Optimal. Leaf size=8 \[ -\frac {\tanh ^{-1}(\cos (x))}{a} \]
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Rubi [A] time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 0.01, 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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fricas [B] time = 0.63, size = 22, normalized size = 2.75 \[ -\frac {\log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 23, normalized size = 2.88 \[ -\frac {\log \left (\cos \relax (x) + 1\right )}{2 \, a} + \frac {\log \left (-\cos \relax (x) + 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 9, normalized size = 1.12 \[ -\frac {\arctanh \left (\cos \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 21, normalized size = 2.62 \[ -\frac {\log \left (\cos \relax (x) + 1\right )}{2 \, a} + \frac {\log \left (\cos \relax (x) - 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.11, size = 8, normalized size = 1.00 \[ -\frac {\mathrm {atanh}\left (\cos \relax (x)\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.22, size = 19, normalized size = 2.38 \[ \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{2 a} - \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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