3}-\frac{a B}{2 c^3 f (\tan (e+f x)+i)^2} \]
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Rubi [A] time = 0.0867701, 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} \[ -\frac{a (A-i B)}{3 c^3 f (\tan (e+f x)+i)^3}-\frac{a B}{2 c^3 f (\tan (e+f x)+i)^2} \]
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))^3} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(c-i c x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{A-i B}{c^4 (i+x)^4}+\frac{B}{c^4 (i+x)^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{a (A-i B)}{3 c^3 f (i+\tan (e+f x))^3}-\frac{a B}{2 c^3 f (i+\tan (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 1.29339, size = 72, normalized size = 1.31 \[ \frac{a (\cos (4 (e+f x))+i \sin (4 (e+f x))) (-2 (A+2 i B) \sin (2 (e+f x))+2 (B-2 i A) \cos (2 (e+f x))-3 i A)}{24 c^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 43, normalized size = 0.8 \begin{align*}{\frac{a}{f{c}^{3}} \left ( -{\frac{B}{2\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}-{\frac{A-iB}{3\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}} \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 = 1.40369, size = 159, normalized size = 2.89 \begin{align*} \frac{{\left (-i \, A - B\right )} a e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, A a e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-3 i \, A + 3 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.94791, size = 202, normalized size = 3.67 \begin{align*} \begin{cases} \frac{- 192 i A a c^{6} f^{2} e^{4 i e} e^{4 i f x} + \left (- 192 i A a c^{6} f^{2} e^{2 i e} + 192 B a c^{6} f^{2} e^{2 i e}\right ) e^{2 i f x} + \left (- 64 i A a c^{6} f^{2} e^{6 i e} - 64 B a c^{6} f^{2} e^{6 i e}\right ) e^{6 i f x}}{1536 c^{9} f^{3}} & \text{for}\: 1536 c^{9} f^{3} \neq 0 \\\frac{x \left (A a e^{6 i e} + 2 A a e^{4 i e} + A a e^{2 i e} - i B a e^{6 i e} + i B a e^{2 i e}\right )}{4 c^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37792, size = 201, normalized size = 3.65 \begin{align*} -\frac{2 \,{\left (3 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 6 i \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 3 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 10 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 2 i \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 6 i \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{3 \, c^{3} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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