Optimal. Leaf size=17 \[ \frac{1}{2} i (\cos (x)-i \sin (x))^2 \]
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Rubi [A] time = 0.0400325, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {4385} \[ \frac{1}{2} i (\cos (x)-i \sin (x))^2 \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin{align*} \int \frac{\cos (x)-i \sin (x)}{\cos (x)+i \sin (x)} \, dx &=\frac{1}{2} i (\cos (x)-i \sin (x))^2\\ \end{align*}
Mathematica [A] time = 0.0054046, size = 19, normalized size = 1.12 \[ \frac{1}{2} \sin (2 x)+\frac{1}{2} i \cos (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 8, normalized size = 0.5 \begin{align*} \left ( \tan \left ( x \right ) -i \right ) ^{-1} \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 = 1.78768, size = 24, normalized size = 1.41 \begin{align*} \frac{1}{2} i \, e^{\left (-2 i \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.128324, size = 8, normalized size = 0.47 \begin{align*} \frac{i e^{- 2 i x}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13732, size = 19, normalized size = 1.12 \begin{align*} -\frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{{\left (\tan \left (\frac{1}{2} \, x\right ) - i\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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