Optimal. Leaf size=74 \[ \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {\log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}-\frac {b \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 c^2 d} \]
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Rubi [A] time = 0.12, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5715, 3716, 2190, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^2 d}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {\log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5715
Rubi steps
\begin {align*} \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 d}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^2 d}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 d}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^2 d}+\frac {b \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 d}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^2 d}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^2 d}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^2 d}-\frac {b \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 85, normalized size = 1.15 \[ \frac {\left (a+b \cosh ^{-1}(c x)\right ) \left (a+b \cosh ^{-1}(c x)-2 b \log \left (1-e^{\cosh ^{-1}(c x)}\right )-2 b \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )-2 b^2 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )-2 b^2 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 b c^2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b x \operatorname {arcosh}\left (c x\right ) + a x}{c^{2} d x^{2} - d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{c^{2} d x^{2} - d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 179, normalized size = 2.42 \[ -\frac {a \ln \left (c x -1\right )}{2 c^{2} d}-\frac {a \ln \left (c x +1\right )}{2 c^{2} d}+\frac {b \mathrm {arccosh}\left (c x \right )^{2}}{2 c^{2} d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{2} d}-\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{2} d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{2} d}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, b {\left (\frac {4 \, {\left (\log \left (c x + 1\right ) + \log \left (c x - 1\right )\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) - \log \left (c x + 1\right )^{2} - 2 \, \log \left (c x + 1\right ) \log \left (c x - 1\right ) - \log \left (c x - 1\right )^{2}}{c^{2} d} + 8 \, \int \frac {\log \left (c x + 1\right ) + \log \left (c x - 1\right )}{2 \, {\left (c^{4} d x^{3} - c^{2} d x + {\left (c^{3} d x^{2} - c d\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x}\right )} - \frac {a \log \left (c^{2} d x^{2} - d\right )}{2 \, c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {a x}{c^{2} x^{2} - 1}\, dx + \int \frac {b x \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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