Optimal. Leaf size=17 \[ -\frac{i}{2 (\cos (x)-i \sin (x))^2} \]
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Rubi [A] time = 0.0365433, 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} \[ -\frac{i}{2 (\cos (x)-i \sin (x))^2} \]
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{i}{2 (\cos (x)-i \sin (x))^2}\\ \end{align*}
Mathematica [A] time = 0.0048979, 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 = 0.057, 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 = 1.88227, 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 = 0.10255, size = 10, normalized size = 0.59 \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 = 1.14414, 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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