Optimal. Leaf size=12 \[ -\frac {1}{2 (1-\cos (x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2667, 32} \[ -\frac {1}{2 (1-\cos (x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx &=\operatorname {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,-\cos (x)\right )\\ &=-\frac {1}{2 (1-\cos (x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ -\frac {1}{8} \csc ^4\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.94, size = 14, normalized size = 1.17 \[ -\frac {1}{2 \, {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.48, size = 8, normalized size = 0.67 \[ -\frac {1}{2 \, {\left (\cos \relax (x) - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 11, normalized size = 0.92 \[ -\frac {1}{2 \left (1-\cos \relax (x )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 8, normalized size = 0.67 \[ -\frac {1}{2 \, {\left (\cos \relax (x) - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 8, normalized size = 0.67 \[ -\frac {1}{2\,{\left (\cos \relax (x)-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.55, size = 15, normalized size = 1.25 \[ - \frac {1}{2 \cos ^{2}{\relax (x )} - 4 \cos {\relax (x )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________