Optimal. Leaf size=39 \[ \frac {\cot (x) \log (\cos (x))}{a \sqrt {a \cot ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cot ^2(x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554,
3556} \begin {gather*} \frac {\tan (x)}{2 a \sqrt {a \cot ^2(x)}}+\frac {\cot (x) \log (\cos (x))}{a \sqrt {a \cot ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rule 3739
Rubi steps
\begin {align*} \int \frac {1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx &=\frac {\cot (x) \int \tan ^3(x) \, dx}{a \sqrt {a \cot ^2(x)}}\\ &=\frac {\tan (x)}{2 a \sqrt {a \cot ^2(x)}}-\frac {\cot (x) \int \tan (x) \, dx}{a \sqrt {a \cot ^2(x)}}\\ &=\frac {\cot (x) \log (\cos (x))}{a \sqrt {a \cot ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cot ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 30, normalized size = 0.77 \begin {gather*} \frac {2 \cot (x) \log (\cos (x))+\csc (x) \sec (x)}{2 a \sqrt {a \cot ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 37, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\cot \left (x \right ) \left (2 \ln \left (\cot \left (x \right )\right ) \left (\cot ^{2}\left (x \right )\right )-\ln \left (\cot ^{2}\left (x \right )+1\right ) \left (\cot ^{2}\left (x \right )\right )+1\right )}{2 \left (a \left (\cot ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}\) | \(37\) |
default | \(\frac {\cot \left (x \right ) \left (2 \ln \left (\cot \left (x \right )\right ) \left (\cot ^{2}\left (x \right )\right )-\ln \left (\cot ^{2}\left (x \right )+1\right ) \left (\cot ^{2}\left (x \right )\right )+1\right )}{2 \left (a \left (\cot ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}\) | \(37\) |
risch | \(\frac {\left ({\mathrm e}^{2 i x}+1\right ) x}{a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {2 i {\mathrm e}^{2 i x}}{a \left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {i \left ({\mathrm e}^{2 i x}+1\right ) \ln \left ({\mathrm e}^{2 i x}+1\right )}{a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 22, normalized size = 0.56 \begin {gather*} \frac {\tan \left (x\right )^{2}}{2 \, a^{\frac {3}{2}}} - \frac {\log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs.
\(2 (33) = 66\).
time = 5.25, size = 74, normalized size = 1.90 \begin {gather*} \frac {{\left ({\left (\cos \left (2 \, x\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \sin \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \sqrt {-\frac {a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \, {\left (a^{2} \cos \left (2 \, x\right )^{2} + 2 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 46, normalized size = 1.18 \begin {gather*} -\frac {\frac {\mathrm {sgn}\left (\sin \left (x\right )\right )}{\sqrt {a}} - \frac {2 \, \sqrt {a} \log \left ({\left | \cos \left (x\right ) \right |}\right ) + \frac {\sqrt {a}}{\cos \left (x\right )^{2}}}{a \mathrm {sgn}\left (\cos \left (x\right )\right ) \mathrm {sgn}\left (\sin \left (x\right )\right )}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\left (a\,{\mathrm {cot}\left (x\right )}^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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