Optimal. Leaf size=193 \[ -\frac{b \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac{b \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac{b^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac{b^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac{b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 x^2+1}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac{2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}+\frac{b^2 \log \left (c^2 x^2+1\right )}{2 d^2} \]
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Rubi [A] time = 0.34, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5755, 5720, 5461, 4182, 2531, 2282, 6589, 5687, 260} \[ -\frac{b \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac{b \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac{b^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac{b^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac{b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 x^2+1}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac{2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}+\frac{b^2 \log \left (c^2 x^2+1\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 5755
Rule 5720
Rule 5461
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 5687
Rule 260
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^2} \, dx &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac{(b c) \int \frac{a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac{\int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )} \, dx}{d}\\ &=-\frac{b c x \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}{2 d^2 \left (1+c^2 x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac{\left (b^2 c^2\right ) \int \frac{x}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac{b c x \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}{2 d^2 \left (1+c^2 x^2\right )}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{2 \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c x \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}{2 d^2 \left (1+c^2 x^2\right )}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c x \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}{2 d^2 \left (1+c^2 x^2\right )}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c x \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}{2 d^2 \left (1+c^2 x^2\right )}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac{b c x \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}{2 d^2 \left (1+c^2 x^2\right )}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b^2 \text{Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac{b^2 \text{Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}\\ \end{align*}
Mathematica [C] time = 1.80625, size = 428, normalized size = 2.22 \[ \frac{2 a b \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+a b \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )\right )+a b \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1-i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )\right )+2 b^2 \left (\sinh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+\sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{-2 \sinh ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+\frac{1}{2} \log \left (c^2 x^2+1\right )+\frac{\sinh ^{-1}(c x)^2}{2 c^2 x^2+2}-\frac{c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{2}{3} \sinh ^{-1}(c x)^3-\sinh ^{-1}(c x)^2 \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )+\sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac{i \pi ^3}{24}\right )+\frac{a^2}{c^2 x^2+1}-a^2 \log \left (c^2 x^2+1\right )+2 a^2 \log (c x)-\frac{a b \left (\sqrt{c^2 x^2+1}-i \sinh ^{-1}(c x)\right )}{c x+i}-\frac{a b \left (\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right )}{c x-i}-2 a b \sinh ^{-1}(c x)^2+4 a b \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.128, size = 724, normalized size = 3.8 \begin{align*} \text{result too large to display} \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{1}{c^{2} d^{2} x^{2} + d^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{d^{2}} + \frac{2 \, \log \left (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^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{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^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}, 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^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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