Optimal. Leaf size=143 \[ -\frac {1}{\sqrt {\cot (x)+1}}+\frac {1}{3 (\cot (x)+1)^{3/2}}+\frac {1}{4} \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\cot (x)+1}}\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3542, 3529, 12, 3536, 3535, 203, 207} \[ -\frac {1}{\sqrt {\cot (x)+1}}+\frac {1}{3 (\cot (x)+1)^{3/2}}+\frac {1}{4} \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\cot (x)+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 207
Rule 3529
Rule 3535
Rule 3536
Rule 3542
Rubi steps
\begin {align*} \int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx &=\frac {1}{3 (1+\cot (x))^{3/2}}+\frac {1}{2} \int \frac {-1+\cot (x)}{(1+\cot (x))^{3/2}} \, dx\\ &=\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{\sqrt {1+\cot (x)}}+\frac {1}{4} \int \frac {2 \cot (x)}{\sqrt {1+\cot (x)}} \, dx\\ &=\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{\sqrt {1+\cot (x)}}+\frac {1}{2} \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx\\ &=\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{\sqrt {1+\cot (x)}}+\frac {\int \frac {-1-\left (-1-\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{4 \sqrt {2}}-\frac {\int \frac {-1-\left (-1+\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{4 \sqrt {2}}\\ &=\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{\sqrt {1+\cot (x)}}+\frac {1}{4} \left (-4+3 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (-1+\sqrt {2}\right )-4 \left (-1+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1+\sqrt {2}\right )-\left (-1+\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}}\right )-\frac {1}{4} \left (4+3 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (-1-\sqrt {2}\right )-4 \left (-1-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1-\sqrt {2}\right )-\left (-1-\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}}\right )\\ &=\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{\sqrt {1+\cot (x)}}\\ \end {align*}
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Mathematica [C] time = 0.41, size = 75, normalized size = 0.52 \[ \frac {-3 \cot (x)-2}{3 (\cot (x)+1)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1-i}}\right )}{2 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1+i}}\right )}{2 \sqrt {1+i}} \]
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 \[ \int \frac {\cot \relax (x)^{2}}{{\left (\cot \relax (x) + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 265, normalized size = 1.85 \[ -\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{16}+\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{16}+\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {1}{3 \left (1+\cot \relax (x )\right )^{\frac {3}{2}}}-\frac {1}{\sqrt {1+\cot \relax (x )}} \]
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 \[ \int \frac {\cot \relax (x)^{2}}{{\left (\cot \relax (x) + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 242, normalized size = 1.69 \[ \mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}-1}-\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}-1}\right )\,\left (2\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\right )-\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}+1}+\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}+1}\right )\,\left (2\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\right )-\frac {\mathrm {cot}\relax (x)+\frac {2}{3}}{{\left (\mathrm {cot}\relax (x)+1\right )}^{3/2}} \]
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 \[ \int \frac {\cot ^{2}{\relax (x )}}{\left (\cot {\relax (x )} + 1\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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