Optimal. Leaf size=36 \[ -\frac {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-a \sqrt {a \cot ^2(x)} \log (\sin (x)) \tan (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 36, 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 {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-a \tan (x) \sqrt {a \cot ^2(x)} \log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rule 3739
Rubi steps
\begin {align*} \int \left (a \cot ^2(x)\right )^{3/2} \, dx &=\left (a \sqrt {a \cot ^2(x)} \tan (x)\right ) \int \cot ^3(x) \, dx\\ &=-\frac {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-\left (a \sqrt {a \cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=-\frac {1}{2} a \cot (x) \sqrt {a \cot ^2(x)}-a \sqrt {a \cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 27, normalized size = 0.75 \begin {gather*} -\frac {1}{2} a \sqrt {a \cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \tan (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 29, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {\left (a \left (\cot ^{2}\left (x \right )\right )\right )^{\frac {3}{2}} \left (-\left (\cot ^{2}\left (x \right )\right )+\ln \left (\cot ^{2}\left (x \right )+1\right )\right )}{2 \cot \left (x \right )^{3}}\) | \(29\) |
default | \(\frac {\left (a \left (\cot ^{2}\left (x \right )\right )\right )^{\frac {3}{2}} \left (-\left (\cot ^{2}\left (x \right )\right )+\ln \left (\cot ^{2}\left (x \right )+1\right )\right )}{2 \cot \left (x \right )^{3}}\) | \(29\) |
risch | \(\frac {a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, x}{{\mathrm e}^{2 i x}+1}-\frac {2 i a \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, {\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right )}+\frac {i a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 i x}-1\right )}{{\mathrm e}^{2 i x}+1}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 30, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, a^{\frac {3}{2}} \log \left (\tan \left (x\right )^{2} + 1\right ) - a^{\frac {3}{2}} \log \left (\tan \left (x\right )\right ) - \frac {a^{\frac {3}{2}}}{2 \, \tan \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.27, size = 52, normalized size = 1.44 \begin {gather*} \frac {{\left ({\left (a \cos \left (2 \, x\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) - 2 \, a\right )} \sqrt {-\frac {a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \, \sin \left (2 \, x\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 \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 = 0.42, size = 31, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, a^{\frac {3}{2}} {\left (\frac {1}{\cos \left (x\right )^{2} - 1} - \log \left (-\cos \left (x\right )^{2} + 1\right )\right )} \mathrm {sgn}\left (\cos \left (x\right )\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) \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 {\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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