Optimal. Leaf size=221 \[ -\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3528, 12, 3485, 708, 1094, 634, 618, 204, 628} \[ -\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 204
Rule 618
Rule 628
Rule 634
Rule 708
Rule 1094
Rule 3485
Rule 3528
Rubi steps
\begin {align*} \int \cot (x) (1+\cot (x))^{3/2} \, dx &=-\frac {2}{3} (1+\cot (x))^{3/2}-\int (1-\cot (x)) \sqrt {1+\cot (x)} \, dx\\ &=-2 \sqrt {1+\cot (x)}-\frac {2}{3} (1+\cot (x))^{3/2}-\int \frac {2}{\sqrt {1+\cot (x)}} \, dx\\ &=-2 \sqrt {1+\cot (x)}-\frac {2}{3} (1+\cot (x))^{3/2}-2 \int \frac {1}{\sqrt {1+\cot (x)}} \, dx\\ &=-2 \sqrt {1+\cot (x)}-\frac {2}{3} (1+\cot (x))^{3/2}+2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-2 \sqrt {1+\cot (x)}-\frac {2}{3} (1+\cot (x))^{3/2}+4 \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 {2}{3} (1+\cot (x))^{3/2}+\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 )}{\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 )}{\sqrt {1+\sqrt {2}}}\\ &=-2 \sqrt {1+\cot (x)}-\frac {2}{3} (1+\cot (x))^{3/2}+\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 )}{\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 )}{\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 )}{2 \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 )}{2 \sqrt {1+\sqrt {2}}}\\ &=-2 \sqrt {1+\cot (x)}-\frac {2}{3} (1+\cot (x))^{3/2}-\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {1+\sqrt {2}}}-\sqrt {2} \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} \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 )\\ &=-\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 )}{\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 )}{\sqrt {-1+\sqrt {2}}}-2 \sqrt {1+\cot (x)}-\frac {2}{3} (1+\cot (x))^{3/2}-\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{2 \sqrt {1+\sqrt {2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.26, size = 98, normalized size = 0.44 \[ \frac {\sin (x) \left (-\frac {2}{3} (\cot (x)+1)^{3/2} (\cot (x)+4) (\sin (x)+\cos (x))+(1+i) \sin (x) (\cot (x)+1)^2 \left (\sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1+i}}\right )-i \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1-i}}\right )\right )\right )}{(\sin (x)+\cos (x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\cot \relax (x) + 1\right )}^{\frac {3}{2}} \cot \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.12, size = 452, normalized size = 2.05 \[ -\frac {2 \left (1+\cot \relax (x )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\cot \relax (x )}+\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{4}-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{2}+\frac {\sqrt {2}\, \left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{4}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{2}+\frac {\sqrt {2}\, \left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\cot \relax (x) + 1\right )}^{\frac {3}{2}} \cot \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.67, size = 254, normalized size = 1.15 \[ -\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\relax (x)+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}-\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\relax (x)+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}+\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\relax (x)+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}+\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\relax (x)+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}-\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right )-2\,\sqrt {\mathrm {cot}\relax (x)+1}-\frac {2\,{\left (\mathrm {cot}\relax (x)+1\right )}^{3/2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\cot {\relax (x )} + 1\right )^{\frac {3}{2}} \cot {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________