Optimal. Leaf size=23 \[ -\frac{i a}{f (c-i c \tan (e+f x))} \]
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Rubi [A] time = 0.0773809, 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} \[ -\frac{i a}{f (c-i c \tan (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{a+i a \tan (e+f x)}{c-i c \tan (e+f x)} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(c-i c \tan (e+f x))^2} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{(c+x)^2} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac{i a}{f (c-i c \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.103282, size = 32, normalized size = 1.39 \[ \frac{a (\sin (2 (e+f x))-i \cos (2 (e+f x)))}{2 c f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 20, normalized size = 0.9 \begin{align*}{\frac{a}{cf \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 = 1.39965, size = 49, normalized size = 2.13 \begin{align*} -\frac{i \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{2 \, c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.404668, size = 39, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{i a e^{2 i e} e^{2 i f x}}{2 c f} & \text{for}\: 2 c f \neq 0 \\\frac{a x e^{2 i e}}{c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31153, size = 45, normalized size = 1.96 \begin{align*} -\frac{2 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{c 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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