Optimal. Leaf size=108 \[ \frac{i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac{i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac{2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d}-\frac{b \sqrt{c^2 x^2+1}}{c^3 d} \]
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Rubi [A] time = 0.139653, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5767, 5693, 4180, 2279, 2391, 261} \[ \frac{i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac{i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac{2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d}-\frac{b \sqrt{c^2 x^2+1}}{c^3 d} \]
Antiderivative was successfully verified.
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Rule 5767
Rule 5693
Rule 4180
Rule 2279
Rule 2391
Rule 261
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac{\int \frac{a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx}{c^2}-\frac{b \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{c d}\\ &=-\frac{b \sqrt{1+c^2 x^2}}{c^3 d}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac{\operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 d}\\ &=-\frac{b \sqrt{1+c^2 x^2}}{c^3 d}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac{(i b) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 d}-\frac{(i b) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 d}\\ &=-\frac{b \sqrt{1+c^2 x^2}}{c^3 d}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^3 d}\\ &=-\frac{b \sqrt{1+c^2 x^2}}{c^3 d}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac{i b \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac{i b \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}\\ \end{align*}
Mathematica [A] time = 0.155034, size = 121, normalized size = 1.12 \[ \frac{i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )-i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )+a c x-a \tan ^{-1}(c x)-b \sqrt{c^2 x^2+1}+b c x \sinh ^{-1}(c x)-i b \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )+i b \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{c^3 d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.009, size = 215, normalized size = 2. \begin{align*}{\frac{ax}{{c}^{2}d}}-{\frac{a\arctan \left ( cx \right ) }{{c}^{3}d}}-{\frac{b\arctan \left ( cx \right ) }{{c}^{3}d}\ln \left ( 1+{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{b\arctan \left ( cx \right ) }{{c}^{3}d}\ln \left ( 1-{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{ib}{{c}^{3}d}{\it dilog} \left ( 1+{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{ib}{{c}^{3}d}{\it dilog} \left ( 1-{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{b{\it Arcsinh} \left ( cx \right ) \arctan \left ( cx \right ) }{{c}^{3}d}}+{\frac{b{\it Arcsinh} \left ( cx \right ) x}{{c}^{2}d}}-{\frac{b}{{c}^{3}d}\sqrt{{c}^{2}{x}^{2}+1}} \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*} a{\left (\frac{x}{c^{2} d} - \frac{\arctan \left (c x\right )}{c^{3} d}\right )} + b \int \frac{x^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2} d x^{2} + d}\,{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{b x^{2} \operatorname{arsinh}\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{asinh}{\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{arsinh}\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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