Optimal. Leaf size=82 \[ \frac{\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b c^3}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b c^3}-\frac{\log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c^3} \]
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Rubi [A] time = 0.273493, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {5779, 5448, 3303, 3298, 3301} \[ \frac{\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^3}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^3}-\frac{\log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c^3} \]
Antiderivative was successfully verified.
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Rule 5779
Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2 \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{8 (a+b x)}+\frac{\cosh (4 x)}{8 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3}\\ &=-\frac{\log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}\\ &=-\frac{\log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{\cosh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}-\frac{\sinh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}\\ &=\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^3}-\frac{\log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c^3}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^3}\\ \end{align*}
Mathematica [A] time = 0.174887, size = 65, normalized size = 0.79 \[ \frac{\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-\log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 79, normalized size = 1. \begin{align*} -{\frac{\ln \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) }{8\,{c}^{3}b}}-{\frac{1}{16\,{c}^{3}b}{{\rm e}^{4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,4\,{\it Arcsinh} \left ( cx \right ) +4\,{\frac{a}{b}} \right ) }-{\frac{1}{16\,{c}^{3}b}{{\rm e}^{-4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-4\,{\it Arcsinh} \left ( cx \right ) -4\,{\frac{a}{b}} \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*} \int \frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{b \operatorname{arsinh}\left (c x\right ) + a}\,{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{\sqrt{c^{2} x^{2} + 1} x^{2}}{b \operatorname{arsinh}\left (c x\right ) + a}, 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{x^{2} \sqrt{c^{2} x^{2} + 1}}{a + b \operatorname{asinh}{\left (c 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{\sqrt{c^{2} x^{2} + 1} x^{2}}{b \operatorname{arsinh}\left (c x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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