Optimal. Leaf size=223 \[ -\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {2}{3} (1+\cot (x))^{3/2}+\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
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Rubi [A]
time = 0.20, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3624, 3566,
714, 1141, 1175, 632, 210, 1178, 642} \begin {gather*} -\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {2}{3} (\cot (x)+1)^{3/2}+\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 714
Rule 1141
Rule 1175
Rule 1178
Rule 3566
Rule 3624
Rubi steps
\begin {align*} \int \cot ^2(x) \sqrt {1+\cot (x)} \, dx &=-\frac {2}{3} (1+\cot (x))^{3/2}-\int \sqrt {1+\cot (x)} \, dx\\ &=-\frac {2}{3} (1+\cot (x))^{3/2}+\text {Subst}\left (\int \frac {\sqrt {1+x}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {2}{3} (1+\cot (x))^{3/2}+2 \text {Subst}\left (\int \frac {x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\cot (x)}\right )\\ &=-\frac {2}{3} (1+\cot (x))^{3/2}-\text {Subst}\left (\int \frac {\sqrt {2}-x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\cot (x)}\right )+\text {Subst}\left (\int \frac {\sqrt {2}+x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\cot (x)}\right )\\ &=-\frac {2}{3} (1+\cot (x))^{3/2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{-\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x-x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{-\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x-x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ &=-\frac {2}{3} (1+\cot (x))^{3/2}+\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}\right )-\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}\right )\\ &=\frac {\tan ^{-1}\left (\frac {-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {2 \left (-1+\sqrt {2}\right )}}-\frac {2}{3} (1+\cot (x))^{3/2}+\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 69, normalized size = 0.31 \begin {gather*} -i \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )+i \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )-\frac {2}{3} (1+\cot (x))^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 197, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}{3}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (-\frac {\ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (\frac {\ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}\) | \(197\) |
default | \(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}{3}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (-\frac {\ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (\frac {\ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}\) | \(197\) |
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 \begin {gather*} \text {Failed to integrate} \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 \sqrt {\cot {\left (x \right )} + 1} \cot ^{2}{\left (x \right )}\, 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 [B]
time = 0.63, size = 119, normalized size = 0.53 \begin {gather*} \mathrm {atanh}\left (4\,\sqrt {\mathrm {cot}\left (x\right )+1}\,{\left (\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}+\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}+2\,\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )-\frac {2\,{\left (\mathrm {cot}\left (x\right )+1\right )}^{3/2}}{3}+\mathrm {atanh}\left (4\,\sqrt {\mathrm {cot}\left (x\right )+1}\,{\left (\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}-\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}-2\,\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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