Optimal. Leaf size=57 \[ -\frac{a \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)},x\right )}{c}-\frac{1}{a c \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.101692, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^2} \, dx &=-\frac{1}{a c \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}-\frac{a \int \frac{x}{\left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)} \, dx}{c}\\ \end{align*}
Mathematica [A] time = 1.4745, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({a}^{2}c{x}^{2}+c \right ) \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a x + \sqrt{a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} c x + a c\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )} - \int \frac{a^{4} x^{4} +{\left (a^{2} x^{2} + 1\right )}^{2} +{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} + 1} - 1}{{\left (a^{6} c x^{6} + 3 \, a^{4} c x^{4} + 3 \, a^{2} c x^{2} +{\left (a^{4} c x^{4} + a^{2} c x^{2}\right )}{\left (a^{2} x^{2} + 1\right )} + 2 \,{\left (a^{5} c x^{5} + 2 \, a^{3} c x^{3} + a c x\right )} \sqrt{a^{2} x^{2} + 1} + c\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{2} c x^{2} + c\right )} \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{2} x^{2} \operatorname{asinh}^{2}{\left (a x \right )} + \operatorname{asinh}^{2}{\left (a x \right )}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )} \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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