Optimal. Leaf size=14 \[ \frac {2}{\cos (x)+1}+\log (\cos (x)+1) \]
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Rubi [A] time = 0.05, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4392, 2667, 43} \[ \frac {2}{\cos (x)+1}+\log (\cos (x)+1) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rule 4392
Rubi steps
\begin {align*} \int \frac {1}{(\cot (x)+\csc (x))^3} \, dx &=\int \frac {\sin ^3(x)}{(1+\cos (x))^3} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {2}{(1+x)^2}\right ) \, dx,x,\cos (x)\right )\\ &=\frac {2}{1+\cos (x)}+\log (1+\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.29 \[ \sec ^2\left (\frac {x}{2}\right )+2 \log \left (\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 21, normalized size = 1.50 \[ \frac {{\left (\cos \relax (x) + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 14, normalized size = 1.00 \[ \frac {2}{\cos \relax (x) + 1} + \log \left (\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 15, normalized size = 1.07 \[ \frac {2}{1+\cos \relax (x )}+\ln \left (1+\cos \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 28, normalized size = 2.00 \[ \frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.35, size = 18, normalized size = 1.29 \[ {\mathrm {tan}\left (\frac {x}{2}\right )}^2-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\cot {\relax (x )} + \csc {\relax (x )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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