Optimal. Leaf size=14 \[ -\frac{\log (\cos (a+b x))}{2 b} \]
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Rubi [A] time = 0.026394, antiderivative size = 14, normalized size of antiderivative = 1., 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 4288
Rule 3475
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*}
Mathematica [A] time = 0.011125, size = 14, normalized size = 1. \[ -\frac{\log (\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 13, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( \cos \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12982, size = 74, normalized size = 5.29 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.497735, size = 36, normalized size = 2.57 \begin{align*} -\frac{\log \left (-\cos \left (b x + a\right )\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.50726, size = 128, normalized size = 9.14 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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