Optimal. Leaf size=25 \[ -\frac{i a}{2 f (c-i c \tan (e+f x))^2} \]
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Rubi [A] time = 0.0735024, antiderivative size = 25, 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} \[ -\frac{i a}{2 f (c-i c \tan (e+f x))^2} \]
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))^2} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(c-i c \tan (e+f x))^3} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{(c+x)^3} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac{i a}{2 f (c-i c \tan (e+f x))^2}\\ \end{align*}
Mathematica [B] time = 0.627989, size = 51, normalized size = 2.04 \[ \frac{a (3 \cos (e+f x)-i \sin (e+f x)) (\sin (3 (e+f x))-i \cos (3 (e+f x)))}{8 c^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 22, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{2}}a}{f{c}^{2} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}} \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 = 1.32476, size = 92, normalized size = 3.68 \begin{align*} \frac{-i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 2 i \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{8 \, c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.744005, size = 90, normalized size = 3.6 \begin{align*} \begin{cases} \frac{- 4 i a c^{2} f e^{4 i e} e^{4 i f x} - 8 i a c^{2} f e^{2 i e} e^{2 i f x}}{32 c^{4} f^{2}} & \text{for}\: 32 c^{4} f^{2} \neq 0 \\\frac{x \left (a e^{4 i e} + a e^{2 i e}\right )}{2 c^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26699, size = 88, normalized size = 3.52 \begin{align*} -\frac{2 \,{\left (a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + i \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{c^{2} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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