Optimal. Leaf size=45 \[ \frac{A x}{2 a c}-\frac{\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f} \]
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Rubi [A] time = 0.127554, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {3588, 73, 639, 205} \[ \frac{A x}{2 a c}-\frac{\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 73
Rule 639
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^2 (c-i c x)^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{\left (a c+a c x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f}+\frac{A \operatorname{Subst}\left (\int \frac{1}{a c+a c x^2} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac{A x}{2 a c}-\frac{\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f}\\ \end{align*}
Mathematica [A] time = 0.0813218, size = 43, normalized size = 0.96 \[ \frac{A (2 (e+f x)+\sin (2 (e+f x)))-2 B \cos ^2(e+f x)}{4 a c f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.062, size = 142, normalized size = 3.2 \begin{align*}{\frac{-{\frac{i}{4}}A\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{afc}}+{\frac{A}{4\,afc \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{i}{4}}B}{afc \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{i}{4}}A\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{afc}}+{\frac{A}{4\,afc \left ( \tan \left ( fx+e \right ) +i \right ) }}-{\frac{{\frac{i}{4}}B}{afc \left ( \tan \left ( fx+e \right ) +i \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 [C] time = 1.69623, size = 144, normalized size = 3.2 \begin{align*} \frac{{\left (4 \, A f x e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + i \, A - B\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.933316, size = 167, normalized size = 3.71 \begin{align*} \frac{A x}{2 a c} + \begin{cases} \frac{\left (\left (8 i A a c f - 8 B a c f\right ) e^{- 2 i f x} + \left (- 8 i A a c f e^{4 i e} - 8 B a c f e^{4 i e}\right ) e^{2 i f x}\right ) e^{- 2 i e}}{64 a^{2} c^{2} f^{2}} & \text{for}\: 64 a^{2} c^{2} f^{2} e^{2 i e} \neq 0 \\x \left (- \frac{A}{2 a c} + \frac{\left (A e^{4 i e} + 2 A e^{2 i e} + A - i B e^{4 i e} + i B\right ) e^{- 2 i e}}{4 a c}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45036, size = 72, normalized size = 1.6 \begin{align*} \frac{\frac{{\left (f x + e\right )} A}{a c} + \frac{A \tan \left (f x + e\right ) - B}{{\left (\tan \left (f x + e\right )^{2} + 1\right )} a c}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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