Optimal. Leaf size=14 \[ \frac{\log (\sin (a+b x))}{2 b} \]
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Rubi [A] time = 0.02515, 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 = {4287, 3475} \[ \frac{\log (\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 4287
Rule 3475
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 = 0.0146544, size = 22, normalized size = 1.57 \[ \frac{\log (\tan (a+b x))+\log (\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 13, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( \sin \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 = 1.1804, size = 111, normalized size = 7.93 \begin{align*} \frac{\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (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 = 0.491007, size = 39, normalized size = 2.79 \begin{align*} \frac{\log \left (\frac{1}{2} \, \sin \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 = 1.28143, size = 76, normalized size = 5.43 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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