Optimal. Leaf size=14 \[ \frac{2}{\cos (x)+1}+\log (\cos (x)+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0465222, antiderivative size = 14, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 4392
Rule 2667
Rule 43
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*}
Mathematica [A] time = 0.0139392, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.073, size = 15, normalized size = 1.1 \begin{align*} 2\, \left ( 1+\cos \left ( x \right ) \right ) ^{-1}+\ln \left ( 1+\cos \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51203, size = 38, normalized size = 2.71 \begin{align*} \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.93883, size = 74, normalized size = 5.29 \begin{align*} \frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2}{\cos \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13826, size = 19, normalized size = 1.36 \begin{align*} \frac{2}{\cos \left (x\right ) + 1} + \log \left (\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]