Integrand size = 10, antiderivative size = 268 \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}-8 a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )+\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right ) \]
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Time = 0.56 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5883, 5933, 5947, 4265, 2611, 6744, 2320, 6724, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )-8 a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )+\frac {2 a^2 \text {arccosh}(a x)^2}{x}-\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{3 x^2} \]
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Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4265
Rule 5883
Rule 5933
Rule 5947
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {1}{3} (4 a) \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}-\left (2 a^2\right ) \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\text {arccosh}(a x)^3}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {1}{3} \left (2 a^3\right ) \text {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\text {arccosh}(a x)\right )-\left (4 a^3\right ) \int \frac {\text {arccosh}(a x)}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}+\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )-\left (2 i a^3\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )+\left (2 i a^3\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )-\left (4 a^3\right ) \text {Subst}(\int x \text {sech}(x) \, dx,x,\text {arccosh}(a x)) \\ & = \frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}-8 a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )+\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )-2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )+\left (4 i a^3\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )-\left (4 i a^3\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = \frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}-8 a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )+\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )-2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^x\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = \frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}-8 a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )+\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right ) \\ & = \frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}-8 a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )+\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(595\) vs. \(2(268)=536\).
Time = 2.11 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.22 \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=a^3 \left (\frac {1}{2} i \left (8+\pi ^2-4 i \pi \text {arccosh}(a x)-4 \text {arccosh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-\frac {1}{96} i \left (7 \pi ^4+8 i \pi ^3 \text {arccosh}(a x)+24 \pi ^2 \text {arccosh}(a x)^2+\frac {192 i \text {arccosh}(a x)^2}{a x}-32 i \pi \text {arccosh}(a x)^3+\frac {64 i \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^3}{a^2 x^2}-16 \text {arccosh}(a x)^4-\frac {32 i \text {arccosh}(a x)^4}{a^3 x^3}-384 \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\text {arccosh}(a x)}\right )+384 \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )+48 \pi ^2 \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-96 i \pi \text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-64 \text {arccosh}(a x)^3 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-48 \pi ^2 \text {arccosh}(a x) \log \left (1-i e^{\text {arccosh}(a x)}\right )+96 i \pi \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\text {arccosh}(a x)}\right )+64 \text {arccosh}(a x)^3 \log \left (1+i e^{\text {arccosh}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 i \text {arccosh}(a x))\right )\right )+384 \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )+192 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-48 \pi ^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )+384 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-384 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-192 i \pi \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{-\text {arccosh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )\right )\right ) \]
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\[\int \frac {\operatorname {arccosh}\left (a x \right )^{4}}{x^{4}}d x\]
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\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^4}{x^4} \,d x \]
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