Optimal. Leaf size=102 \[ \frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac{b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac{2 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d}+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c^3 d} \]
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Rubi [A] time = 0.13777, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {5766, 74, 5694, 4182, 2279, 2391} \[ \frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac{b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac{2 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d}+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c^3 d} \]
Antiderivative was successfully verified.
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Rule 5766
Rule 74
Rule 5694
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{c^2}+\frac{b \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c d}\\ &=\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}-\frac{\operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 d}\\ &=\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}+\frac{b \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 d}-\frac{b \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 d}\\ &=\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^3 d}\\ &=\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}+\frac{b \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac{b \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}\\ \end{align*}
Mathematica [A] time = 0.149289, size = 155, normalized size = 1.52 \[ \frac{-2 b \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )-2 b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )-2 a c x-a \log (1-c x)+a \log (c x+1)+2 b \sqrt{\frac{c x-1}{c x+1}}+2 b c x \sqrt{\frac{c x-1}{c x+1}}+b \cosh ^{-1}(c x)^2-2 b c x \cosh ^{-1}(c x)+2 b \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right )-2 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.073, size = 208, normalized size = 2. \begin{align*} -{\frac{ax}{{c}^{2}d}}-{\frac{a\ln \left ( cx-1 \right ) }{2\,{c}^{3}d}}+{\frac{a\ln \left ( cx+1 \right ) }{2\,{c}^{3}d}}+{\frac{b{\rm arccosh} \left (cx\right )}{{c}^{3}d}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b{\rm arccosh} \left (cx\right )}{{c}^{3}d}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b{\rm arccosh} \left (cx\right )x}{{c}^{2}d}}+{\frac{b}{{c}^{3}d}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b}{{c}^{3}d}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b}{{c}^{3}d}{\it polylog} \left ( 2,cx+\sqrt{cx-1}\sqrt{cx+1} \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{1}{8} \,{\left (4 \, c^{2}{\left (\frac{2 \, x}{c^{4} d} - \frac{\log \left (c x + 1\right )}{c^{5} d} + \frac{\log \left (c x - 1\right )}{c^{5} d}\right )} + 24 \, c \int \frac{x \log \left (c x - 1\right )}{4 \,{\left (c^{4} d x^{2} - c^{2} d\right )}}\,{d x} - \frac{4 \,{\left (2 \, c x - \log \left (c x + 1\right ) + \log \left (c x - 1\right )\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + \log \left (c x + 1\right )^{2} + 2 \, \log \left (c x + 1\right ) \log \left (c x - 1\right )}{c^{3} d} + 8 \, \int -\frac{2 \, c x - \log \left (c x + 1\right ) + \log \left (c x - 1\right )}{2 \,{\left (c^{5} d x^{3} - c^{3} d x +{\left (c^{4} d x^{2} - c^{2} d\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}}\,{d x} - 8 \, \int \frac{\log \left (c x - 1\right )}{4 \,{\left (c^{4} d x^{2} - c^{2} d\right )}}\,{d x}\right )} b - \frac{1}{2} \, a{\left (\frac{2 \, x}{c^{2} d} - \frac{\log \left (c x + 1\right )}{c^{3} d} + \frac{\log \left (c x - 1\right )}{c^{3} d}\right )} \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{b x^{2} \operatorname{arcosh}\left (c x\right ) + a x^{2}}{c^{2} d x^{2} - d}, 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*} - \frac{\int \frac{a x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac{b x^{2} \operatorname{acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \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 (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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