Optimal. Leaf size=37 \[ \frac{\sin (e+f x) \cos (e+f x)}{2 a c f}+\frac{x}{2 a c} \]
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Rubi [A] time = 0.0725315, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 2635, 8} \[ \frac{\sin (e+f x) \cos (e+f x)}{2 a c f}+\frac{x}{2 a c} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx &=\frac{\int \cos ^2(e+f x) \, dx}{a c}\\ &=\frac{\cos (e+f x) \sin (e+f x)}{2 a c f}+\frac{\int 1 \, dx}{2 a c}\\ &=\frac{x}{2 a c}+\frac{\cos (e+f x) \sin (e+f x)}{2 a c f}\\ \end{align*}
Mathematica [A] time = 0.0283078, size = 29, normalized size = 0.78 \[ \frac{2 (e+f x)+\sin (2 (e+f x))}{4 a c f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.036, size = 90, normalized size = 2.4 \begin{align*}{\frac{-{\frac{i}{4}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{fac}}+{\frac{1}{4\,fac \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{i}{4}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{fac}}+{\frac{1}{4\,fac \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 = 1.37227, size = 122, normalized size = 3.3 \begin{align*} \frac{{\left (4 \, f x e^{\left (2 i \, f x + 2 i \, e\right )} - i \, e^{\left (4 i \, f x + 4 i \, e\right )} + i\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 = 0.623378, size = 119, normalized size = 3.22 \begin{align*} \begin{cases} \frac{\left (- 8 i a c f e^{4 i e} e^{2 i f x} + 8 i a c f 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{\left (e^{4 i e} + 2 e^{2 i e} + 1\right ) e^{- 2 i e}}{4 a c} - \frac{1}{2 a c}\right ) & \text{otherwise} \end{cases} + \frac{x}{2 a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.43097, size = 62, normalized size = 1.68 \begin{align*} \frac{\frac{f x + e}{a c} + \frac{\tan \left (f x + e\right )}{{\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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