Optimal. Leaf size=73 \[ \frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac{x}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.160922, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5774, 5657, 3303, 3298, 3301} \[ \frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac{x}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5774
Rule 5657
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\int \frac{1}{a+b \sinh ^{-1}(c x)} \, dx}{b c}\\ &=-\frac{x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2 c^2}\\ &=-\frac{x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\cosh \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2 c^2}-\frac{\sinh \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2 c^2}\\ &=-\frac{x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}\\ \end{align*}
Mathematica [A] time = 0.123444, size = 60, normalized size = 0.82 \[ \frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-\frac{b c x}{a+b \sinh ^{-1}(c x)}}{b^2 c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 151, normalized size = 2.1 \begin{align*} -{\frac{1}{2\,b{c}^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ( cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{1}{2\,{c}^{2}{b}^{2}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\it Arcsinh} \left ( cx \right ) +{\frac{a}{b}} \right ) }-{\frac{1}{2\,{c}^{2}{b}^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ({\it Arcsinh} \left ( cx \right ){\it Ei} \left ( 1,-{\it Arcsinh} \left ( cx \right ) -{\frac{a}{b}} \right ){{\rm e}^{-{\frac{a}{b}}}}b+{\it Ei} \left ( 1,-{\it Arcsinh} \left ( cx \right ) -{\frac{a}{b}} \right ){{\rm e}^{-{\frac{a}{b}}}}a+xbc+\sqrt{{c}^{2}{x}^{2}+1}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*} -\frac{c^{3} x^{4} + c x^{2} +{\left (c^{2} x^{3} + x\right )} \sqrt{c^{2} x^{2} + 1}}{{\left (c^{2} x^{2} + 1\right )} a b c^{2} x +{\left ({\left (c^{2} x^{2} + 1\right )} b^{2} c^{2} x +{\left (b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a b c^{3} x^{2} + a b c\right )} \sqrt{c^{2} x^{2} + 1}} + \int \frac{c^{5} x^{5} +{\left (c^{2} x^{2} + 1\right )} c^{3} x^{3} + 3 \, c^{3} x^{3} + 2 \, c x +{\left (2 \, c^{4} x^{4} + 3 \, c^{2} x^{2} + 1\right )} \sqrt{c^{2} x^{2} + 1}}{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b c^{3} x^{2} + 2 \,{\left (a b c^{4} x^{3} + a b c^{2} x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left ({\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} c^{3} x^{2} + 2 \,{\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (b^{2} c^{5} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a b c^{5} x^{4} + 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt{c^{2} x^{2} + 1}}\,{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}{a^{2} c^{2} x^{2} +{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + a^{2} + 2 \,{\left (a b c^{2} x^{2} + a b\right )} \operatorname{arsinh}\left (c 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*} \int \frac{x}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2} \sqrt{c^{2} x^{2} + 1}}\, 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{x}{\sqrt{c^{2} x^{2} + 1}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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