Integrand size = 22, antiderivative size = 167 \[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {-1+a x}}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 x^2}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}} \]
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Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5932, 5946, 4265, 2317, 2438, 30} \[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {a^2 \sqrt {a x-1} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {a x-1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {i a^2 \sqrt {a x-1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 x^2}+\frac {a \sqrt {a x-1}}{2 x \sqrt {1-a x}} \]
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Rule 30
Rule 2317
Rule 2438
Rule 4265
Rule 5932
Rule 5946
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 x^2}+\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx-\frac {\left (a \sqrt {-1+a x}\right ) \int \frac {1}{x^2} \, dx}{2 \sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x}}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 x^2}+\frac {\left (a^2 \sqrt {-1+a x}\right ) \text {Subst}(\int x \text {sech}(x) \, dx,x,\text {arccosh}(a x))}{2 \sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x}}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 x^2}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\left (i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {1-a x}}+\frac {\left (i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x}}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 x^2}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\left (i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {\left (i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}} \\ & = \frac {a \sqrt {-1+a x}}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 x^2}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.40 \[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {(1+a x) \left (a x \sqrt {\frac {-1+a x}{1+a x}}-\text {arccosh}(a x)+a x \text {arccosh}(a x)-i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )+i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )+i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right )}{2 x^2 \sqrt {1-a^2 x^2}} \]
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Time = 0.99 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.09
method | result | size |
default | \(-\frac {\left (a^{2} x^{2} \operatorname {arccosh}\left (a x \right )+\sqrt {a x -1}\, \sqrt {a x +1}\, a x -\operatorname {arccosh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 \left (a^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}\) | \(349\) |
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\[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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\[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]
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