Optimal. Leaf size=17 \[ -\frac {i}{2 (\cos (x)-i \sin (x))^2} \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.65, size = 6, normalized size = 0.35 \[ -\frac {1}{2} i \, e^{\left (2 i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 14, normalized size = 0.82 \[ -\frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{{\left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 8, normalized size = 0.47 \[ \frac {1}{\tan \relax (x )+i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.74, size = 13, normalized size = 0.76 \[ \frac {\sin \relax (x)}{\cos \relax (x)-\sin \relax (x)\,1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 10, normalized size = 0.59 \[ - \frac {i e^{2 i x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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