Optimal. Leaf size=12 \[ \frac {2 \log (\sin (a+b x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 12, 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 = {4288, 3475} \[ \frac {2 \log (\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4288
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cot (a+b x) \, dx\\ &=\frac {2 \log (\sin (a+b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.67 \[ \frac {2 (\log (\tan (a+b x))+\log (\cos (a+b x)))}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 14, normalized size = 1.17 \[ \frac {2 \, \log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.99, size = 55, normalized size = 4.58 \[ \frac {\log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 13, normalized size = 1.08 \[ \frac {2 \ln \left (\sin \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 81, normalized size = 6.75 \[ \frac {\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 13, normalized size = 1.08 \[ \frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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