Optimal. Leaf size=23 \[ \frac{i c}{f (a+i a \tan (e+f x))} \]
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Rubi [A] time = 0.0771408, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3522, 3487, 32} \[ \frac{i c}{f (a+i a \tan (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{c-i c \tan (e+f x)}{a+i a \tan (e+f x)} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(a+i a \tan (e+f x))^2} \, dx\\ &=-\frac{(i c) \operatorname{Subst}\left (\int \frac{1}{(a+x)^2} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{i c}{f (a+i a \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.0989466, size = 32, normalized size = 1.39 \[ \frac{c (\sin (2 (e+f x))+i \cos (2 (e+f x)))}{2 a f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 20, normalized size = 0.9 \begin{align*}{\frac{c}{fa \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 [A] time = 1.12565, size = 49, normalized size = 2.13 \begin{align*} \frac{i \, c e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.238739, size = 44, normalized size = 1.91 \begin{align*} \begin{cases} \frac{i c e^{- 2 i e} e^{- 2 i f x}}{2 a f} & \text{for}\: 2 a f e^{2 i e} \neq 0 \\\frac{c x e^{- 2 i e}}{a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3554, size = 45, normalized size = 1.96 \begin{align*} -\frac{2 \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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