Optimal. Leaf size=158 \[ -\frac{5}{3} b^2 c^3 d \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{5}{3} b^2 c^3 d \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )-\frac{b c d \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{2 c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{10}{3} b c^3 d \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{b^2 c^2 d}{3 x} \]
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Rubi [A] time = 0.393379, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5739, 5661, 5760, 4182, 2279, 2391, 5737, 30} \[ -\frac{5}{3} b^2 c^3 d \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{5}{3} b^2 c^3 d \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )-\frac{b c d \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{2 c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{10}{3} b c^3 d \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{b^2 c^2 d}{3 x} \]
Antiderivative was successfully verified.
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Rule 5739
Rule 5661
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 5737
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} (2 b c d) \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx+\frac{1}{3} \left (2 c^2 d\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx\\ &=-\frac{b c d \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{2 c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d\right ) \int \frac{1}{x^2} \, dx+\frac{1}{3} \left (b c^3 d\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx+\frac{1}{3} \left (4 b c^3 d\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{b^2 c^2 d}{3 x}-\frac{b c d \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{2 c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b c^3 d\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (4 b c^3 d\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac{b^2 c^2 d}{3 x}-\frac{b c d \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{2 c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{10}{3} b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{1}{3} \left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )-\frac{1}{3} \left (4 b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (4 b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac{b^2 c^2 d}{3 x}-\frac{b c d \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{2 c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{10}{3} b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{1}{3} \left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\frac{1}{3} \left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )-\frac{1}{3} \left (4 b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\frac{1}{3} \left (4 b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=-\frac{b^2 c^2 d}{3 x}-\frac{b c d \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{2 c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac{d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{10}{3} b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{5}{3} b^2 c^3 d \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac{5}{3} b^2 c^3 d \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.793836, size = 245, normalized size = 1.55 \[ -\frac{d \left (-5 b^2 c^3 x^3 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )+5 b^2 c^3 x^3 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+3 a^2 c^2 x^2+a^2+a b c x \sqrt{c^2 x^2+1}+6 a b c^2 x^2 \sinh ^{-1}(c x)+5 a b c^3 x^3 \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )+2 a b \sinh ^{-1}(c x)+b^2 c^2 x^2+3 b^2 c^2 x^2 \sinh ^{-1}(c x)^2+b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-5 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+5 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+b^2 \sinh ^{-1}(c x)^2\right )}{3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.238, size = 278, normalized size = 1.8 \begin{align*} -{\frac{{c}^{2}d{a}^{2}}{x}}-{\frac{d{a}^{2}}{3\,{x}^{3}}}-{\frac{{c}^{2}d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{x}}-{\frac{cd{b}^{2}{\it Arcsinh} \left ( cx \right ) }{3\,{x}^{2}}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{{c}^{2}d{b}^{2}}{3\,x}}-{\frac{5\,{c}^{3}d{b}^{2}{\it Arcsinh} \left ( cx \right ) }{3}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{5\,{c}^{3}d{b}^{2}}{3}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{5\,{c}^{3}d{b}^{2}{\it Arcsinh} \left ( cx \right ) }{3}\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{5\,{c}^{3}d{b}^{2}}{3}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-2\,{\frac{{c}^{2}dab{\it Arcsinh} \left ( cx \right ) }{x}}-{\frac{2\,dab{\it Arcsinh} \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{5\,{c}^{3}dab}{3}{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ) }-{\frac{cdab}{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*} -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 c^{2} d + \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 - \frac{a^{2} c^{2} d}{x} - \frac{a^{2} d}{3 \, x^{3}} - \frac{{\left (3 \, b^{2} c^{2} d x^{2} + b^{2} d\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{3 \, x^{3}} + \int \frac{2 \,{\left (3 \, b^{2} c^{5} d x^{4} + 4 \, b^{2} c^{3} d x^{2} + b^{2} c d +{\left (3 \, b^{2} c^{4} d x^{3} + b^{2} c^{2} d 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^{2} d x^{2} + a^{2} d +{\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} + a b d\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 \left (\int \frac{a^{2}}{x^{4}}\, dx + \int \frac{a^{2} c^{2}}{x^{2}}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{b^{2} c^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{2 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 )}{\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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