Optimal. Leaf size=9 \[ a \log (\sin (x))+b x \]
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Rubi [A] time = 0.0190863, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3085, 3475} \[ a \log (\sin (x))+b x \]
Antiderivative was successfully verified.
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Rule 3085
Rule 3475
Rubi steps
\begin{align*} \int \csc (x) (a \cos (x)+b \sin (x)) \, dx &=\int (b+a \cot (x)) \, dx\\ &=b x+a \int \cot (x) \, dx\\ &=b x+a \log (\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0051956, size = 9, normalized size = 1. \[ a \log (\sin (x))+b x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 10, normalized size = 1.1 \begin{align*} bx+a\ln \left ( \sin \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16693, size = 12, normalized size = 1.33 \begin{align*} b x + a \log \left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.489148, size = 34, normalized size = 3.78 \begin{align*} b x + a \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.55686, size = 8, normalized size = 0.89 \begin{align*} a \log{\left (\sin{\left (x \right )} \right )} + b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0948, size = 32, normalized size = 3.56 \begin{align*} b x - a \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) + a \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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