Optimal. Leaf size=118 \[ -\frac{b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac{b^2 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 b d e}+\frac{\log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e} \]
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Rubi [A] time = 0.190075, antiderivative size = 117, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {5866, 12, 5660, 3718, 2190, 2531, 2282, 6589} \[ \frac{b \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac{b^2 \text{PolyLog}\left (3,-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 b d e}+\frac{\log \left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e} \]
Warning: Unable to verify antiderivative.
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Rule 5866
Rule 12
Rule 5660
Rule 3718
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{c e+d e x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^2 \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 b d e}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^2}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{b \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{b \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{b \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{b^2 \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}\\ \end{align*}
Mathematica [A] time = 0.392605, size = 140, normalized size = 1.19 \[ \frac{-b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )-\frac{1}{2} b^2 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right )+a^2 \log (c+d x)+a b \cosh ^{-1}(c+d x)^2+2 a b \cosh ^{-1}(c+d x) \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )+\frac{1}{3} b^2 \cosh ^{-1}(c+d x)^3+b^2 \cosh ^{-1}(c+d x)^2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )}{d e} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.031, size = 263, normalized size = 2.2 \begin{align*}{\frac{{a}^{2}\ln \left ( dx+c \right ) }{de}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{3}}{3\,de}}+{\frac{{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{de}\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }+{\frac{{b}^{2}{\rm arccosh} \left (dx+c\right )}{de}{\it polylog} \left ( 2,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }-{\frac{{b}^{2}}{2\,de}{\it polylog} \left ( 3,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }-{\frac{ab \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{de}}+2\,{\frac{ab{\rm arccosh} \left (dx+c\right )\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }{de}}+{\frac{ab}{de}{\it polylog} \left ( 2,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{arcosh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcosh}\left (d x + c\right ) + a^{2}}{d e x + c e}, 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 + d x}\, dx + \int \frac{b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac{2 a b \operatorname{acosh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \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{arcosh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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