Optimal. Leaf size=9 \[ \log (\cos (x))-\log (\sin (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4334, 266, 36, 31, 29} \[ \log (\cos (x))-\log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4334
Rubi steps
\begin {align*} \int \frac {\cos (x)}{-\sin (x)+\sin ^3(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (-1+x^2\right )} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(-1+x) x} \, dx,x,\sin ^2(x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\sin ^2(x)\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\sin ^2(x)\right )\\ &=\log (\cos (x))-\log (\sin (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 9, normalized size = 1.00 \[ \log (\cos (x))-\log (\sin (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 19, normalized size = 2.11 \[ \frac {1}{2} \, \log \left (\cos \relax (x)^{2}\right ) - \frac {1}{2} \, \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 18, normalized size = 2.00 \[ \frac {1}{2} \, \log \left (-\sin \relax (x)^{2} + 1\right ) - \log \left ({\left | \sin \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 21, normalized size = 2.33 \[ -\ln \left (\sin \relax (x )\right )+\frac {\ln \left (\sin \relax (x )-1\right )}{2}+\frac {\ln \left (1+\sin \relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 20, normalized size = 2.22 \[ \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\sin \relax (x) - 1\right ) - \log \left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.01, size = 13, normalized size = 1.44 \[ \frac {\ln \left ({\cos \relax (x)}^2\right )}{2}-\ln \left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.32, size = 20, normalized size = 2.22 \[ \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{2} - \log {\left (\sin {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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