Optimal. Leaf size=22 \[ -\frac {1}{2} \sinh ^{-1}(\cot (x))-\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3738, 4207,
201, 221} \begin {gather*} -\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{2} \sinh ^{-1}(\cot (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 3738
Rule 4207
Rubi steps
\begin {align*} \int \left (1+\cot ^2(x)\right )^{3/2} \, dx &=\int \csc ^2(x)^{3/2} \, dx\\ &=-\text {Subst}\left (\int \sqrt {1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{2} \sinh ^{-1}(\cot (x))-\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
time = 0.11, size = 51, normalized size = 2.32 \begin {gather*} \frac {1}{8} \sqrt {\csc ^2(x)} \left (-\csc ^2\left (\frac {x}{2}\right )-4 \log \left (\cos \left (\frac {x}{2}\right )\right )+4 \log \left (\sin \left (\frac {x}{2}\right )\right )+\sec ^2\left (\frac {x}{2}\right )\right ) \sin (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 19, normalized size = 0.86
method | result | size |
derivativedivides | \(-\frac {\cot \left (x \right ) \sqrt {1+\cot ^{2}\left (x \right )}}{2}-\frac {\arcsinh \left (\cot \left (x \right )\right )}{2}\) | \(19\) |
default | \(-\frac {\cot \left (x \right ) \sqrt {1+\cot ^{2}\left (x \right )}}{2}-\frac {\arcsinh \left (\cot \left (x \right )\right )}{2}\) | \(19\) |
risch | \(-\frac {i \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}-\sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )+\sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 300 vs.
\(2 (16) = 32\).
time = 0.55, size = 300, normalized size = 13.64 \begin {gather*} -\frac {4 \, {\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \cos \left (4 \, x\right ) - 4 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \left (x\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \, {\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \cos \left (x\right )}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \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. 91 vs.
\(2 (16) = 32\).
time = 3.02, size = 91, normalized size = 4.14 \begin {gather*} -\frac {2 \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) + 1\right )} + \log \left (\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \sin \left (2 \, x\right ) - \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \sin \left (2 \, x\right )}{4 \, \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 (\cot ^{2}{\left (x \right )} + 1\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 = 32, normalized size = 1.45 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - \log \left (\cos \left (x\right ) + 1\right ) + \log \left (-\cos \left (x\right ) + 1\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 [B]
time = 0.37, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\mathrm {asinh}\left (\mathrm {cot}\left (x\right )\right )}{2}-\frac {\mathrm {cot}\left (x\right )\,\sqrt {{\mathrm {cot}\left (x\right )}^2+1}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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