Integrand size = 23, antiderivative size = 74 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {(a+b \text {arccosh}(c x))^2}{2 b c^2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^2 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^2 d} \]
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Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5913, 3797, 2221, 2317, 2438} \[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {(a+b \text {arccosh}(c x))^2}{2 b c^2 d}-\frac {\log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c^2 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^2 d} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5913
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}(\int (a+b x) \coth (x) \, dx,x,\text {arccosh}(c x))}{c^2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{2 b c^2 d}+\frac {2 \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\text {arccosh}(c x)\right )}{c^2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{2 b c^2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^2 d}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{2 b c^2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^2 d}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 c^2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{2 b c^2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^2 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^2 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {(a+b \text {arccosh}(c x)) \left (a+b \text {arccosh}(c x)-2 b \log \left (1-e^{\text {arccosh}(c x)}\right )-2 b \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-2 b^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-2 b^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 b c^2 d} \]
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Time = 0.55 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.85
method | result | size |
parts | \(-\frac {a \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{2}}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \,c^{2}}\) | \(137\) |
derivativedivides | \(\frac {-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}}{c^{2}}\) | \(142\) |
default | \(\frac {-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}}{c^{2}}\) | \(142\) |
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a x}{c^{2} x^{2} - 1}\, dx + \int \frac {b x \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{c^{2} d x^{2} - d} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]
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