Optimal. Leaf size=239 \[ -\frac{1}{2} b d^3 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^3 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{36} b c d^3 x (c x-1)^{5/2} (c x+1)^{5/2}-\frac{7}{72} b c d^3 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac{19}{48} b c d^3 x \sqrt{c x-1} \sqrt{c x+1}-\frac{19}{48} b d^3 \cosh ^{-1}(c x) \]
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Rubi [A] time = 0.300358, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {5727, 5660, 3718, 2190, 2279, 2391, 38, 52} \[ \frac{1}{2} b d^3 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^3 \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{36} b c d^3 x (c x-1)^{5/2} (c x+1)^{5/2}-\frac{7}{72} b c d^3 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac{19}{48} b c d^3 x \sqrt{c x-1} \sqrt{c x+1}-\frac{19}{48} b d^3 \cosh ^{-1}(c x) \]
Warning: Unable to verify antiderivative.
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Rule 5727
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 38
Rule 52
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )+d \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx+\frac{1}{6} \left (b c d^3\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx\\ &=\frac{1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx-\frac{1}{36} \left (5 b c d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx-\frac{1}{4} \left (b c d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=-\frac{7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx+\frac{1}{48} \left (5 b c d^3\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx+\frac{1}{16} \left (3 b c d^3\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx+\frac{1}{2} \left (b c d^3\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx\\ &=\frac{19}{48} b c d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )-\frac{1}{96} \left (5 b c d^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{32} \left (3 b c d^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{4} \left (b c d^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{19}{48} b c d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac{19}{48} b d^3 \cosh ^{-1}(c x)+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac{19}{48} b c d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac{19}{48} b d^3 \cosh ^{-1}(c x)+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\left (b d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac{19}{48} b c d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac{19}{48} b d^3 \cosh ^{-1}(c x)+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac{1}{2} \left (b d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac{19}{48} b c d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac{19}{48} b d^3 \cosh ^{-1}(c x)+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac{1}{2} b d^3 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.370836, size = 207, normalized size = 0.87 \[ -\frac{1}{144} d^3 \left (72 b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+24 a c^6 x^6-108 a c^4 x^4+216 a c^2 x^2-144 a \log (x)-4 b c^5 x^5 \sqrt{c x-1} \sqrt{c x+1}+22 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+12 b \cosh ^{-1}(c x) \left (2 c^6 x^6-9 c^4 x^4+18 c^2 x^2-12 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )-75 b c x \sqrt{c x-1} \sqrt{c x+1}-72 b \cosh ^{-1}(c x)^2-150 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.162, size = 255, normalized size = 1.1 \begin{align*} -{\frac{{d}^{3}a{c}^{6}{x}^{6}}{6}}+{\frac{3\,{d}^{3}a{c}^{4}{x}^{4}}{4}}-{\frac{3\,{d}^{3}a{c}^{2}{x}^{2}}{2}}+{d}^{3}a\ln \left ( cx \right ) +{\frac{25\,b{d}^{3}{\rm arccosh} \left (cx\right )}{48}}-{\frac{3\,{d}^{3}b{\rm arccosh} \left (cx\right ){c}^{2}{x}^{2}}{2}}-{\frac{{d}^{3}b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}-{\frac{{d}^{3}b{\rm arccosh} \left (cx\right ){c}^{6}{x}^{6}}{6}}+{\frac{3\,{d}^{3}b{\rm arccosh} \left (cx\right ){c}^{4}{x}^{4}}{4}}+{\frac{{d}^{3}b{c}^{5}{x}^{5}}{36}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{11\,{d}^{3}b{c}^{3}{x}^{3}}{72}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{25\,{d}^{3}bcx}{48}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{{d}^{3}b}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }+{d}^{3}b{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) \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}{6} \, a c^{6} d^{3} x^{6} + \frac{3}{4} \, a c^{4} d^{3} x^{4} - \frac{3}{2} \, a c^{2} d^{3} x^{2} + a d^{3} \log \left (x\right ) - \int b c^{6} d^{3} x^{5} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) - 3 \, b c^{4} d^{3} x^{3} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 3 \, b c^{2} d^{3} x \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) - \frac{b d^{3} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{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{a c^{6} d^{3} x^{6} - 3 \, a c^{4} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2} - a d^{3} +{\left (b c^{6} d^{3} x^{6} - 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} - b d^{3}\right )} \operatorname{arcosh}\left (c x\right )}{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*} - d^{3} \left (\int - \frac{a}{x}\, dx + \int 3 a c^{2} x\, dx + \int - 3 a c^{4} x^{3}\, dx + \int a c^{6} x^{5}\, dx + \int - \frac{b \operatorname{acosh}{\left (c x \right )}}{x}\, dx + \int 3 b c^{2} x \operatorname{acosh}{\left (c x \right )}\, dx + \int - 3 b c^{4} x^{3} \operatorname{acosh}{\left (c x \right )}\, dx + \int b c^{6} x^{5} \operatorname{acosh}{\left (c x \right )}\, dx\right ) \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 (c^{2} d x^{2} - d\right )}^{3}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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