Integrand size = 15, antiderivative size = 186 \[ \int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx=\frac {1}{2} \sqrt {a-a x^2}+\frac {1}{2} x \sqrt {a-a x^2} \coth ^{-1}(x)-\frac {a \sqrt {1-x^2} \coth ^{-1}(x) \arctan \left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}-\frac {i a \sqrt {1-x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{2 \sqrt {a-a x^2}}+\frac {i a \sqrt {1-x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{2 \sqrt {a-a x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6090, 6102, 6098} \[ \int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx=-\frac {a \sqrt {1-x^2} \arctan \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \coth ^{-1}(x)}{\sqrt {a-a x^2}}-\frac {i a \sqrt {1-x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{2 \sqrt {a-a x^2}}+\frac {i a \sqrt {1-x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{2 \sqrt {a-a x^2}}+\frac {1}{2} \sqrt {a-a x^2}+\frac {1}{2} x \sqrt {a-a x^2} \coth ^{-1}(x) \]
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Rule 6090
Rule 6098
Rule 6102
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \sqrt {a-a x^2}+\frac {1}{2} x \sqrt {a-a x^2} \coth ^{-1}(x)+\frac {1}{2} a \int \frac {\coth ^{-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} \coth ^{-1}(x)+\frac {\left (a \sqrt {1-x^2}\right ) \int \frac {\coth ^{-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} \coth ^{-1}(x)-\frac {a \sqrt {1-x^2} \coth ^{-1}(x) \arctan \left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}-\frac {i a \sqrt {1-x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{2 \sqrt {a-a x^2}}+\frac {i a \sqrt {1-x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{2 \sqrt {a-a x^2}} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.67 \[ \int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx=-\frac {\sqrt {a-a x^2} \left (-2 \coth \left (\frac {1}{2} \coth ^{-1}(x)\right )-\coth ^{-1}(x) \text {csch}^2\left (\frac {1}{2} \coth ^{-1}(x)\right )-4 \coth ^{-1}(x) \log \left (1-e^{-\coth ^{-1}(x)}\right )+4 \coth ^{-1}(x) \log \left (1+e^{-\coth ^{-1}(x)}\right )-4 \operatorname {PolyLog}\left (2,-e^{-\coth ^{-1}(x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-\coth ^{-1}(x)}\right )-\coth ^{-1}(x) \text {sech}^2\left (\frac {1}{2} \coth ^{-1}(x)\right )+2 \tanh \left (\frac {1}{2} \coth ^{-1}(x)\right )\right )}{8 \sqrt {1-\frac {1}{x^2}} x} \]
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Time = 0.48 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {\left (\operatorname {arccoth}\left (x \right ) x +1\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{2}+\frac {\sqrt {-\left (1+x \right ) \left (x -1\right ) a}\, \sqrt {\frac {x -1}{1+x}}\, \operatorname {arccoth}\left (x \right ) \ln \left (1-\frac {1}{\sqrt {\frac {x -1}{1+x}}}\right )}{2 x -2}+\frac {\sqrt {-\left (1+x \right ) \left (x -1\right ) a}\, \sqrt {\frac {x -1}{1+x}}\, \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {x -1}{1+x}}}\right )}{2 x -2}-\frac {\sqrt {-\left (1+x \right ) \left (x -1\right ) a}\, \sqrt {\frac {x -1}{1+x}}\, \operatorname {arccoth}\left (x \right ) \ln \left (\frac {1}{\sqrt {\frac {x -1}{1+x}}}+1\right )}{2 \left (x -1\right )}-\frac {\sqrt {-\left (1+x \right ) \left (x -1\right ) a}\, \sqrt {\frac {x -1}{1+x}}\, \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {x -1}{1+x}}}\right )}{2 \left (x -1\right )}\) | \(199\) |
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\[ \int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx=\int { \sqrt {-a x^{2} + a} \operatorname {arcoth}\left (x\right ) \,d x } \]
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\[ \int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx=\int \sqrt {- a \left (x - 1\right ) \left (x + 1\right )} \operatorname {acoth}{\left (x \right )}\, dx \]
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\[ \int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx=\int { \sqrt {-a x^{2} + a} \operatorname {arcoth}\left (x\right ) \,d x } \]
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\[ \int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx=\int { \sqrt {-a x^{2} + a} \operatorname {arcoth}\left (x\right ) \,d x } \]
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Timed out. \[ \int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx=\int \mathrm {acoth}\left (x\right )\,\sqrt {a-a\,x^2} \,d x \]
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