Optimal. Leaf size=9 \[ a \log (\sin (x))+b x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3085, 3475} \[ a \log (\sin (x))+b x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3085
Rule 3475
Rubi steps
\begin {align*} \int \csc (x) (a \cos (x)+b \sin (x)) \, dx &=\int (b+a \cot (x)) \, dx\\ &=b x+a \int \cot (x) \, dx\\ &=b x+a \log (\sin (x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ a \log (\sin (x))+b x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 11, normalized size = 1.22 \[ b x + a \log \left (\frac {1}{2} \, \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.21, size = 24, normalized size = 2.67 \[ b x - a \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + a \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.45, size = 10, normalized size = 1.11 \[ b x +a \ln \left (\sin \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 9, normalized size = 1.00 \[ b x + a \log \left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.54, size = 54, normalized size = 6.00 \[ a\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-a\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )-a\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )-b\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )\,1{}\mathrm {i}+b\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.14, size = 8, normalized size = 0.89 \[ a \log {\left (\sin {\relax (x )} \right )} + b x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________