Optimal. Leaf size=8 \[ -\frac{\tanh ^{-1}(\cos (x))}{a} \]
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Rubi [A] time = 0.0249527, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3175, 3770} \[ -\frac{\tanh ^{-1}(\cos (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sin (x)}{a-a \cos ^2(x)} \, dx &=\frac{\int \csc (x) \, dx}{a}\\ &=-\frac{\tanh ^{-1}(\cos (x))}{a}\\ \end{align*}
Mathematica [B] time = 0.0067421, size = 21, normalized size = 2.62 \[ \frac{\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 9, normalized size = 1.1 \begin{align*} -{\frac{{\it Artanh} \left ( \cos \left ( x \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.952486, size = 28, normalized size = 3.5 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{2 \, a} + \frac{\log \left (\cos \left (x\right ) - 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01169, size = 77, normalized size = 9.62 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.264378, size = 19, normalized size = 2.38 \begin{align*} \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{2 a} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16478, size = 31, normalized size = 3.88 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{2 \, a} + \frac{\log \left (-\cos \left (x\right ) + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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