Integrand size = 14, antiderivative size = 195 \[ \int \sqrt {a+a x^2} \cot ^{-1}(x) \, dx=\frac {1}{2} \sqrt {a+a x^2}+\frac {1}{2} x \sqrt {a+a x^2} \cot ^{-1}(x)-\frac {i a \sqrt {1+x^2} \cot ^{-1}(x) \arctan \left (\frac {\sqrt {1+i x}}{\sqrt {1-i x}}\right )}{\sqrt {a+a x^2}}-\frac {i a \sqrt {1+x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i x}}{\sqrt {1-i x}}\right )}{2 \sqrt {a+a x^2}}+\frac {i a \sqrt {1+x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i x}}{\sqrt {1-i x}}\right )}{2 \sqrt {a+a x^2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4999, 5011, 5007} \[ \int \sqrt {a+a x^2} \cot ^{-1}(x) \, dx=-\frac {i a \sqrt {x^2+1} \arctan \left (\frac {\sqrt {1+i x}}{\sqrt {1-i x}}\right ) \cot ^{-1}(x)}{\sqrt {a x^2+a}}-\frac {i a \sqrt {x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i x+1}}{\sqrt {1-i x}}\right )}{2 \sqrt {a x^2+a}}+\frac {i a \sqrt {x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i x+1}}{\sqrt {1-i x}}\right )}{2 \sqrt {a x^2+a}}+\frac {1}{2} \sqrt {a x^2+a}+\frac {1}{2} x \sqrt {a x^2+a} \cot ^{-1}(x) \]
[In]
[Out]
Rule 4999
Rule 5007
Rule 5011
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \sqrt {a+a x^2}+\frac {1}{2} x \sqrt {a+a x^2} \cot ^{-1}(x)+\frac {1}{2} a \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx \\ & = \frac {1}{2} \sqrt {a+a x^2}+\frac {1}{2} x \sqrt {a+a x^2} \cot ^{-1}(x)+\frac {\left (a \sqrt {1+x^2}\right ) \int \frac {\cot ^{-1}(x)}{\sqrt {1+x^2}} \, dx}{2 \sqrt {a+a x^2}} \\ & = \frac {1}{2} \sqrt {a+a x^2}+\frac {1}{2} x \sqrt {a+a x^2} \cot ^{-1}(x)-\frac {i a \sqrt {1+x^2} \cot ^{-1}(x) \arctan \left (\frac {\sqrt {1+i x}}{\sqrt {1-i x}}\right )}{\sqrt {a+a x^2}}-\frac {i a \sqrt {1+x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i x}}{\sqrt {1-i x}}\right )}{2 \sqrt {a+a x^2}}+\frac {i a \sqrt {1+x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i x}}{\sqrt {1-i x}}\right )}{2 \sqrt {a+a x^2}} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.70 \[ \int \sqrt {a+a x^2} \cot ^{-1}(x) \, dx=-\frac {\left (a \left (1+x^2\right )\right )^{3/2} \left (-2 \cot \left (\frac {1}{2} \cot ^{-1}(x)\right )-\cot ^{-1}(x) \csc ^2\left (\frac {1}{2} \cot ^{-1}(x)\right )+4 \cot ^{-1}(x) \log \left (1-e^{i \cot ^{-1}(x)}\right )-4 \cot ^{-1}(x) \log \left (1+e^{i \cot ^{-1}(x)}\right )+4 i \operatorname {PolyLog}\left (2,-e^{i \cot ^{-1}(x)}\right )-4 i \operatorname {PolyLog}\left (2,e^{i \cot ^{-1}(x)}\right )+\cot ^{-1}(x) \sec ^2\left (\frac {1}{2} \cot ^{-1}(x)\right )-2 \tan \left (\frac {1}{2} \cot ^{-1}(x)\right )\right )}{8 a \left (1+\frac {1}{x^2}\right )^{3/2} x^3} \]
[In]
[Out]
Time = 0.87 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {\sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x \,\operatorname {arccot}\left (x \right )+1\right )}{2}-\frac {i \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (i \operatorname {arccot}\left (x \right ) \ln \left (\frac {i+x}{\sqrt {x^{2}+1}}+1\right )-i \operatorname {arccot}\left (x \right ) \ln \left (1-\frac {i+x}{\sqrt {x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i+x}{\sqrt {x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i+x}{\sqrt {x^{2}+1}}\right )\right )}{2 \sqrt {x^{2}+1}}\) | \(117\) |
[In]
[Out]
\[ \int \sqrt {a+a x^2} \cot ^{-1}(x) \, dx=\int { \sqrt {a x^{2} + a} \operatorname {arccot}\left (x\right ) \,d x } \]
[In]
[Out]
\[ \int \sqrt {a+a x^2} \cot ^{-1}(x) \, dx=\int \sqrt {a \left (x^{2} + 1\right )} \operatorname {acot}{\left (x \right )}\, dx \]
[In]
[Out]
\[ \int \sqrt {a+a x^2} \cot ^{-1}(x) \, dx=\int { \sqrt {a x^{2} + a} \operatorname {arccot}\left (x\right ) \,d x } \]
[In]
[Out]
\[ \int \sqrt {a+a x^2} \cot ^{-1}(x) \, dx=\int { \sqrt {a x^{2} + a} \operatorname {arccot}\left (x\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {a+a x^2} \cot ^{-1}(x) \, dx=\int \mathrm {acot}\left (x\right )\,\sqrt {a\,x^2+a} \,d x \]
[In]
[Out]