Optimal. Leaf size=54 \[ -\frac{c \left (a^2 x^2+1\right )^{3/2}}{a \sinh ^{-1}(a x)}+\frac{3 c \text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a}+\frac{3 c \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a} \]
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Rubi [A] time = 0.135066, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {5696, 5779, 5448, 3298} \[ -\frac{c \left (a^2 x^2+1\right )^{3/2}}{a \sinh ^{-1}(a x)}+\frac{3 c \text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a}+\frac{3 c \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 5696
Rule 5779
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int \frac{c+a^2 c x^2}{\sinh ^{-1}(a x)^2} \, dx &=-\frac{c \left (1+a^2 x^2\right )^{3/2}}{a \sinh ^{-1}(a x)}+(3 a c) \int \frac{x \sqrt{1+a^2 x^2}}{\sinh ^{-1}(a x)} \, dx\\ &=-\frac{c \left (1+a^2 x^2\right )^{3/2}}{a \sinh ^{-1}(a x)}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=-\frac{c \left (1+a^2 x^2\right )^{3/2}}{a \sinh ^{-1}(a x)}+\frac{(3 c) \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 x}+\frac{\sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=-\frac{c \left (1+a^2 x^2\right )^{3/2}}{a \sinh ^{-1}(a x)}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a}\\ &=-\frac{c \left (1+a^2 x^2\right )^{3/2}}{a \sinh ^{-1}(a x)}+\frac{3 c \text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a}+\frac{3 c \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.211704, size = 54, normalized size = 1. \[ \frac{c \left (-4 \left (a^2 x^2+1\right )^{3/2}+3 \sinh ^{-1}(a x) \text{Shi}\left (\sinh ^{-1}(a x)\right )+3 \sinh ^{-1}(a x) \text{Shi}\left (3 \sinh ^{-1}(a x)\right )\right )}{4 a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 60, normalized size = 1.1 \begin{align*}{\frac{c}{4\,a{\it Arcsinh} \left ( ax \right ) } \left ( 3\,{\it Shi} \left ({\it Arcsinh} \left ( ax \right ) \right ){\it Arcsinh} \left ( ax \right ) +3\,{\it Shi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ){\it Arcsinh} \left ( ax \right ) -3\,\sqrt{{a}^{2}{x}^{2}+1}-\cosh \left ( 3\,{\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^{5} c x^{5} + 2 \, a^{3} c x^{3} + a c x +{\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\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{3 \, a^{6} c x^{6} + 7 \, a^{4} c x^{4} + 5 \, a^{2} c x^{2} +{\left (3 \, a^{4} c x^{4} + 2 \, a^{2} c x^{2} - c\right )}{\left (a^{2} x^{2} + 1\right )} + 3 \,{\left (2 \, a^{5} c x^{5} + 3 \, a^{3} c x^{3} + a c x\right )} \sqrt{a^{2} x^{2} + 1} + c}{{\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{a^{2} c x^{2} + c}{\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*} c \left (\int \frac{a^{2} x^{2}}{\operatorname{asinh}^{2}{\left (a x \right )}}\, dx + \int \frac{1}{\operatorname{asinh}^{2}{\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 )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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