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 = 0.204585, antiderivative size = 143, normalized size of antiderivative = 1., 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 3542
Rule 3529
Rule 12
Rule 3536
Rule 3535
Rule 203
Rule 207
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*}
Mathematica [C] time = 0.384202, 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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Maple [B] time = 0.036, size = 265, normalized size = 1.9 \begin{align*}{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{16}\ln \left ( 1+\cot \left ( x \right ) +\sqrt{2}+\sqrt{1+\cot \left ( x \right ) }\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{\sqrt{2}}{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{1}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{16}\ln \left ( 1+\cot \left ( x \right ) +\sqrt{2}-\sqrt{1+\cot \left ( x \right ) }\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{\sqrt{2}}{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{1}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{1}{3} \left ( 1+\cot \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{\sqrt{1+\cot \left ( x \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )^{2}}{{\left (\cot \left (x\right ) + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{2}{\left (x \right )}}{\left (\cot{\left (x \right )} + 1\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )^{2}}{{\left (\cot \left (x\right ) + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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