Optimal. Leaf size=248 \[ -\frac{11}{3} b^2 c^3 d^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{11}{3} b^2 c^3 d^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )-\frac{5}{3} b c^3 d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{4 c^2 d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{22}{3} b c^3 d^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+2 b^2 c^4 d^2 x-\frac{b^2 c^2 d^2}{3 x} \]
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Rubi [A] time = 0.706998, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {5739, 5653, 5717, 8, 5742, 5760, 4182, 2279, 2391, 14} \[ -\frac{11}{3} b^2 c^3 d^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{11}{3} b^2 c^3 d^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )-\frac{5}{3} b c^3 d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{4 c^2 d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{22}{3} b c^3 d^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+2 b^2 c^4 d^2 x-\frac{b^2 c^2 d^2}{3 x} \]
Antiderivative was successfully verified.
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Rule 5739
Rule 5653
Rule 5717
Rule 8
Rule 5742
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 14
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^4} \, dx &=-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (4 c^2 d\right ) \int \frac{\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac{1}{3} \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^3} \, dx\\ &=-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d^2\right ) \int \frac{1+c^2 x^2}{x^2} \, dx+\left (b c^3 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 b c^3 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^4 d^2\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ &=\frac{11}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d^2\right ) \int \left (c^2+\frac{1}{x^2}\right ) \, dx+\left (b c^3 d^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx+\frac{1}{3} \left (8 b c^3 d^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx-\left (b^2 c^4 d^2\right ) \int 1 \, dx-\frac{1}{3} \left (8 b^2 c^4 d^2\right ) \int 1 \, dx-\frac{1}{3} \left (16 b c^5 d^2\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{b^2 c^2 d^2}{3 x}-\frac{10}{3} b^2 c^4 d^2 x-\frac{5}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (b c^3 d^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (8 b c^3 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^4 d^2\right ) \int 1 \, dx\\ &=-\frac{b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac{5}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{22}{3} b c^3 d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )-\frac{1}{3} \left (8 b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (8 b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac{b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac{5}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{22}{3} b c^3 d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )-\frac{1}{3} \left (8 b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\frac{1}{3} \left (8 b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=-\frac{b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac{5}{3} b c^3 d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{22}{3} b c^3 d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{11}{3} b^2 c^3 d^2 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac{11}{3} b^2 c^3 d^2 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.90395, size = 357, normalized size = 1.44 \[ \frac{d^2 \left (11 b^2 c^3 x^3 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-11 b^2 c^3 x^3 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+3 a^2 c^4 x^4-6 a^2 c^2 x^2-a^2-6 a b c^3 x^3 \sqrt{c^2 x^2+1}-a b c x \sqrt{c^2 x^2+1}+6 a b c^4 x^4 \sinh ^{-1}(c x)-12 a b c^2 x^2 \sinh ^{-1}(c x)-11 a b c^3 x^3 \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-2 a b \sinh ^{-1}(c x)+6 b^2 c^4 x^4-b^2 c^2 x^2+3 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-6 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-6 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+11 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-11 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-b^2 \sinh ^{-1}(c x)^2\right )}{3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.281, size = 408, normalized size = 1.7 \begin{align*}{c}^{4}{d}^{2}{a}^{2}x-2\,{\frac{{c}^{2}{d}^{2}{a}^{2}}{x}}-{\frac{{d}^{2}{a}^{2}}{3\,{x}^{3}}}+{c}^{4}{d}^{2}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}x-2\,{c}^{3}{d}^{2}{b}^{2}{\it Arcsinh} \left ( cx \right ) \sqrt{{c}^{2}{x}^{2}+1}+2\,{b}^{2}{c}^{4}{d}^{2}x-2\,{\frac{{d}^{2}{b}^{2}{c}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{x}}-{\frac{{d}^{2}{b}^{2}c{\it Arcsinh} \left ( cx \right ) }{3\,{x}^{2}}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{{d}^{2}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{{d}^{2}{b}^{2}{c}^{2}}{3\,x}}-{\frac{11\,{c}^{3}{d}^{2}{b}^{2}{\it Arcsinh} \left ( cx \right ) }{3}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{11\,{c}^{3}{d}^{2}{b}^{2}}{3}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{11\,{c}^{3}{d}^{2}{b}^{2}{\it Arcsinh} \left ( cx \right ) }{3}\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{11\,{c}^{3}{d}^{2}{b}^{2}}{3}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }+2\,{c}^{4}{d}^{2}ab{\it Arcsinh} \left ( cx \right ) x-4\,{\frac{{d}^{2}ab{c}^{2}{\it Arcsinh} \left ( cx \right ) }{x}}-{\frac{2\,{d}^{2}ab{\it Arcsinh} \left ( cx \right ) }{3\,{x}^{3}}}-2\,{c}^{3}{d}^{2}ab\sqrt{{c}^{2}{x}^{2}+1}-{\frac{11\,{c}^{3}{d}^{2}ab}{3}{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ) }-{\frac{c{d}^{2}ab}{3\,{x}^{2}}\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*} b^{2} c^{4} d^{2} x \operatorname{arsinh}\left (c x\right )^{2} + 2 \, b^{2} c^{4} d^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + a^{2} c^{4} d^{2} x + 2 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b c^{3} d^{2} - 4 \,{\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 c^{2} d^{2} + \frac{1}{3} \,{\left ({\left (c^{2} \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac{2 \, \operatorname{arsinh}\left (c x\right )}{x^{3}}\right )} a b d^{2} - \frac{2 \, a^{2} c^{2} d^{2}}{x} - \frac{a^{2} d^{2}}{3 \, x^{3}} - \frac{{\left (6 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{3 \, x^{3}} + \int \frac{2 \,{\left (6 \, b^{2} c^{5} d^{2} x^{4} + 7 \, b^{2} c^{3} d^{2} x^{2} + b^{2} c d^{2} +{\left (6 \, b^{2} c^{4} d^{2} x^{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^{6} + c x^{4} +{\left (c^{2} x^{5} + x^{3}\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^{4}}, 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 a^{2} c^{4}\, dx + \int \frac{a^{2}}{x^{4}}\, dx + \int \frac{2 a^{2} c^{2}}{x^{2}}\, dx + \int b^{2} c^{4} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int 2 a b c^{4} \operatorname{asinh}{\left (c x \right )}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{2 b^{2} c^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{4 a b c^{2} \operatorname{asinh}{\left (c x \right )}}{x^{2}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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