Optimal. Leaf size=268 \[ -2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac {2 a^2 \cosh ^{-1}(a x)^2}{x}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}+\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{3 x^2} \]
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Rubi [A] time = 0.80, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5662, 5748, 5761, 4180, 2531, 6609, 2282, 6589, 2279, 2391} \[ -2 i a^3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )+\frac {2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac {\cosh ^{-1}(a x)^4}{3 x^3}+\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{3 x^2} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4180
Rule 5662
Rule 5748
Rule 5761
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^4}{x^4} \, dx &=-\frac {\cosh ^{-1}(a x)^4}{3 x^3}+\frac {1}{3} (4 a) \int \frac {\cosh ^{-1}(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} \cosh ^{-1}(a x)^3}{3 x^2}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}-\left (2 a^2\right ) \int \frac {\cosh ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\cosh ^{-1}(a x)^3}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}+\frac {1}{3} \left (2 a^3\right ) \operatorname {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 a^3\right ) \int \frac {\cosh ^{-1}(a x)}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}+\frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\left (2 i a^3\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (2 i a^3\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname {Subst}\left (\int x \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 i a^3\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (4 i a^3\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 i a^3\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (4 i a^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=\frac {2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [B] time = 3.41, size = 595, normalized size = 2.22 \[ a^3 \left (\frac {1}{2} i \left (-4 \cosh ^{-1}(a x)^2-4 i \pi \cosh ^{-1}(a x)+\pi ^2+8\right ) \text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )-\frac {1}{96} i \left (-\frac {32 i \cosh ^{-1}(a x)^4}{a^3 x^3}+\frac {64 i \sqrt {\frac {a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^3}{a^2 x^2}+192 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+384 \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{-\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x) \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )+384 \text {Li}_2\left (i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \text {Li}_3\left (-i e^{-\cosh ^{-1}(a x)}\right )-192 i \pi \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )+384 \text {Li}_4\left (-i e^{-\cosh ^{-1}(a x)}\right )+384 \text {Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )-16 \cosh ^{-1}(a x)^4-32 i \pi \cosh ^{-1}(a x)^3+\frac {192 i \cosh ^{-1}(a x)^2}{a x}+24 \pi ^2 \cosh ^{-1}(a x)^2+8 i \pi ^3 \cosh ^{-1}(a x)-64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+64 \cosh ^{-1}(a x)^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )-96 i \pi \cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+96 i \pi \cosh ^{-1}(a x)^2 \log \left (1-i e^{\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )+48 \pi ^2 \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+384 \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \cosh ^{-1}(a x) \log \left (1-i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \cosh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )^{4}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{4}}{3 \, x^{3}} + \int \frac {4 \, {\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3}}{3 \, {\left (a^{3} x^{6} - a x^{4} + {\left (a^{2} x^{5} - x^{3}\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^4}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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