Optimal. Leaf size=29 \[ \frac {x}{2}+\frac {1}{3} \log (2-\sin (2 x))-\frac {1}{6} \log (\sin (x)+\cos (x)) \]
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Rubi [A] time = 0.13, antiderivative size = 37, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2074, 635, 203, 260, 628} \[ \frac {x}{2}+\frac {1}{3} \log \left (\tan ^2(x)-\tan (x)+1\right )-\frac {1}{6} \log (\tan (x)+1)+\frac {1}{2} \log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 628
Rule 635
Rule 2074
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{\cos ^3(x)+\sin ^3(x)} \, dx &=\operatorname {Subst}\left (\int \frac {x^3}{1+x^2+x^3+x^5} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{6 (1+x)}+\frac {1-x}{2 \left (1+x^2\right )}+\frac {-1+2 x}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tan (x)\right )\\ &=-\frac {1}{6} \log (1+\tan (x))+\frac {1}{3} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\tan (x)\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x}{1+x^2} \, dx,x,\tan (x)\right )\\ &=-\frac {1}{6} \log (1+\tan (x))+\frac {1}{3} \log \left (1-\tan (x)+\tan ^2(x)\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {x}{2}+\frac {1}{2} \log (\cos (x))-\frac {1}{6} \log (1+\tan (x))+\frac {1}{3} \log \left (1-\tan (x)+\tan ^2(x)\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 29, normalized size = 1.00 \[ \frac {x}{2}+\frac {1}{3} \log (2-\sin (2 x))-\frac {1}{6} \log (\sin (x)+\cos (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 2.01, size = 26, normalized size = 0.90 \[ \frac {1}{2} \, x - \frac {1}{12} \, \log \left (2 \, \cos \relax (x) \sin \relax (x) + 1\right ) + \frac {1}{3} \, \log \left (-\cos \relax (x) \sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 34, normalized size = 1.17 \[ \frac {1}{2} \, x + \frac {1}{3} \, \log \left (\tan \relax (x)^{2} - \tan \relax (x) + 1\right ) - \frac {1}{4} \, \log \left (\tan \relax (x)^{2} + 1\right ) - \frac {1}{6} \, \log \left ({\left | \tan \relax (x) + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 34, normalized size = 1.17 \[ \frac {\ln \left (1-\tan \relax (x )+\tan ^{2}\relax (x )\right )}{3}-\frac {\ln \left (1+\tan ^{2}\relax (x )\right )}{4}-\frac {\ln \left (1+\tan \relax (x )\right )}{6}+\frac {x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 103, normalized size = 3.55 \[ \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) + \frac {1}{3} \, \log \left (-\frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {2 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {2 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {\sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + 1\right ) - \frac {1}{6} \, \log \left (-\frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - 1\right ) - \frac {1}{2} \, \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.31, size = 45, normalized size = 1.55 \[ \frac {x}{2}-\frac {\ln \left (\frac {1}{{\cos \left (\frac {x}{2}\right )}^2}\right )}{2}+\frac {\ln \left (\frac {\sin \left (2\,x\right )-2}{{\cos \left (\frac {x}{2}\right )}^4}\right )}{3}-\frac {\ln \left (\frac {\sin \left (x+\frac {\pi }{4}\right )}{{\cos \left (\frac {x}{2}\right )}^2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 32, normalized size = 1.10 \[ \frac {x}{2} - \frac {\log {\left (\sin {\relax (x )} + \cos {\relax (x )} \right )}}{6} + \frac {\log {\left (\sin ^{2}{\relax (x )} - \sin {\relax (x )} \cos {\relax (x )} + \cos ^{2}{\relax (x )} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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