Integrand size = 10, antiderivative size = 114 \[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 x^2}-\frac {\text {arccosh}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5883, 5933, 5947, 4265, 2317, 2438, 30} \[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\frac {2}{3} a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+\frac {a^2}{3 x}-\frac {\text {arccosh}(a x)^2}{3 x^3}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 x^2} \]
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Rule 30
Rule 2317
Rule 2438
Rule 4265
Rule 5883
Rule 5933
Rule 5947
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 x^2}-\frac {\text {arccosh}(a x)^2}{3 x^3}-\frac {1}{3} a^2 \int \frac {1}{x^2} \, dx+\frac {1}{3} a^3 \int \frac {\text {arccosh}(a x)}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 x^2}-\frac {\text {arccosh}(a x)^2}{3 x^3}+\frac {1}{3} a^3 \text {Subst}(\int x \text {sech}(x) \, dx,x,\text {arccosh}(a x)) \\ & = \frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 x^2}-\frac {\text {arccosh}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )+\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = \frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 x^2}-\frac {\text {arccosh}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )+\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right ) \\ & = \frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 x^2}-\frac {\text {arccosh}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right ) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.26 \[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\frac {1}{3} a^3 \left (\frac {1}{a x}+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a^2 x^2}-\frac {\text {arccosh}(a x)^2}{a^3 x^3}-i \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )+i \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right ) \]
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Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.50
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {-a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+\operatorname {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {i \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}\right )\) | \(171\) |
default | \(a^{3} \left (-\frac {-a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+\operatorname {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {i \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}\right )\) | \(171\) |
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\[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^4} \,d x \]
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