Optimal. Leaf size=139 \[ \frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \cot (x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\cot (x)}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \cot (x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\cot (x)}}\right )+\frac {1}{\sqrt {1+\cot (x)}} \]
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Rubi [A]
time = 0.15, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3623, 3617,
3616, 209, 213} \begin {gather*} \frac {1}{2} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \text {ArcTan}\left (\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\cot (x)+1}}\right )+\frac {1}{\sqrt {\cot (x)+1}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\cot (x)+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 213
Rule 3616
Rule 3617
Rule 3623
Rubi steps
\begin {align*} \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx &=\frac {1}{\sqrt {1+\cot (x)}}+\frac {1}{2} \int \frac {-1+\cot (x)}{\sqrt {1+\cot (x)}} \, dx\\ &=\frac {1}{\sqrt {1+\cot (x)}}+\frac {\int \frac {-\sqrt {2}-\left (2-\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{4 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{4 \sqrt {2}}\\ &=\frac {1}{\sqrt {1+\cot (x)}}-\frac {1}{2} \left (-4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {2} \left (2-\sqrt {2}\right )-4 \left (2-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {\sqrt {2}-2 \left (2-\sqrt {2}\right )-\left (2-\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}}\right )+\frac {1}{2} \left (4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2 \sqrt {2} \left (2+\sqrt {2}\right )-4 \left (2+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {-\sqrt {2}-2 \left (2+\sqrt {2}\right )-\left (2+\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}}\right )\\ &=\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \cot (x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\cot (x)}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \cot (x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\cot (x)}}\right )+\frac {1}{\sqrt {1+\cot (x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 65, normalized size = 0.47 \begin {gather*} \frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )+\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )+\frac {1}{\sqrt {1+\cot (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 173, normalized size = 1.24
method | result | size |
derivativedivides | \(-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}+\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {1}{\sqrt {1+\cot \left (x \right )}}\) | \(173\) |
default | \(-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}+\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {1}{\sqrt {1+\cot \left (x \right )}}\) | \(173\) |
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 \frac {\cot ^{2}{\left (x \right )}}{\left (\cot {\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 [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 = 0.50, size = 208, normalized size = 1.50 \begin {gather*} \frac {1}{\sqrt {\mathrm {cot}\left (x\right )+1}}-\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}-\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{16\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{16\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}\right )\,\left (2\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}-2\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}\right )+\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}-\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{16\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{16\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}\right )\,\left (2\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}+2\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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