Optimal. Leaf size=61 \[ -\frac {1}{2} \cot (x) \sqrt {\cot ^2(x)-1}+\frac {5}{2} \tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {\cot ^2(x)-1}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3661, 416, 523, 217, 206, 377} \[ -\frac {1}{2} \cot (x) \sqrt {\cot ^2(x)-1}+\frac {5}{2} \tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {\cot ^2(x)-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 377
Rule 416
Rule 523
Rule 3661
Rubi steps
\begin {align*} \int \left (-1+\cot ^2(x)\right )^{3/2} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^{3/2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{2} \cot (x) \sqrt {-1+\cot ^2(x)}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {3-5 x^2}{\sqrt {-1+x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{2} \cot (x) \sqrt {-1+\cot ^2(x)}+\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,\cot (x)\right )-4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{2} \cot (x) \sqrt {-1+\cot ^2(x)}+\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-4 \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )\\ &=\frac {5}{2} \tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-\frac {1}{2} \cot (x) \sqrt {-1+\cot ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 121, normalized size = 1.98 \[ \frac {1}{2} \left (\cot ^2(x)-1\right )^{3/2} \sec ^2(2 x) \left (-\frac {1}{4} \sin (4 x)-4 \sqrt {2} \sin ^3(x) \sqrt {\cos (2 x)} \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )+\sin ^3(x) \sqrt {-\cos (2 x)} \tan ^{-1}\left (\frac {\cos (x)}{\sqrt {-\cos (2 x)}}\right )+4 \sin ^3(x) \sqrt {\cos (2 x)} \tanh ^{-1}\left (\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 170, normalized size = 2.79 \[ \frac {4 \, \sqrt {2} \log \left (2 \, \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - 1\right ) \sin \left (2 \, x\right ) - 2 \, \sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) + 1\right )} + 5 \, \log \left (\frac {\sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right ) - 5 \, \log \left (\frac {\sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right )}{4 \, \sin \left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 179, normalized size = 2.93 \[ \frac {1}{4} \, {\left (4 \, \sqrt {2} \log \left ({\left (\sqrt {2} \cos \relax (x) - \sqrt {2 \, \cos \relax (x)^{2} - 1}\right )}^{2}\right ) - \frac {4 \, \sqrt {2} {\left (3 \, {\left (\sqrt {2} \cos \relax (x) - \sqrt {2 \, \cos \relax (x)^{2} - 1}\right )}^{2} - 1\right )}}{{\left (\sqrt {2} \cos \relax (x) - \sqrt {2 \, \cos \relax (x)^{2} - 1}\right )}^{4} - 6 \, {\left (\sqrt {2} \cos \relax (x) - \sqrt {2 \, \cos \relax (x)^{2} - 1}\right )}^{2} + 1} + 5 \, \log \left (\frac {{\left | 2 \, {\left (\sqrt {2} \cos \relax (x) - \sqrt {2 \, \cos \relax (x)^{2} - 1}\right )}^{2} - 4 \, \sqrt {2} - 6 \right |}}{{\left | 2 \, {\left (\sqrt {2} \cos \relax (x) - \sqrt {2 \, \cos \relax (x)^{2} - 1}\right )}^{2} + 4 \, \sqrt {2} - 6 \right |}}\right )\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 48, normalized size = 0.79 \[ -\frac {\cot \relax (x ) \sqrt {-1+\cot ^{2}\relax (x )}}{2}+\frac {5 \ln \left (\cot \relax (x )+\sqrt {-1+\cot ^{2}\relax (x )}\right )}{2}-2 \arctanh \left (\frac {\cot \relax (x ) \sqrt {2}}{\sqrt {-1+\cot ^{2}\relax (x )}}\right ) \sqrt {2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\cot \relax (x)^{2} - 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left ({\mathrm {cot}\relax (x)}^2-1\right )}^{3/2} \,d x \]
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 \[ \int \left (\cot ^{2}{\relax (x )} - 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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