Optimal. Leaf size=121 \[ \frac {1}{2} \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\cot (x)+1}}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3536, 3535, 203, 207} \[ \frac {1}{2} \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\cot (x)+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 3535
Rule 3536
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx &=\frac {\int \frac {-1-\left (-1-\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{2 \sqrt {2}}-\frac {\int \frac {-1-\left (-1+\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{2 \sqrt {2}}\\ &=\frac {1}{2} \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}{2} \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}{2} \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}{2} \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 )\\ \end {align*}
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Mathematica [C] time = 0.08, size = 51, normalized size = 0.42 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1+i}}\right )}{\sqrt {1+i}} \]
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 \[ \int \frac {\cot \relax (x)}{\sqrt {\cot \relax (x) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 249, normalized size = 2.06 \[ -\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}+\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}+\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}} \]
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 \[ \int \frac {\cot \relax (x)}{\sqrt {\cot \relax (x) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 230, normalized size = 1.90 \[ \mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{128\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}-8}-\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{128\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}-8}\right )\,\left (2\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}+2\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}\right )-\mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{128\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}+8}+\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}\,\sqrt {\mathrm {cot}\relax (x)+1}}{128\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}+8}\right )\,\left (2\,\sqrt {\frac {1}{16}-\frac {\sqrt {2}}{16}}-2\,\sqrt {\frac {\sqrt {2}}{16}+\frac {1}{16}}\right ) \]
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 \[ \int \frac {\cot {\relax (x )}}{\sqrt {\cot {\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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