Optimal. Leaf size=184 \[ -\frac{1}{2} b d^2 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{16} b c d^2 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac{11}{32} b c d^2 x \sqrt{c x-1} \sqrt{c x+1}-\frac{11}{32} b d^2 \cosh ^{-1}(c x) \]
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Rubi [A] time = 0.204203, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{1}{2} b d^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )+\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{16} b c d^2 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac{11}{32} b c d^2 x \sqrt{c x-1} \sqrt{c x+1}-\frac{11}{32} b d^2 \cosh ^{-1}(c x) \]
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 )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+d \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx-\frac{1}{4} \left (b c d^2\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=-\frac{1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx+\frac{1}{16} \left (3 b c d^2\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx+\frac{1}{2} \left (b c d^2\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx\\ &=\frac{11}{32} b c d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac{1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )-\frac{1}{32} \left (3 b c d^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{4} \left (b c d^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{11}{32} b c d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac{11}{32} b d^2 \cosh ^{-1}(c x)+\frac{1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac{11}{32} b c d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac{11}{32} b d^2 \cosh ^{-1}(c x)+\frac{1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\left (b d^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac{11}{32} b c d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac{11}{32} b d^2 \cosh ^{-1}(c x)+\frac{1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac{1}{2} \left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac{11}{32} b c d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac{11}{32} b d^2 \cosh ^{-1}(c x)+\frac{1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac{1}{2} b d^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.260816, size = 162, normalized size = 0.88 \[ \frac{1}{32} d^2 \left (-16 b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+8 a c^4 x^4-32 a c^2 x^2+32 a \log (x)-2 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+8 b \cosh ^{-1}(c x) \left (c^4 x^4-4 c^2 x^2+4 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+13 b c x \sqrt{c x-1} \sqrt{c x+1}+16 b \cosh ^{-1}(c x)^2+26 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.145, size = 201, normalized size = 1.1 \begin{align*}{\frac{{d}^{2}a{c}^{4}{x}^{4}}{4}}-{d}^{2}a{c}^{2}{x}^{2}+{d}^{2}a\ln \left ( cx \right ) +{\frac{13\,b{d}^{2}{\rm arccosh} \left (cx\right )}{32}}+{\frac{{d}^{2}b{\rm arccosh} \left (cx\right ){c}^{4}{x}^{4}}{4}}+{d}^{2}b{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) -{\frac{{d}^{2}b{c}^{3}{x}^{3}}{16}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{13\,{d}^{2}bcx}{32}\sqrt{cx-1}\sqrt{cx+1}}-{d}^{2}b{\rm arccosh} \left (cx\right ){c}^{2}{x}^{2}-{\frac{{d}^{2}b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}+{\frac{{d}^{2}b}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \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^{4} d^{2} x^{4} - a c^{2} d^{2} x^{2} + a d^{2} \log \left (x\right ) + \int b c^{4} d^{2} x^{3} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) - 2 \, b c^{2} d^{2} x \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + \frac{b d^{2} \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^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\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^{2} \left (\int \frac{a}{x}\, dx + \int - 2 a c^{2} x\, dx + \int a c^{4} x^{3}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{x}\, dx + \int - 2 b c^{2} x \operatorname{acosh}{\left (c x \right )}\, dx + \int b c^{4} 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 )}^{2}{\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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