Optimal. Leaf size=200 \[ \frac {2}{3} a \sqrt {a \cot ^3(x)}+\frac {a \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {a \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \sqrt {a \cot ^3(x)} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {a \sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \]
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Rubi [A]
time = 0.06, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3739, 3554,
3557, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {a \sqrt {a \cot ^3(x)} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {a \sqrt {a \cot ^3(x)} \text {ArcTan}\left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \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 {a \sqrt {a \cot ^3(x)} \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rule 3739
Rubi steps
\begin {align*} \int \left (a \cot ^3(x)\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \int \cot ^{\frac {9}{2}}(x) \, dx}{\cot ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \int \cot ^{\frac {5}{2}}(x) \, dx}{\cot ^{\frac {3}{2}}(x)}\\ &=\frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}+\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \int \sqrt {\cot (x)} \, dx}{\cot ^{\frac {3}{2}}(x)}\\ &=\frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\cot (x)\right )}{\cot ^{\frac {3}{2}}(x)}\\ &=\frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {\left (2 a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}\\ &=\frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}+\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}\\ &=\frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \cot ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \cot ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {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 {\left (a \sqrt {a \cot ^3(x)}\right ) \text {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 {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \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 {a \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 {\left (a \sqrt {a \cot ^3(x)}\right ) \text {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 {\left (a \sqrt {a \cot ^3(x)}\right ) \text {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 {2}{3} a \sqrt {a \cot ^3(x)}+\frac {a \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {a \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \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 {a \sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 39, normalized size = 0.20 \begin {gather*} -\frac {2}{21} a \sqrt {a \cot ^3(x)} \left (-7+3 \cot ^2(x)+7 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 186, normalized size = 0.93
method | result | size |
derivativedivides | \(-\frac {\left (a \left (\cot ^{3}\left (x \right )\right )\right )^{\frac {3}{2}} \left (24 \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \left (a^{2}\right )^{\frac {1}{4}}+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+21 a^{4} \sqrt {2}\, \ln \left (\frac {a \cot \left (x \right )-\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )-56 a^{2} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} \left (a^{2}\right )^{\frac {1}{4}}\right )}{84 \cot \left (x \right )^{3} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} a^{2} \left (a^{2}\right )^{\frac {1}{4}}}\) | \(186\) |
default | \(-\frac {\left (a \left (\cot ^{3}\left (x \right )\right )\right )^{\frac {3}{2}} \left (24 \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \left (a^{2}\right )^{\frac {1}{4}}+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+21 a^{4} \sqrt {2}\, \ln \left (\frac {a \cot \left (x \right )-\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )-56 a^{2} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} \left (a^{2}\right )^{\frac {1}{4}}\right )}{84 \cot \left (x \right )^{3} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} a^{2} \left (a^{2}\right )^{\frac {1}{4}}}\) | \(186\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 113, normalized size = 0.56 \begin {gather*} \frac {1}{4} \, {\left (2 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt {2} \sqrt {a} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + \sqrt {2} \sqrt {a} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \sqrt {2} \sqrt {a} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right )\right )} a + \frac {2 \, a^{\frac {3}{2}}}{3 \, \tan \left (x\right )^{\frac {3}{2}}} - \frac {2 \, a^{\frac {3}{2}}}{7 \, \tan \left (x\right )^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 ^{3}{\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int {\left (a\,{\mathrm {cot}\left (x\right )}^3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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