Optimal. Leaf size=214 \[ -2 \sqrt {\cot (x)+1}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{2} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3543, 3485, 708, 1094, 634, 618, 204, 628} \[ -2 \sqrt {\cot (x)+1}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{2} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 708
Rule 1094
Rule 3485
Rule 3543
Rubi steps
\begin {align*} \int \frac {\cot ^2(x)}{\sqrt {1+\cot (x)}} \, dx &=-2 \sqrt {1+\cot (x)}-\int \frac {1}{\sqrt {1+\cot (x)}} \, dx\\ &=-2 \sqrt {1+\cot (x)}+\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-2 \sqrt {1+\cot (x)}+2 \operatorname {Subst}\left (\int \frac {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\cot (x)}\right )\\ &=-2 \sqrt {1+\cot (x)}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {1+\sqrt {2}}}\\ &=-2 \sqrt {1+\cot (x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {2}}-\frac {\operatorname {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 )}{4 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {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 )}{4 \sqrt {1+\sqrt {2}}}\\ &=-2 \sqrt {1+\cot (x)}-\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\operatorname {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 )}{\sqrt {2}}-\frac {\operatorname {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 )}{\sqrt {2}}\\ &=-\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 )}{2 \sqrt {-1+\sqrt {2}}}+\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 )}{2 \sqrt {-1+\sqrt {2}}}-2 \sqrt {1+\cot (x)}-\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 67, normalized size = 0.31 \[ -2 \sqrt {\cot (x)+1}+\frac {1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1-i}}\right )+\frac {1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1+i}}\right ) \]
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 \[ \int \frac {\cot \relax (x)^{2}}{\sqrt {\cot \relax (x) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 442, normalized size = 2.07 \[ -2 \sqrt {1+\cot \relax (x )}-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{4}+\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}+\frac {\sqrt {2}\, \left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{4}-\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}+\frac {\sqrt {2}\, \left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}} \]
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 \[ \int \frac {\cot \relax (x)^{2}}{\sqrt {\cot \relax (x) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 238, normalized size = 1.11 \[ \mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-8}-\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-8}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+2\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\right )-\mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+8}+\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+8}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-2\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\right )-2\,\sqrt {\mathrm {cot}\relax (x)+1} \]
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 \[ \int \frac {\cot ^{2}{\relax (x )}}{\sqrt {\cot {\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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