Optimal. Leaf size=29 \[ \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^4} \]
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Rubi [A] time = 0.07, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5670, 5448, 3298} \[ \frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5448
Rule 5670
Rubi steps
\begin {align*} \int \frac {x^3}{\cosh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^4}\\ &=\frac {\text {Shi}\left (2 \cosh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 24, normalized size = 0.83 \[ \frac {2 \text {Shi}\left (2 \cosh ^{-1}(a x)\right )+\text {Shi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\operatorname {arcosh}\left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 24, normalized size = 0.83 \[ \frac {\frac {\Shi \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{8}+\frac {\Shi \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{4}}{a^{4}} \]
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 \[ \int \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^3}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
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 \[ \int \frac {x^{3}}{\operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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