Optimal. Leaf size=29 \[ \frac{3 c \text{Chi}\left (\sinh ^{-1}(a x)\right )}{4 a}+\frac{c \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{4 a} \]
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Rubi [A] time = 0.0705044, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5699, 3312, 3301} \[ \frac{3 c \text{Chi}\left (\sinh ^{-1}(a x)\right )}{4 a}+\frac{c \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 5699
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{c+a^2 c x^2}{\sinh ^{-1}(a x)} \, dx &=\frac{c \operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{4 x}+\frac{\cosh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a}\\ &=\frac{3 c \text{Chi}\left (\sinh ^{-1}(a x)\right )}{4 a}+\frac{c \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0126044, size = 23, normalized size = 0.79 \[ \frac{c \left (3 \text{Chi}\left (\sinh ^{-1}(a x)\right )+\text{Chi}\left (3 \sinh ^{-1}(a x)\right )\right )}{4 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 22, normalized size = 0.8 \begin{align*}{\frac{c \left ( 3\,{\it Chi} \left ({\it Arcsinh} \left ( ax \right ) \right ) +{\it Chi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) \right ) }{4\,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{a^{2} c x^{2} + c}{\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^{2} c x^{2} + c}{\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 \left (\int \frac{a^{2} x^{2}}{\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{a^{2} c x^{2} + c}{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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