Optimal. Leaf size=34 \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a}-\frac{\sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.0833157, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5655, 5779, 3298} \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a}-\frac{\sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5655
Rule 5779
Rule 3298
Rubi steps
\begin{align*} \int \frac{1}{\sinh ^{-1}(a x)^2} \, dx &=-\frac{\sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}+a \int \frac{x}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx\\ &=-\frac{\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 )}{a}\\ &=-\frac{\sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0478953, size = 31, normalized size = 0.91 \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )-\frac{\sqrt{a^2 x^2+1}}{\sinh ^{-1}(a x)}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 30, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{{\it Arcsinh} \left ( ax \right ) }\sqrt{{a}^{2}{x}^{2}+1}}+{\it Shi} \left ({\it Arcsinh} \left ( ax \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} x^{3} + a x +{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\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{a^{4} x^{4} + 2 \, a^{2} x^{2} +{\left (a^{2} x^{2} + 1\right )}{\left (a^{2} x^{2} - 1\right )} +{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1} + 1}{{\left (a^{4} x^{4} +{\left (a^{2} x^{2} + 1\right )} a^{2} x^{2} + 2 \, a^{2} x^{2} + 2 \,{\left (a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1} + 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\operatorname{arsinh}\left (a x\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{asinh}^{2}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{arsinh}\left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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