Optimal. Leaf size=159 \[ -\frac{3 b^2 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d e}-\frac{3 b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e}-\frac{3 b^3 \text{PolyLog}\left (4,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{4 d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac{\log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e} \]
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Rubi [A] time = 0.216049, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {5866, 12, 5660, 3718, 2190, 2531, 6609, 2282, 6589} \[ -\frac{3 b^2 \text{PolyLog}\left (3,-e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d e}+\frac{3 b \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e}+\frac{3 b^3 \text{PolyLog}\left (4,-e^{2 \cosh ^{-1}(c+d x)}\right )}{4 d e}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac{\log \left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e} \]
Warning: Unable to verify antiderivative.
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Rule 5866
Rule 12
Rule 5660
Rule 3718
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{c e+d e x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^3 \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^3}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \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 )^4}{4 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \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 )^4}{4 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c+d x)}\right )}{4 d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}+\frac{3 b^3 \text{Li}_4\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{4 d e}\\ \end{align*}
Mathematica [A] time = 0.531425, size = 217, normalized size = 1.36 \[ \frac{-6 b^2 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )-6 b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2-3 b^3 \text{PolyLog}\left (4,-e^{-2 \cosh ^{-1}(c+d x)}\right )+6 a^2 b \cosh ^{-1}(c+d x)^2+12 a^2 b \cosh ^{-1}(c+d x) \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )+4 a^3 \log (c+d x)+4 a b^2 \cosh ^{-1}(c+d x)^3+12 a b^2 \cosh ^{-1}(c+d x)^2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )+b^3 \cosh ^{-1}(c+d x)^4+4 b^3 \cosh ^{-1}(c+d x)^3 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )}{4 d e} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.035, size = 471, normalized size = 3. \begin{align*}{\frac{{a}^{3}\ln \left ( dx+c \right ) }{de}}-{\frac{{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{4}}{4\,de}}+{\frac{{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{3}}{de}\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }+{\frac{3\,{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{2\,de}{\it polylog} \left ( 2,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }-{\frac{3\,{b}^{3}{\rm arccosh} \left (dx+c\right )}{2\,de}{\it polylog} \left ( 3,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }+{\frac{3\,{b}^{3}}{4\,de}{\it polylog} \left ( 4,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }-{\frac{a{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{3}}{de}}+3\,{\frac{a{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }{de}}+3\,{\frac{a{b}^{2}{\rm arccosh} \left (dx+c\right ){\it polylog} \left ( 2,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }{de}}-{\frac{3\,a{b}^{2}}{2\,de}{\it polylog} \left ( 3,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }-{\frac{3\,{a}^{2}b \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{2\,de}}+3\,{\frac{{a}^{2}b{\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{3\,{a}^{2}b}{2\,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^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (d x + c\right ) + a^{3}}{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^{3}}{c + d x}\, dx + \int \frac{b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac{3 a b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac{3 a^{2} 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 )}^{3}}{d e x + c e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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