Optimal. Leaf size=93 \[ \frac {3}{40} a^5 \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )+\frac {3 a^3 \sqrt {a x-1} \sqrt {a x+1}}{40 x^2}-\frac {\cosh ^{-1}(a x)}{5 x^5}+\frac {a \sqrt {a x-1} \sqrt {a x+1}}{20 x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5662, 103, 12, 92, 205} \[ \frac {3 a^3 \sqrt {a x-1} \sqrt {a x+1}}{40 x^2}+\frac {3}{40} a^5 \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )+\frac {a \sqrt {a x-1} \sqrt {a x+1}}{20 x^4}-\frac {\cosh ^{-1}(a x)}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 103
Rule 205
Rule 5662
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{x^6} \, dx &=-\frac {\cosh ^{-1}(a x)}{5 x^5}+\frac {1}{5} a \int \frac {1}{x^5 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}-\frac {\cosh ^{-1}(a x)}{5 x^5}+\frac {1}{20} a \int \frac {3 a^2}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}-\frac {\cosh ^{-1}(a x)}{5 x^5}+\frac {1}{20} \left (3 a^3\right ) \int \frac {1}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\cosh ^{-1}(a x)}{5 x^5}+\frac {1}{40} \left (3 a^3\right ) \int \frac {a^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\cosh ^{-1}(a x)}{5 x^5}+\frac {1}{40} \left (3 a^5\right ) \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\cosh ^{-1}(a x)}{5 x^5}+\frac {1}{40} \left (3 a^6\right ) \operatorname {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right )\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\cosh ^{-1}(a x)}{5 x^5}+\frac {3}{40} a^5 \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 104, normalized size = 1.12 \[ -\frac {-3 a^5 x^5+a^3 x^3-3 a^5 x^5 \sqrt {a^2 x^2-1} \tan ^{-1}\left (\sqrt {a^2 x^2-1}\right )+2 a x+8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{40 x^5 \sqrt {a x-1} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 101, normalized size = 1.09 \[ \frac {6 \, a^{5} x^{5} \arctan \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + 8 \, x^{5} \log \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + 8 \, {\left (x^{5} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (3 \, a^{3} x^{3} + 2 \, a x\right )} \sqrt {a^{2} x^{2} - 1}}{40 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 85, normalized size = 0.91 \[ \frac {3 \, a^{6} \arctan \left (\sqrt {a^{2} x^{2} - 1}\right ) + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{6} + 5 \, \sqrt {a^{2} x^{2} - 1} a^{6}}{a^{4} x^{4}}}{40 \, a} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 95, normalized size = 1.02 \[ -\frac {\mathrm {arccosh}\left (a x \right )}{5 x^{5}}-\frac {3 a^{5} \sqrt {a x -1}\, \sqrt {a x +1}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{40 \sqrt {a^{2} x^{2}-1}}+\frac {3 a^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{40 x^{2}}+\frac {a \sqrt {a x -1}\, \sqrt {a x +1}}{20 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 63, normalized size = 0.68 \[ -\frac {1}{40} \, {\left (3 \, a^{4} \arcsin \left (\frac {1}{a {\left | x \right |}}\right ) - \frac {3 \, \sqrt {a^{2} x^{2} - 1} a^{2}}{x^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} - 1}}{x^{4}}\right )} a - \frac {\operatorname {arcosh}\left (a x\right )}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{x^6} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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