Optimal. Leaf size=210 \[ -\frac{i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c d^2}+\frac{i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c d^2}+\frac{i b^2 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac{i b^2 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{c d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c d^2 \sqrt{c^2 x^2+1}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 x^2+1\right )}+\frac{\tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c d^2}-\frac{b^2 \tan ^{-1}(c x)}{c d^2} \]
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Rubi [A] time = 0.245082, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {5690, 5693, 4180, 2531, 2282, 6589, 5717, 203} \[ -\frac{i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c d^2}+\frac{i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c d^2}+\frac{i b^2 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac{i b^2 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{c d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c d^2 \sqrt{c^2 x^2+1}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 x^2+1\right )}+\frac{\tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c d^2}-\frac{b^2 \tan ^{-1}(c x)}{c d^2} \]
Antiderivative was successfully verified.
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Rule 5690
Rule 5693
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5717
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac{(b c) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac{\int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{2 d}\\ &=\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c d^2 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{d^2}+\frac{\operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c d^2}\\ &=\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c d^2 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac{b^2 \tan ^{-1}(c x)}{c d^2}-\frac{(i b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c d^2}+\frac{(i b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c d^2}\\ &=\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c d^2 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac{b^2 \tan ^{-1}(c x)}{c d^2}-\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c d^2}+\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c d^2}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c d^2}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c d^2}\\ &=\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c d^2 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac{b^2 \tan ^{-1}(c x)}{c d^2}-\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c d^2}+\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c d^2}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c d^2}\\ &=\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c d^2 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac{b^2 \tan ^{-1}(c x)}{c d^2}-\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c d^2}+\frac{i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c d^2}+\frac{i b^2 \text{Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac{i b^2 \text{Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{c d^2}\\ \end{align*}
Mathematica [A] time = 1.35834, size = 403, normalized size = 1.92 \[ \frac{\frac{2 a b \left (-i \left (c^2 x^2+1\right ) \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )+i \left (c^2 x^2+1\right ) \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1}+i c^2 x^2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-i c^2 x^2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )+c x \sinh ^{-1}(c x)+i \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-i \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )}{c^3 x^2+c}+\frac{2 b^2 \left (-\frac{1}{2} i \left (2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )-2 \text{PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )+\sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\frac{c x \sinh ^{-1}(c x)^2}{2 c^2 x^2+2}+\frac{\sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}\right )}{c}+\frac{a^2 x}{c^2 x^2+1}+\frac{a^2 \tan ^{-1}(c x)}{c}}{2 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{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{x}{c^{2} d^{2} x^{2} + d^{2}} + \frac{\arctan \left (c x\right )}{c d^{2}}\right )} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{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^{4} + 2 \, c^{2} d^{2} x^{2} + d^{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^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, 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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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