Optimal. Leaf size=14 \[ \frac {\log (\sin (a+b x))}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4287, 3475} \[ \frac {\log (\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 4287
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \csc (2 a+2 b x) \, dx &=\frac {1}{2} \int \cot (a+b x) \, dx\\ &=\frac {\log (\sin (a+b x))}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 22, normalized size = 1.57 \[ \frac {\log (\tan (a+b x))+\log (\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 14, normalized size = 1.00 \[ \frac {\log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.22, size = 56, normalized size = 4.00 \[ \frac {\log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.64, size = 13, normalized size = 0.93 \[ \frac {\ln \left (\sin \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.33, size = 82, normalized size = 5.86 \[ \frac {\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 14, normalized size = 1.00 \[ \frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________