Optimal. Leaf size=37 \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac {x \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5665, 3301} \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac {x \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5665
Rubi steps
\begin {align*} \int \frac {x}{\sinh ^{-1}(a x)^2} \, dx &=-\frac {x \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 0.86 \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac {\sinh \left (2 \sinh ^{-1}(a x)\right )}{2 a^2 \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\operatorname {arsinh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {x}{\operatorname {arsinh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 28, normalized size = 0.76 \[ \frac {-\frac {\sinh \left (2 \arcsinh \left (a x \right )\right )}{2 \arcsinh \left (a x \right )}+\Chi \left (2 \arcsinh \left (a x \right )\right )}{a^{2}} \]
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 \[ -\frac {a^{3} x^{4} + a x^{2} + {\left (a^{2} x^{3} + x\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )} + \int \frac {2 \, a^{5} x^{5} + 2 \, {\left (a^{2} x^{2} + 1\right )} a^{3} x^{3} + 4 \, a^{3} x^{3} + 2 \, a x + {\left (4 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 1\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{5} x^{4} + {\left (a^{2} x^{2} + 1\right )} a^{3} x^{2} + 2 \, a^{3} x^{2} + 2 \, {\left (a^{4} x^{3} + a^{2} x\right )} \sqrt {a^{2} x^{2} + 1} + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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