Optimal. Leaf size=50 \[ \frac{5 c^2 \text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a}+\frac{5 c^2 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{16 a}+\frac{c^2 \text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.100021, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {5699, 3312, 3301} \[ \frac{5 c^2 \text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a}+\frac{5 c^2 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{16 a}+\frac{c^2 \text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Rule 5699
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2}{\sinh ^{-1}(a x)} \, dx &=\frac{c^2 \operatorname{Subst}\left (\int \frac{\cosh ^5(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=\frac{c^2 \operatorname{Subst}\left (\int \left (\frac{5 \cosh (x)}{8 x}+\frac{5 \cosh (3 x)}{16 x}+\frac{\cosh (5 x)}{16 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=\frac{c^2 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a}+\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a}+\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a}\\ &=\frac{5 c^2 \text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a}+\frac{5 c^2 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{16 a}+\frac{c^2 \text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{16 a}\\ \end{align*}
Mathematica [A] time = 0.0758856, size = 34, normalized size = 0.68 \[ \frac{c^2 \left (10 \text{Chi}\left (\sinh ^{-1}(a x)\right )+5 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )+\text{Chi}\left (5 \sinh ^{-1}(a x)\right )\right )}{16 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 33, normalized size = 0.7 \begin{align*}{\frac{{c}^{2} \left ( 10\,{\it Chi} \left ({\it Arcsinh} \left ( ax \right ) \right ) +5\,{\it Chi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) +{\it Chi} \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) \right ) }{16\,a}} \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*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname{arsinh}\left (a x\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{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}{\operatorname{arsinh}\left (a x\right )}, 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*} c^{2} \left (\int \frac{2 a^{2} x^{2}}{\operatorname{asinh}{\left (a x \right )}}\, dx + \int \frac{a^{4} x^{4}}{\operatorname{asinh}{\left (a x \right )}}\, dx + \int \frac{1}{\operatorname{asinh}{\left (a x \right )}}\, dx\right ) \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{{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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