Optimal. Leaf size=12 \[ -\frac {\sin (x)}{1-\cos (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2727}
\begin {gather*} -\frac {\sin (x)}{1-\cos (x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2727
Rubi steps
\begin {align*} \int \frac {1}{1-\cos (x)} \, dx &=-\frac {\sin (x)}{1-\cos (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 8, normalized size = 0.67 \begin {gather*} -\cot \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 9, normalized size = 0.75
method | result | size |
default | \(-\frac {1}{\tan \left (\frac {x}{2}\right )}\) | \(9\) |
norman | \(-\frac {1}{\tan \left (\frac {x}{2}\right )}\) | \(9\) |
risch | \(-\frac {2 i}{{\mathrm e}^{i x}-1}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 2.00, size = 10, normalized size = 0.83 \begin {gather*} -\frac {\cos \left (x\right ) + 1}{\sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.02, size = 10, normalized size = 0.83 \begin {gather*} -\frac {\cos \left (x\right ) + 1}{\sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.16, size = 7, normalized size = 0.58 \begin {gather*} - \frac {1}{\tan {\left (\frac {x}{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.89, size = 8, normalized size = 0.67 \begin {gather*} -\frac {1}{\tan \left (\frac {1}{2} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.00, size = 6, normalized size = 0.50 \begin {gather*} -\mathrm {cot}\left (\frac {x}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________