Optimal. Leaf size=14 \[ -\frac {\log (\cos (a+b x))}{2 b} \]
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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 = {4288, 3475} \[ -\frac {\log (\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4288
Rubi steps
\begin {align*} \int \csc (2 a+2 b x) \sin ^2(a+b x) \, dx &=\frac {1}{2} \int \tan (a+b x) \, dx\\ &=-\frac {\log (\cos (a+b x))}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ -\frac {\log (\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 14, normalized size = 1.00 \[ -\frac {\log \left (-\cos \left (b x + a\right )\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.77, size = 95, normalized size = 6.79 \[ \frac {\log \left (\tan \left (b x + 4 \, a\right )^{2} + 1\right ) - 2 \, \log \left ({\left | 6 \, \tan \left (b x + 4 \, a\right ) \tan \left (\frac {1}{2} \, a\right )^{5} - \tan \left (\frac {1}{2} \, a\right )^{6} - 20 \, \tan \left (b x + 4 \, a\right ) \tan \left (\frac {1}{2} \, a\right )^{3} + 15 \, \tan \left (\frac {1}{2} \, a\right )^{4} + 6 \, \tan \left (b x + 4 \, a\right ) \tan \left (\frac {1}{2} \, a\right ) - 15 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 1 \right |}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 13, normalized size = 0.93 \[ -\frac {\ln \left (\cos \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 55, normalized size = 3.93 \[ -\frac {\log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 12, normalized size = 0.86 \[ -\frac {\ln \left (\cos \left (a+b\,x\right )\right )}{2\,b} \]
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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