Optimal. Leaf size=268 \[ \frac {2 a^2 \cosh ^{-1}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 x^2}-\frac {\cosh ^{-1}(a x)^4}{3 x^3}-8 a^3 \cosh ^{-1}(a x) \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )+\frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )-2 i a^3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+2 i a^3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \cosh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )-4 i a^3 \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )+4 i a^3 \text {PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.53, 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 = {5883, 5933,
5947, 4265, 2611, 6744, 2320, 6724, 2317, 2438} \begin {gather*} \frac {4}{3} a^3 \cosh ^{-1}(a x)^3 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )-8 a^3 \cosh ^{-1}(a x) \text {ArcTan}\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 )+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 {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} \end {gather*}
Antiderivative was successfully verified.
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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*} \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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 a^3\right ) \text {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 ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (4 i a^3\right ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (4 i a^3\right ) \text {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 ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )+\left (4 i a^3\right ) \text {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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(595\) vs. \(2(268)=536\).
time = 2.15, size = 595, normalized size = 2.22 \begin {gather*} a^3 \left (\frac {1}{2} i \left (8+\pi ^2-4 i \pi \cosh ^{-1}(a x)-4 \cosh ^{-1}(a x)^2\right ) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\frac {1}{96} i \left (7 \pi ^4+8 i \pi ^3 \cosh ^{-1}(a x)+24 \pi ^2 \cosh ^{-1}(a x)^2+\frac {192 i \cosh ^{-1}(a x)^2}{a x}-32 i \pi \cosh ^{-1}(a x)^3+\frac {64 i \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)^3}{a^2 x^2}-16 \cosh ^{-1}(a x)^4-\frac {32 i \cosh ^{-1}(a x)^4}{a^3 x^3}-384 \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 )+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 )-96 i \pi \cosh ^{-1}(a x)^2 \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 )-48 \pi ^2 \cosh ^{-1}(a x) \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 )-8 i \pi ^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 )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \cosh ^{-1}(a x)\right )\right )\right )+384 \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )+192 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )-48 \pi ^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )+384 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-192 i \pi \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )+384 \text {PolyLog}\left (4,-i e^{-\cosh ^{-1}(a x)}\right )+384 \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 3.15, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{4}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^4}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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