Optimal. Leaf size=19 \[ -\frac {1}{3} \cos ^3(x)-\frac {1}{3} i \sin ^3(x) \]
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Rubi [A]
time = 0.07, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {3599, 3187,
3186, 2645, 30, 2644} \begin {gather*} -\frac {1}{3} \cos ^3(x)-\frac {1}{3} i \sin ^3(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rule 2645
Rule 3186
Rule 3187
Rule 3599
Rubi steps
\begin {align*} \int \frac {\cos (x)}{i+\cot (x)} \, dx &=-\int \frac {\cos (x) \sin (x)}{-\cos (x)-i \sin (x)} \, dx\\ &=i \int \cos (x) (-i \cos (x)-\sin (x)) \sin (x) \, dx\\ &=i \int \left (-i \cos ^2(x) \sin (x)-\cos (x) \sin ^2(x)\right ) \, dx\\ &=-\left (i \int \cos (x) \sin ^2(x) \, dx\right )+\int \cos ^2(x) \sin (x) \, dx\\ &=-\left (i \text {Subst}\left (\int x^2 \, dx,x,\sin (x)\right )\right )-\text {Subst}\left (\int x^2 \, dx,x,\cos (x)\right )\\ &=-\frac {1}{3} \cos ^3(x)-\frac {1}{3} i \sin ^3(x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{3} \left (-\cos ^3(x)-i \sin ^3(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 48 vs. \(2 (14 ) = 28\).
time = 0.21, size = 49, normalized size = 2.58
method | result | size |
risch | \(-\frac {{\mathrm e}^{i x}}{4}-\frac {{\mathrm e}^{-3 i x}}{12}\) | \(16\) |
default | \(\frac {i}{-2 i+2 \tan \left (\frac {x}{2}\right )}-\frac {2 i}{3 \left (-i+\tan \left (\frac {x}{2}\right )\right )^{3}}-\frac {1}{\left (-i+\tan \left (\frac {x}{2}\right )\right )^{2}}-\frac {i}{2 \left (\tan \left (\frac {x}{2}\right )+i\right )}\) | \(49\) |
norman | \(\frac {\frac {2 \left (\tan ^{2}\left (x \right )\right )}{3}+\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3}+\frac {2 i \tan \left (x \right )}{3}-\frac {4 i \tan \left (\frac {x}{2}\right )}{3}-\frac {2 \tan \left (x \right ) \tan \left (\frac {x}{2}\right )}{3}-\frac {2 i \tan \left (x \right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3}-\frac {2 i \left (\tan ^{2}\left (x \right )\right ) \tan \left (\frac {x}{2}\right )}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (x \right )\right )}\) | \(77\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.94, size = 14, normalized size = 0.74 \begin {gather*} -\frac {1}{12} \, {\left (3 \, e^{\left (4 i \, x\right )} + 1\right )} e^{\left (-3 i \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 17, normalized size = 0.89 \begin {gather*} - \frac {e^{i x}}{4} - \frac {e^{- 3 i x}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 31 vs. \(2 (13) = 26\).
time = 0.42, size = 31, normalized size = 1.63 \begin {gather*} -\frac {i}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}} - \frac {-3 i \, \tan \left (\frac {1}{2} \, x\right )^{2} + i}{6 \, {\left (\tan \left (\frac {1}{2} \, x\right ) - i\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 40, normalized size = 2.11 \begin {gather*} -\frac {\left (-3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )\,2{}\mathrm {i}+1\right )\,2{}\mathrm {i}}{3\,{\left (1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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