Optimal. Leaf size=117 \[ \frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{2} b d \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+\frac {1}{4} b c d x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{4} b d \cosh ^{-1}(c x) \]
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Rubi [A] time = 0.12, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5727, 5660, 3718, 2190, 2279, 2391, 38, 52} \[ \frac {1}{2} b d \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} b c d x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{4} b d \cosh ^{-1}(c x) \]
Warning: Unable to verify antiderivative.
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Rule 38
Rule 52
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5727
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+d \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx+\frac {1}{2} (b c d) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+d \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )-\frac {1}{4} (b c d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+(2 d) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-(b d) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac {1}{2} (b d) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac {1}{2} b d \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 130, normalized size = 1.11 \[ -\frac {1}{2} a c^2 d x^2+a d \log (x)-\frac {1}{2} b c^2 d x^2 \cosh ^{-1}(c x)+\frac {1}{2} b d \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )-\text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )\right )+\frac {1}{4} b c d x \sqrt {c x-1} \sqrt {c x+1}+\frac {1}{2} b d \tanh ^{-1}\left (\frac {\sqrt {c x-1}}{\sqrt {c x+1}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a c^{2} d x^{2} - a d + {\left (b c^{2} d x^{2} - b d\right )} \operatorname {arcosh}\left (c x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 131, normalized size = 1.12 \[ -\frac {d a \,c^{2} x^{2}}{2}+d a \ln \left (c x \right )-\frac {d b \mathrm {arccosh}\left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {d b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b d \,\mathrm {arccosh}\left (c x \right )}{4}+d b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2} \]
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 \[ -\frac {1}{2} \, a c^{2} d x^{2} + a d \log \relax (x) - \int b c^{2} d x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) - \frac {b d \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x} \,d x \]
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 \[ - d \left (\int \left (- \frac {a}{x}\right )\, dx + \int a c^{2} x\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x}\right )\, dx + \int b c^{2} x \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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