Optimal. Leaf size=267 \[ \frac{3}{2} b c^2 d^3 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{7}{16} b c^3 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}-\frac{3}{32} b c^3 d^3 x \sqrt{c x-1} \sqrt{c x+1}+\frac{3}{32} b c^2 d^3 \cosh ^{-1}(c x)+\frac{b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{2 x} \]
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Rubi [A] time = 0.317803, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {5729, 97, 12, 38, 52, 5727, 5660, 3718, 2190, 2279, 2391} \[ -\frac{3}{2} b c^2 d^3 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{7}{16} b c^3 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}-\frac{3}{32} b c^3 d^3 x \sqrt{c x-1} \sqrt{c x+1}+\frac{3}{32} b c^2 d^3 \cosh ^{-1}(c x)+\frac{b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{2 x} \]
Warning: Unable to verify antiderivative.
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Rule 5729
Rule 97
Rule 12
Rule 38
Rule 52
Rule 5727
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d\right ) \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx-\frac{1}{2} \left (b c d^3\right ) \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2}}{x^2} \, dx\\ &=\frac{b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^2\right ) \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx-\frac{1}{2} \left (b c d^3\right ) \int 5 c^2 (-1+c x)^{3/2} (1+c x)^{3/2} \, dx+\frac{1}{4} \left (3 b c^3 d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=\frac{3}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx-\frac{1}{16} \left (9 b c^3 d^3\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx-\frac{1}{2} \left (3 b c^3 d^3\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx-\frac{1}{2} \left (5 b c^3 d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=-\frac{33}{32} b c^3 d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )+\frac{1}{32} \left (9 b c^3 d^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx+\frac{1}{4} \left (3 b c^3 d^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx+\frac{1}{8} \left (15 b c^3 d^3\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx\\ &=-\frac{3}{32} b c^3 d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac{33}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-\left (6 c^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{1}{16} \left (15 b c^3 d^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{3}{32} b c^3 d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac{3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (3 b c^2 d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac{3}{32} b c^3 d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac{3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac{1}{2} \left (3 b c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=-\frac{3}{32} b c^3 d^3 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac{3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac{3}{2} b c^2 d^3 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.351571, size = 226, normalized size = 0.85 \[ -\frac{d^3 \left (-48 b c^2 x^2 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+8 a c^6 x^6-48 a c^4 x^4+96 a c^2 x^2 \log (x)+16 a-2 b c^5 x^5 \sqrt{c x-1} \sqrt{c x+1}+21 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+48 b c^2 x^2 \cosh ^{-1}(c x)^2+42 b c^2 x^2 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )+8 b \cosh ^{-1}(c x) \left (c^6 x^6-6 c^4 x^4+12 c^2 x^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )+2\right )-16 b c x \sqrt{c x-1} \sqrt{c x+1}\right )}{32 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.306, size = 275, normalized size = 1. \begin{align*} -{\frac{{c}^{6}{d}^{3}a{x}^{4}}{4}}+{\frac{3\,{c}^{4}{d}^{3}a{x}^{2}}{2}}-3\,{c}^{2}{d}^{3}a\ln \left ( cx \right ) -{\frac{{d}^{3}a}{2\,{x}^{2}}}-{\frac{{d}^{3}b{c}^{2}}{2}}+{\frac{3\,{c}^{2}{d}^{3}b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}-{\frac{21\,b{c}^{2}{d}^{3}{\rm arccosh} \left (cx\right )}{32}}-{\frac{3\,{d}^{3}b{c}^{2}}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }+{\frac{{d}^{3}bc}{2\,x}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{3\,{c}^{4}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{2}}{2}}-{\frac{b{d}^{3}{\rm arccosh} \left (cx\right )}{2\,{x}^{2}}}-{\frac{{c}^{6}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{4}}{4}}+{\frac{{c}^{5}{d}^{3}b{x}^{3}}{16}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{21\,{d}^{3}b{c}^{3}x}{32}\sqrt{cx-1}\sqrt{cx+1}}-3\,{c}^{2}{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}{4} \, a c^{6} d^{3} x^{4} + \frac{3}{2} \, a c^{4} d^{3} x^{2} - 3 \, a c^{2} d^{3} \log \left (x\right ) + \frac{1}{2} \, b d^{3}{\left (\frac{\sqrt{c^{2} x^{2} - 1} c}{x} - \frac{\operatorname{arcosh}\left (c x\right )}{x^{2}}\right )} - \frac{a d^{3}}{2 \, x^{2}} - \int b c^{6} d^{3} x^{3} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) - 3 \, b c^{4} d^{3} x \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + \frac{3 \, b c^{2} 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^{3}}, 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^{3}}\, dx + \int \frac{3 a c^{2}}{x}\, dx + \int - 3 a c^{4} x\, dx + \int a c^{6} x^{3}\, dx + \int - \frac{b \operatorname{acosh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{3 b c^{2} \operatorname{acosh}{\left (c x \right )}}{x}\, dx + \int - 3 b c^{4} x \operatorname{acosh}{\left (c x \right )}\, dx + \int b c^{6} x^{3} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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