Optimal. Leaf size=23 \[ B \log (1-\cos (x))-\frac {A \sin (x)}{1-\cos (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4401, 2648, 2667, 31} \[ B \log (1-\cos (x))-\frac {A \sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 2648
Rule 2667
Rule 4401
Rubi steps
\begin {align*} \int \frac {A+B \sin (x)}{1-\cos (x)} \, dx &=\int \left (-\frac {A}{-1+\cos (x)}-\frac {B \sin (x)}{-1+\cos (x)}\right ) \, dx\\ &=-\left (A \int \frac {1}{-1+\cos (x)} \, dx\right )-B \int \frac {\sin (x)}{-1+\cos (x)} \, dx\\ &=-\frac {A \sin (x)}{1-\cos (x)}+B \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\cos (x)\right )\\ &=B \log (1-\cos (x))-\frac {A \sin (x)}{1-\cos (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 20, normalized size = 0.87 \[ 2 B \log \left (\sin \left (\frac {x}{2}\right )\right )-A \cot \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 25, normalized size = 1.09 \[ \frac {B \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \sin \relax (x) - A \cos \relax (x) - A}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.94, size = 39, normalized size = 1.70 \[ -B \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + 2 \, B \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) - \frac {2 \, B \tan \left (\frac {1}{2} \, x\right ) + A}{\tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 31, normalized size = 1.35 \[ -\frac {A}{\tan \left (\frac {x}{2}\right )}+2 B \ln \left (\tan \left (\frac {x}{2}\right )\right )-B \ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 19, normalized size = 0.83 \[ B \log \left (\cos \relax (x) - 1\right ) - \frac {A {\left (\cos \relax (x) + 1\right )}}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.21, size = 30, normalized size = 1.30 \[ 2\,B\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\frac {A}{\mathrm {tan}\left (\frac {x}{2}\right )}-B\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.45, size = 27, normalized size = 1.17 \[ - \frac {A}{\tan {\left (\frac {x}{2} \right )}} - B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} + 2 B \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________