Optimal. Leaf size=9 \[ \frac {\log (\sin (x)+1)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 9, 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 = {3879, 31} \[ \frac {\log (\sin (x)+1)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3879
Rubi steps
\begin {align*} \int \frac {\cot (x)}{a+a \csc (x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\sin (x)\right )\\ &=\frac {\log (1+\sin (x))}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ \frac {\log (\sin (x)+1)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 9, normalized size = 1.00 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 9, normalized size = 1.00 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.32, size = 19, normalized size = 2.11 \[ -\frac {\ln \left (\csc \relax (x )\right )}{a}+\frac {\ln \left (1+\csc \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 9, normalized size = 1.00 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.29, size = 25, normalized size = 2.78 \[ \frac {2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot {\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________