Optimal. Leaf size=297 \[ -\frac{2 i b c^3 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}+\frac{2 i b c^3 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}+\frac{7 b^2 c^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac{7 b^2 c^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac{2 i b^2 c^3 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{2 i b^2 c^3 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac{2 c^3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d}+\frac{14 b c^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}-\frac{b^2 c^2}{3 d x} \]
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Rubi [A] time = 0.638874, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {5747, 5693, 4180, 2531, 2282, 6589, 5760, 4182, 2279, 2391, 30} \[ -\frac{2 i b c^3 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}+\frac{2 i b c^3 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}+\frac{7 b^2 c^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac{7 b^2 c^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac{2 i b^2 c^3 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{2 i b^2 c^3 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac{2 c^3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d}+\frac{14 b c^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}-\frac{b^2 c^2}{3 d x} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5693
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^4 \left (d+c^2 d x^2\right )} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}-c^2 \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )} \, dx+\frac{(2 b c) \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \sqrt{1+c^2 x^2}} \, dx}{3 d}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+c^4 \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx+\frac{\left (b^2 c^2\right ) \int \frac{1}{x^2} \, dx}{3 d}-\frac{\left (b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{3 d}-\frac{\left (2 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac{c^3 \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d}-\frac{\left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac{2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac{\left (2 i b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac{\left (2 i b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d}-\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d}+\frac{\left (2 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac{\left (2 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac{2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac{2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{\left (2 i b^2 c^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac{\left (2 i b^2 c^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac{\left (2 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{\left (2 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac{2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac{7 b^2 c^3 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac{2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{7 b^2 c^3 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac{\left (2 i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{\left (2 i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac{2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac{7 b^2 c^3 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac{2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{7 b^2 c^3 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac{2 i b^2 c^3 \text{Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{2 i b^2 c^3 \text{Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{d}\\ \end{align*}
Mathematica [B] time = 8.01868, size = 602, normalized size = 2.03 \[ \frac{2 a b \left (-\frac{1}{2} i c^4 \left (\frac{2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{c}\right )+\frac{1}{2} i c^4 \left (\frac{2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )}{c}\right )-\frac{c \sqrt{c^2 x^2+1}}{6 x^2}+\frac{1}{6} c^3 \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-c^2 \left (-c \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-\frac{\sinh ^{-1}(c x)}{x}\right )-\frac{\sinh ^{-1}(c x)}{3 x^3}\right )}{d}+\frac{b^2 c^3 \left (-56 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-48 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+48 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+56 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-48 i \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )+48 i \text{PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )-\frac{8 \sinh ^{-1}(c x)^2 \sinh ^4\left (\frac{1}{2} \sinh ^{-1}(c x)\right )}{c^3 x^3}-56 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-24 i \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+24 i \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+56 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-14 \sinh ^{-1}(c x)^2 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+4 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+14 \sinh ^{-1}(c x)^2 \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-4 \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\frac{1}{2} c x \sinh ^{-1}(c x)^2 \text{csch}^4\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}{24 d}+\frac{a^2 c^2}{d x}+\frac{a^2 c^3 \tan ^{-1}(c x)}{d}-\frac{a^2}{3 d x^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.228, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{{x}^{4} \left ({c}^{2}d{x}^{2}+d \right ) }}\, 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}{3} \,{\left (\frac{3 \, c^{3} \arctan \left (c x\right )}{d} + \frac{3 \, c^{2} x^{2} - 1}{d x^{3}}\right )} a^{2} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{2} d x^{6} + d x^{4}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2} d x^{6} + d x^{4}}\,{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} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{c^{2} d x^{6} + d 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*} \frac{\int \frac{a^{2}}{c^{2} x^{6} + x^{4}}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, 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 )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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