Optimal. Leaf size=176 \[ -2 \tan (x) \sqrt {a \cot ^3(x)}-\frac {\sqrt {a \cot ^3(x)} \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {a \cot ^3(x)} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)} \]
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Rubi [A] time = 0.09, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3658, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\sqrt {a \cot ^3(x)} \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {a \cot ^3(x)} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \tan (x) \sqrt {a \cot ^3(x)} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3473
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \sqrt {a \cot ^3(x)} \, dx &=\frac {\sqrt {a \cot ^3(x)} \int \cot ^{\frac {3}{2}}(x) \, dx}{\cot ^{\frac {3}{2}}(x)}\\ &=-2 \sqrt {a \cot ^3(x)} \tan (x)-\frac {\sqrt {a \cot ^3(x)} \int \frac {1}{\sqrt {\cot (x)}} \, dx}{\cot ^{\frac {3}{2}}(x)}\\ &=-2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )}{\cot ^{\frac {3}{2}}(x)}\\ &=-2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\left (2 \sqrt {a \cot ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}\\ &=-2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}\\ &=-2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}\\ &=-\frac {\sqrt {a \cot ^3(x)} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {a \cot ^3(x)} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {a \cot ^3(x)} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \sqrt {a \cot ^3(x)} \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 122, normalized size = 0.69 \[ -\frac {\sqrt {a \cot ^3(x)} \left (8 \sqrt {\cot (x)}+\sqrt {2} \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )-\sqrt {2} \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )+2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (x)}\right )-2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (x)}+1\right )\right )}{4 \cot ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \cot \relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 165, normalized size = 0.94 \[ \frac {\sqrt {a \left (\cot ^{3}\relax (x )\right )}\, \left (\left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {a \cot \relax (x )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \relax (x )}\, \sqrt {2}+\sqrt {a^{2}}}{\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \relax (x )}\, \sqrt {2}-a \cot \relax (x )-\sqrt {a^{2}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \relax (x )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \relax (x )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {a \cot \relax (x )}\right )}{4 \cot \relax (x ) \sqrt {a \cot \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 94, normalized size = 0.53 \[ -\frac {1}{4} \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \relax (x)}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \relax (x)}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right )\right )} \sqrt {a} - \frac {2 \, \sqrt {a}}{\sqrt {\tan \relax (x)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a\,{\mathrm {cot}\relax (x)}^3} \,d x \]
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 \[ \int \sqrt {a \cot ^{3}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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