Optimal. Leaf size=318 \[ -\frac{i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{i b^2 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{i b^2 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{c^2 x^2+1}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (c^2 x^2+1\right )}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac{\tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^3 d^3}+\frac{b^2 x}{12 c^2 d^3 \left (c^2 x^2+1\right )}-\frac{b^2 \tan ^{-1}(c x)}{6 c^3 d^3} \]
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Rubi [A] time = 0.417253, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {5751, 5690, 5693, 4180, 2531, 2282, 6589, 5717, 203, 199} \[ -\frac{i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{i b^2 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{i b^2 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{c^2 x^2+1}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (c^2 x^2+1\right )}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac{\tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^3 d^3}+\frac{b^2 x}{12 c^2 d^3 \left (c^2 x^2+1\right )}-\frac{b^2 \tan ^{-1}(c x)}{6 c^3 d^3} \]
Antiderivative was successfully verified.
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Rule 5751
Rule 5690
Rule 5693
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5717
Rule 203
Rule 199
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^3} \, dx &=-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{b \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}+\frac{\int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=-\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac{b^2 \int \frac{1}{\left (1+c^2 x^2\right )^2} \, dx}{6 c^2 d^3}-\frac{b \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{4 c d^3}+\frac{\int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{8 c^2 d^2}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1+c^2 x^2}}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3 d^3}+\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{12 c^2 d^3}-\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{4 c^2 d^3}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1+c^2 x^2}}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \tan ^{-1}(c x)}{6 c^3 d^3}-\frac{(i b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{(i b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3 d^3}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1+c^2 x^2}}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \tan ^{-1}(c x)}{6 c^3 d^3}-\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3 d^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3 d^3}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1+c^2 x^2}}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \tan ^{-1}(c x)}{6 c^3 d^3}-\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1+c^2 x^2}}-\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \tan ^{-1}(c x)}{6 c^3 d^3}-\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{i b^2 \text{Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{i b^2 \text{Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}\\ \end{align*}
Mathematica [A] time = 2.39617, size = 550, normalized size = 1.73 \[ \frac{\frac{3}{2} i a b \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )\right )-\frac{3}{2} i a b \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1-i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )\right )+b^2 \left (-6 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+6 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )-6 i \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )+6 i \text{PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )+\frac{2 c x}{c^2 x^2+1}+\frac{3 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}-\frac{6 c x \sinh ^{-1}(c x)^2}{\left (c^2 x^2+1\right )^2}+\frac{6 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{4 \sinh ^{-1}(c x)}{\left (c^2 x^2+1\right )^{3/2}}-3 i \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+3 i \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )-8 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\frac{3 a^2 c x}{c^2 x^2+1}-\frac{6 a^2 c x}{\left (c^2 x^2+1\right )^2}+3 a^2 \tan ^{-1}(c x)+\frac{a b \left (\sqrt{c^2 x^2+1} (2+i c x)+3 i \sinh ^{-1}(c x)\right )}{(c x-i)^2}+\frac{3 a b \left (\sinh ^{-1}(c x)-i \sqrt{c^2 x^2+1}\right )}{c x-i}+\frac{3 a b \left (\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right )}{c x+i}-\frac{i a b \left (3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)\right )}{(c x+i)^2}}{24 c^3 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.326, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{ \left ({c}^{2}d{x}^{2}+d \right ) ^{3}}}\, dx \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} \, a^{2}{\left (\frac{c^{2} x^{3} - x}{c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}} + \frac{\arctan \left (c x\right )}{c^{3} d^{3}}\right )} + \int \frac{b^{2} x^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}} + \frac{2 \, a b x^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}\,{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^{2} x^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{2}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, 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^{2} x^{2}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac{b^{2} x^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac{2 a b x^{2} \operatorname{asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \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 )}^{2} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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