Optimal. Leaf size=204 \[ \frac{2 i b c \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{2 i b c \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{2 b^2 c \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 b^2 c \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{2 i b^2 c \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b^2 c \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac{2 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d}-\frac{4 b c \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d} \]
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Rubi [A] time = 0.335271, antiderivative size = 204, 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 = {5747, 5693, 4180, 2531, 2282, 6589, 5760, 4182, 2279, 2391} \[ \frac{2 i b c \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{2 i b c \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{2 b^2 c \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 b^2 c \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{2 i b^2 c \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b^2 c \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac{2 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d}-\frac{4 b c \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d} \]
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
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-c^2 \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx+\frac{(2 b c) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{d}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac{c \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac{(2 b c) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac{2 c \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{4 b c \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{(2 i b c) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac{(2 i b c) \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 b^2 c\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\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac{2 c \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{4 b c \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b c \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 \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\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\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac{\left (2 b^2 c\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\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac{2 c \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{4 b c \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{2 b^2 c \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b c \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 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 b^2 c \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{\left (2 i b^2 c\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\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac{2 c \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{4 b c \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{2 b^2 c \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b c \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 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 b^2 c \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac{2 i b^2 c \text{Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac{2 i b^2 c \text{Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{d}\\ \end{align*}
Mathematica [A] time = 1.06068, size = 363, normalized size = 1.78 \[ -\frac{\frac{1}{2} i a b c \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )\right )-\frac{1}{2} i a b c \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1-i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )\right )-b^2 c \left (2 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-2 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+2 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-2 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+2 i \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )-2 i \text{PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )-\frac{\sinh ^{-1}(c x)^2}{c x}+i \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-i \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+2 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-2 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )+a^2 c \tan ^{-1}(c x)+\frac{a^2}{x}+2 a b c \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )+\frac{2 a b \sinh ^{-1}(c x)}{x}}{d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.201, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{{x}^{2} \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*} -a^{2}{\left (\frac{c \arctan \left (c x\right )}{d} + \frac{1}{d x}\right )} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{2} d x^{4} + d x^{2}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2} d x^{4} + d x^{2}}\,{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^{4} + d 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*} \frac{\int \frac{a^{2}}{c^{2} x^{4} + x^{2}}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{2} x^{4} + x^{2}}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{c^{2} x^{4} + x^{2}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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