Optimal. Leaf size=54 \[ -\frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {3 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5665, 3298} \[ -\frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {3 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5665
Rubi steps
\begin {align*} \int \frac {x^2}{\sinh ^{-1}(a x)^2} \, dx &=-\frac {x^2 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \left (-\frac {\sinh (x)}{4 x}+\frac {3 \sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {x^2 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {x^2 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}-\frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {3 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 49, normalized size = 0.91 \[ -\frac {\frac {4 a^2 x^2 \sqrt {a^2 x^2+1}}{\sinh ^{-1}(a x)}+\text {Shi}\left (\sinh ^{-1}(a x)\right )-3 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 56, normalized size = 1.04 \[ \frac {\frac {\sqrt {a^{2} x^{2}+1}}{4 \arcsinh \left (a x \right )}-\frac {\Shi \left (\arcsinh \left (a x \right )\right )}{4}-\frac {\cosh \left (3 \arcsinh \left (a x \right )\right )}{4 \arcsinh \left (a x \right )}+\frac {3 \Shi \left (3 \arcsinh \left (a x \right )\right )}{4}}{a^{3}} \]
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 \[ -\frac {a^{3} x^{5} + a x^{3} + {\left (a^{2} x^{4} + x^{2}\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )} + \int \frac {3 \, a^{5} x^{6} + 6 \, a^{3} x^{4} + 3 \, a x^{2} + {\left (3 \, a^{3} x^{4} + a x^{2}\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (6 \, a^{4} x^{5} + 7 \, a^{2} x^{3} + 2 \, x\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{5} x^{4} + {\left (a^{2} x^{2} + 1\right )} a^{3} x^{2} + 2 \, a^{3} x^{2} + 2 \, {\left (a^{4} x^{3} + a^{2} x\right )} \sqrt {a^{2} x^{2} + 1} + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{2}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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