Optimal. Leaf size=17 \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {a \cot ^2(x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3658, 3475} \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {a \cot ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \cot ^2(x)}} \, dx &=\frac {\cot (x) \int \tan (x) \, dx}{\sqrt {a \cot ^2(x)}}\\ &=-\frac {\cot (x) \log (\cos (x))}{\sqrt {a \cot ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {a \cot ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 45, normalized size = 2.65 \[ -\frac {\sqrt {-\frac {a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}} \log \left (\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \sin \left (2 \, x\right )}{2 \, {\left (a \cos \left (2 \, x\right ) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 19, normalized size = 1.12 \[ -\frac {\log \left ({\left | \cos \relax (x) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \relax (x)\right ) \mathrm {sgn}\left (\sin \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 28, normalized size = 1.65 \[ -\frac {\cot \relax (x ) \left (2 \ln \left (\cot \relax (x )\right )-\ln \left (\cot ^{2}\relax (x )+1\right )\right )}{2 \sqrt {a \left (\cot ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 12, normalized size = 0.71 \[ \frac {\log \left (\tan \relax (x)^{2} + 1\right )}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 25, normalized size = 1.47 \[ -\frac {\mathrm {atan}\left (\frac {\sqrt {-a}\,\mathrm {cot}\relax (x)}{\sqrt {a}\,\sqrt {{\mathrm {cot}\relax (x)}^2}}\right )}{\sqrt {-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cot ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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