Optimal. Leaf size=14 \[ \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^2} \]
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Rubi [A] time = 0.04, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5670, 5448, 12, 3298} \[ \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 5448
Rule 5670
Rubi steps
\begin {align*} \int \frac {x}{\cosh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^2}\\ &=\frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 14, normalized size = 1.00 \[ \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\operatorname {arcosh}\left (a x\right )}, x\right ) \]
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 \[ \int \frac {x}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 13, normalized size = 0.93 \[ \frac {\Shi \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2}} \]
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 \[ \int \frac {x}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {x}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
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 \[ \int \frac {x}{\operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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