Optimal. Leaf size=16 \[ -\frac {\sin ^3(x)}{3 (1-\cos (x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2671} \[ -\frac {\sin ^3(x)}{3 (1-\cos (x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2671
Rubi steps
\begin {align*} \int \frac {\sin ^2(x)}{(1-\cos (x))^3} \, dx &=-\frac {\sin ^3(x)}{3 (1-\cos (x))^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 12, normalized size = 0.75 \[ -\frac {1}{3} \cot ^3\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 22, normalized size = 1.38 \[ \frac {\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1}{3 \, {\left (\cos \relax (x) - 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.98, size = 8, normalized size = 0.50 \[ -\frac {1}{3 \, \tan \left (\frac {1}{2} \, x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 9, normalized size = 0.56 \[ -\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 12, normalized size = 0.75 \[ -\frac {{\left (\cos \relax (x) + 1\right )}^{3}}{3 \, \sin \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.35, size = 8, normalized size = 0.50 \[ -\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.33, size = 10, normalized size = 0.62 \[ - \frac {1}{3 \tan ^{3}{\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________