Optimal. Leaf size=287 \[ \frac{3 i b c \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{3 i b c \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{2 b^2 c \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{2 b^2 c \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{3 i b^2 c \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{3 i b^2 c \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 x^2+1}}-\frac{3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (c^2 x^2+1\right )}-\frac{3 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}-\frac{4 b c \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac{b^2 c \tan ^{-1}(c x)}{d^2} \]
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Rubi [A] time = 0.543779, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 14, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {5747, 5690, 5693, 4180, 2531, 2282, 6589, 5717, 203, 5755, 5760, 4182, 2279, 2391} \[ \frac{3 i b c \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{3 i b c \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{2 b^2 c \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{2 b^2 c \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{3 i b^2 c \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{3 i b^2 c \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 x^2+1}}-\frac{3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (c^2 x^2+1\right )}-\frac{3 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}-\frac{4 b c \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac{b^2 c \tan ^{-1}(c x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5690
Rule 5693
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5717
Rule 203
Rule 5755
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 )^2} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (1+c^2 x^2\right )}-\left (3 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx+\frac{(2 b c) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}\\ &=\frac{2 b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (1+c^2 x^2\right )}-\frac{3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac{(2 b c) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{d^2}-\frac{\left (2 b^2 c^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{d^2}+\frac{\left (3 b c^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}-\frac{\left (3 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{2 d}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (1+c^2 x^2\right )}-\frac{3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac{2 b^2 c \tan ^{-1}(c x)}{d^2}-\frac{(3 c) \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac{\left (3 b^2 c^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (1+c^2 x^2\right )}-\frac{3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac{3 c \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c \tan ^{-1}(c x)}{d^2}-\frac{4 b c \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{(3 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^2}-\frac{(3 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^2}-\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^2}+\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^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (1+c^2 x^2\right )}-\frac{3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac{3 c \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c \tan ^{-1}(c x)}{d^2}-\frac{4 b c \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{3 i b c \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{3 i b c \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{\left (3 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^2}+\frac{\left (3 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^2}-\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^2}+\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^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (1+c^2 x^2\right )}-\frac{3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac{3 c \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c \tan ^{-1}(c x)}{d^2}-\frac{4 b c \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{2 b^2 c \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{3 i b c \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{3 i b c \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{2 b^2 c \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{\left (3 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^2}+\frac{\left (3 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^2}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 x \left (1+c^2 x^2\right )}-\frac{3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac{3 c \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 c \tan ^{-1}(c x)}{d^2}-\frac{4 b c \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{2 b^2 c \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{3 i b c \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{3 i b c \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{2 b^2 c \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac{3 i b^2 c \text{Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac{3 i b^2 c \text{Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 7.19479, size = 549, normalized size = 1.91 \[ \frac{2 a b c \left (\frac{3}{4} i \left (2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )-\frac{1}{2} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )-\frac{3}{4} i \left (2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )-\frac{1}{2} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )\right )+\frac{\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)}{4 (-1-i c x)}-\frac{\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}}{4 (c x+i)}-\tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-\frac{\sinh ^{-1}(c x)}{c x}\right )}{d^2}+\frac{b^2 c \left (6 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-6 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+4 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-4 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+6 i \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )-6 i \text{PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )-\frac{c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}-\frac{2 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+3 i \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-3 i \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+\sinh ^{-1}(c x)^2 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+\sinh ^{-1}(c x)^2 \left (-\coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+4 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{2 d^2}-\frac{a^2 c^2 x}{2 d^2 \left (c^2 x^2+1\right )}-\frac{3 a^2 c \tan ^{-1}(c x)}{2 d^2}-\frac{a^2}{d^2 x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.255, 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 ) ^{2}}}\, 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}{2} \, a^{2}{\left (\frac{3 \, c^{2} x^{2} + 2}{c^{2} d^{2} x^{3} + d^{2} x} + \frac{3 \, c \arctan \left (c x\right )}{d^{2}}\right )} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{4} d^{2} x^{6} + 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{6} + 2 \, c^{2} d^{2} x^{4} + d^{2} 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^{4} d^{2} x^{6} + 2 \, c^{2} d^{2} x^{4} + d^{2} 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^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \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 )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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