Integrand size = 10, antiderivative size = 183 \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\frac {a^2 \text {arccosh}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}+a^3 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-a^3 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+i a^3 \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-i a^3 \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right ) \]
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Time = 0.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5883, 5933, 5947, 4265, 2611, 2320, 6724, 94, 211} \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=a^3 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+i a^3 \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-i a^3 \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-a^3 \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )+\frac {a^2 \text {arccosh}(a x)}{x}-\frac {\text {arccosh}(a x)^3}{3 x^3}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x^2} \]
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Rule 94
Rule 211
Rule 2320
Rule 2611
Rule 4265
Rule 5883
Rule 5933
Rule 5947
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(a x)^3}{3 x^3}+a \int \frac {\text {arccosh}(a x)^2}{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)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}-a^2 \int \frac {\text {arccosh}(a x)}{x^2} \, dx+\frac {1}{2} a^3 \int \frac {\text {arccosh}(a x)^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a^2 \text {arccosh}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}+\frac {1}{2} a^3 \text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\text {arccosh}(a x)\right )-a^3 \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a^2 \text {arccosh}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}+a^3 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-\left (i a^3\right ) \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )+\left (i a^3\right ) \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )-a^4 \text {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right ) \\ & = \frac {a^2 \text {arccosh}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}+a^3 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-a^3 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+\left (i a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )-\left (i a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = \frac {a^2 \text {arccosh}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}+a^3 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-a^3 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+\left (i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )-\left (i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right ) \\ & = \frac {a^2 \text {arccosh}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}+a^3 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-a^3 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+i a^3 \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-i a^3 \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right ) \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.10 \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\frac {1}{6} \left (\frac {6 a^2 \text {arccosh}(a x)}{x}+\frac {3 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^2}{x^2}-\frac {2 \text {arccosh}(a x)^3}{x^3}-3 i a^3 \left (-4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )\right )+\text {arccosh}(a x)^2 \log \left (1-i e^{-\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )+2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(a x)}\right )\right )\right ) \]
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\[\int \frac {\operatorname {arccosh}\left (a x \right )^{3}}{x^{4}}d x\]
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\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^4} \,d x \]
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