Optimal. Leaf size=229 \[ -2 b^2 c d^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{4}{3} c^2 d^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2}{9} b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{10}{3} b c d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{2}{27} b^2 c^4 d^2 x^3+\frac{32}{9} b^2 c^2 d^2 x \]
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Rubi [A] time = 0.522505, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {5739, 5684, 5653, 5717, 8, 5744, 5742, 5760, 4182, 2279, 2391} \[ -2 b^2 c d^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{4}{3} c^2 d^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2}{9} b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{10}{3} b c d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{2}{27} b^2 c^4 d^2 x^3+\frac{32}{9} b^2 c^2 d^2 x \]
Antiderivative was successfully verified.
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Rule 5739
Rule 5684
Rule 5653
Rule 5717
Rule 8
Rule 5744
Rule 5742
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (4 c^2 d\right ) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\left (2 b c d^2\right ) \int \frac{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx\\ &=\frac{2}{3} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac{1}{3} \left (8 c^2 d^2\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{3} \left (2 b^2 c^2 d^2\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac{1}{3} \left (8 b c^3 d^2\right ) \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{2}{3} b^2 c^2 d^2 x-\frac{2}{9} b^2 c^4 d^2 x^3+2 b c d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx+\frac{1}{9} \left (8 b^2 c^2 d^2\right ) \int \left (1+c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^2\right ) \int 1 \, dx-\frac{1}{3} \left (16 b c^3 d^2\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{16}{9} b^2 c^2 d^2 x+\frac{2}{27} b^2 c^4 d^2 x^3-\frac{10}{3} b c d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (16 b^2 c^2 d^2\right ) \int 1 \, dx\\ &=\frac{32}{9} b^2 c^2 d^2 x+\frac{2}{27} b^2 c^4 d^2 x^3-\frac{10}{3} b c d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (2 b^2 c d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (2 b^2 c d^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{32}{9} b^2 c^2 d^2 x+\frac{2}{27} b^2 c^4 d^2 x^3-\frac{10}{3} b c d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (2 b^2 c d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (2 b^2 c d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=\frac{32}{9} b^2 c^2 d^2 x+\frac{2}{27} b^2 c^4 d^2 x^3-\frac{10}{3} b c d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-2 b^2 c d^2 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^2 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 1.15082, size = 306, normalized size = 1.34 \[ \frac{1}{54} d^2 \left (108 b^2 c \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-108 b^2 c \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+18 a^2 c^4 x^3+108 a^2 c^2 x-\frac{54 a^2}{x}-12 a b c \left (c^2 x^2-2\right ) \sqrt{c^2 x^2+1}+36 a b c^4 x^3 \sinh ^{-1}(c x)+216 a b c \left (c x \sinh ^{-1}(c x)-\sqrt{c^2 x^2+1}\right )-\frac{108 a b \left (c x \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )+\sinh ^{-1}(c x)\right )}{x}+2 b^2 c^2 x \left (2 c^2 x^2+9 c^2 x^2 \sinh ^{-1}(c x)^2-12\right )-189 b^2 c \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+108 b^2 c^2 x \left (\sinh ^{-1}(c x)^2+2\right )-\frac{54 b^2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+2 c x \left (\log \left (e^{-\sinh ^{-1}(c x)}+1\right )-\log \left (1-e^{-\sinh ^{-1}(c x)}\right )\right )\right )}{x}-3 b^2 c \sinh ^{-1}(c x) \cosh \left (3 \sinh ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.134, size = 400, normalized size = 1.8 \begin{align*}{\frac{{d}^{2}{a}^{2}{c}^{4}{x}^{3}}{3}}+2\,{d}^{2}{a}^{2}{c}^{2}x-{\frac{{d}^{2}{a}^{2}}{x}}+{\frac{32\,{b}^{2}{c}^{2}{d}^{2}x}{9}}+{\frac{2\,{b}^{2}{c}^{4}{d}^{2}{x}^{3}}{27}}+2\,{b}^{2}c{d}^{2}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) -2\,{b}^{2}c{d}^{2}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +{\frac{{d}^{2}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{4}{x}^{3}}{3}}+2\,{d}^{2}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}x-{\frac{32\,{d}^{2}{b}^{2}c{\it Arcsinh} \left ( cx \right ) }{9}\sqrt{{c}^{2}{x}^{2}+1}}+2\,c{d}^{2}{b}^{2}{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) -{\frac{{d}^{2}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{x}}-2\,c{d}^{2}{b}^{2}{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) -{\frac{2\,{d}^{2}{b}^{2}{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{2}}{9}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{2\,{d}^{2}ab{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{3}}{3}}+4\,{d}^{2}ab{\it Arcsinh} \left ( cx \right ){c}^{2}x-2\,{\frac{{d}^{2}ab{\it Arcsinh} \left ( cx \right ) }{x}}-{\frac{2\,{d}^{2}ab{c}^{3}{x}^{2}}{9}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{32\,c{d}^{2}ab}{9}\sqrt{{c}^{2}{x}^{2}+1}}-2\,c{d}^{2}ab{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+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}{3} \, a^{2} c^{4} d^{2} x^{3} + \frac{2}{9} \,{\left (3 \, x^{3} \operatorname{arsinh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac{2 \, \sqrt{c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{4} d^{2} + 2 \, b^{2} c^{2} d^{2} x \operatorname{arsinh}\left (c x\right )^{2} + 4 \, b^{2} c^{2} d^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + 2 \, a^{2} c^{2} d^{2} x + 4 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b c d^{2} - 2 \,{\left (c \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\operatorname{arsinh}\left (c x\right )}{x}\right )} a b d^{2} - \frac{a^{2} d^{2}}{x} + \frac{{\left (b^{2} c^{4} d^{2} x^{4} - 3 \, b^{2} d^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{3 \, x} - \int \frac{2 \,{\left (b^{2} c^{7} d^{2} x^{6} + b^{2} c^{5} d^{2} x^{4} - 3 \, b^{2} c^{3} d^{2} x^{2} - 3 \, b^{2} c d^{2} +{\left (b^{2} c^{6} d^{2} x^{5} - 3 \, b^{2} c^{2} d^{2} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{3 \,{\left (c^{3} x^{4} + c x^{2} +{\left (c^{2} x^{3} + x\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{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{a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname{arsinh}\left (c x\right )}{x^{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*} d^{2} \left (\int 2 a^{2} c^{2}\, dx + \int \frac{a^{2}}{x^{2}}\, dx + \int a^{2} c^{4} x^{2}\, dx + \int 2 b^{2} c^{2} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int 4 a b c^{2} \operatorname{asinh}{\left (c x \right )}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{4} x^{2} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{4} x^{2} \operatorname{asinh}{\left (c x \right )}\, dx\right ) \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 (c^{2} d x^{2} + d\right )}^{2}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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