Optimal. Leaf size=124 \[ -\frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac{b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d^2}-\frac{b}{2 c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.135397, antiderivative size = 124, 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 = {5750, 74, 5694, 4182, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac{b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d^2}-\frac{b}{2 c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5750
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 )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b \int \frac{x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}-\frac{\int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=-\frac{b}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3 d^2}\\ &=-\frac{b}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d^2}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3 d^2}+\frac{b \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3 d^2}\\ &=-\frac{b}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d^2}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}\\ &=-\frac{b}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d^2}-\frac{b \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac{b \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}\\ \end{align*}
Mathematica [A] time = 0.739994, size = 206, normalized size = 1.66 \[ \frac{-2 b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )+2 b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )-\frac{2 a c x}{c^2 x^2-1}+a \log (1-c x)-a \log (c x+1)+\frac{b c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{b \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+b \sqrt{\frac{c x-1}{c x+1}}+\frac{b \cosh ^{-1}(c x)}{1-c x}-\frac{b \cosh ^{-1}(c x)}{c x+1}+2 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )-2 b \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )}{4 c^3 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.099, size = 255, normalized size = 2.1 \begin{align*} -{\frac{a}{4\,{c}^{3}{d}^{2} \left ( cx-1 \right ) }}+{\frac{a\ln \left ( cx-1 \right ) }{4\,{c}^{3}{d}^{2}}}-{\frac{a}{4\,{c}^{3}{d}^{2} \left ( cx+1 \right ) }}-{\frac{a\ln \left ( cx+1 \right ) }{4\,{c}^{3}{d}^{2}}}-{\frac{b{\rm arccosh} \left (cx\right )x}{2\,{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b}{2\,{c}^{3}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )}{2\,{c}^{3}{d}^{2}}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b}{2\,{c}^{3}{d}^{2}}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b{\rm arccosh} \left (cx\right )}{2\,{c}^{3}{d}^{2}}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b}{2\,{c}^{3}{d}^{2}}{\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}{64} \,{\left (192 \, c^{3} \int \frac{x^{3} \log \left (c x - 1\right )}{8 \,{\left (c^{6} d^{2} x^{4} - 2 \, c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}}\,{d x} + 8 \, c^{2}{\left (\frac{2 \, x}{c^{6} d^{2} x^{2} - c^{4} d^{2}} + \frac{\log \left (c x + 1\right )}{c^{5} d^{2}} - \frac{\log \left (c x - 1\right )}{c^{5} d^{2}}\right )} - 64 \, c^{2} \int \frac{x^{2} \log \left (c x - 1\right )}{8 \,{\left (c^{6} d^{2} x^{4} - 2 \, c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}}\,{d x} + 3 \,{\left (c{\left (\frac{2}{c^{6} d^{2} x - c^{5} d^{2}} - \frac{\log \left (c x + 1\right )}{c^{5} d^{2}} + \frac{\log \left (c x - 1\right )}{c^{5} d^{2}}\right )} + \frac{4 \, \log \left (c x - 1\right )}{c^{6} d^{2} x^{2} - c^{4} d^{2}}\right )} c - \frac{4 \,{\left ({\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )^{2} + 2 \,{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) \log \left (c x - 1\right ) - 4 \,{\left (2 \, c x +{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) -{\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )}}{c^{5} d^{2} x^{2} - c^{3} d^{2}} + 64 \, \int \frac{2 \, c x +{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) -{\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )}{4 \,{\left (c^{7} d^{2} x^{5} - 2 \, c^{5} d^{2} x^{3} + c^{3} d^{2} x +{\left (c^{6} d^{2} x^{4} - 2 \, c^{4} d^{2} x^{2} + c^{2} d^{2}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}}\,{d x} + 64 \, \int \frac{\log \left (c x - 1\right )}{8 \,{\left (c^{6} d^{2} x^{4} - 2 \, c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}}\,{d x}\right )} b - \frac{1}{4} \, a{\left (\frac{2 \, x}{c^{4} d^{2} x^{2} - c^{2} d^{2}} + \frac{\log \left (c x + 1\right )}{c^{3} d^{2}} - \frac{\log \left (c x - 1\right )}{c^{3} d^{2}}\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^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{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*} \frac{\int \frac{a x^{2}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b x^{2} \operatorname{acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \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}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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