Optimal. Leaf size=12 \[ \frac {\tan ^2(x)}{2}+\log (\tan (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.42, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4335, 266, 44} \[ \frac {\sec ^2(x)}{2}+\log (\sin (x))-\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 4335
Rubi steps
\begin {align*} \int \frac {\sin (x)}{\cos ^3(x)-\cos ^5(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^3 \left (1-x^2\right )} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1-x) x^2} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{1-x}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=-\log (\cos (x))+\log (\sin (x))+\frac {\sec ^2(x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.42 \[ \frac {\sec ^2(x)}{2}+\log (\sin (x))-\log (\cos (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 33, normalized size = 2.75 \[ -\frac {\cos \relax (x)^{2} \log \left (\cos \relax (x)^{2}\right ) - \cos \relax (x)^{2} \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right ) - 1}{2 \, \cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 24, normalized size = 2.00 \[ \frac {1}{2 \, \cos \relax (x)^{2}} + \frac {1}{2} \, \log \left (-\cos \relax (x)^{2} + 1\right ) - \log \left ({\left | \cos \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 27, normalized size = 2.25 \[ \frac {1}{2 \cos \relax (x )^{2}}-\ln \left (\cos \relax (x )\right )+\frac {\ln \left (-1+\cos \relax (x )\right )}{2}+\frac {\ln \left (1+\cos \relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 26, normalized size = 2.17 \[ \frac {1}{2 \, \cos \relax (x)^{2}} + \frac {1}{2} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cos \relax (x) - 1\right ) - \log \left (\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 19, normalized size = 1.58 \[ \frac {\ln \left ({\sin \relax (x)}^2\right )}{2}-\ln \left (\cos \relax (x)\right )+\frac {1}{2\,{\cos \relax (x)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.49, size = 29, normalized size = 2.42 \[ \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{2} - \log {\left (\cos {\relax (x )} \right )} + \frac {1}{2 \cos ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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