Integrand size = 23, antiderivative size = 117 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \text {arccosh}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d (a+b \text {arccosh}(c x))^2}{2 b}+d (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5919, 5882, 3799, 2221, 2317, 2438, 38, 54} \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d (a+b \text {arccosh}(c x))^2}{2 b}+d \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))-\frac {1}{2} b d \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{4} b d \text {arccosh}(c x)+\frac {1}{4} b c d x \sqrt {c x-1} \sqrt {c x+1} \]
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Rule 38
Rule 54
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5919
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+d \int \frac {a+b \text {arccosh}(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 ) (a+b \text {arccosh}(c x))-\frac {d \text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b}-\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 \text {arccosh}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d (a+b \text {arccosh}(c x))^2}{2 b}-\frac {(2 d) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b} \\ & = \frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \text {arccosh}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d (a+b \text {arccosh}(c x))^2}{2 b}+d (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-d \text {Subst}\left (\int \log \left (1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arccosh}(c x)\right ) \\ & = \frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \text {arccosh}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d (a+b \text {arccosh}(c x))^2}{2 b}+d (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+\frac {1}{2} (b d) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right ) \\ & = \frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \text {arccosh}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d (a+b \text {arccosh}(c x))^2}{2 b}+d (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {1}{2} a c^2 d x^2+\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{2} b c^2 d x^2 \text {arccosh}(c x)+\frac {1}{2} b d \text {arctanh}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )+a d \log (x)+\frac {1}{2} b d \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]
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Time = 0.88 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.12
method | result | size |
parts | \(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (x \right )\right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {d b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}+\frac {b d \,\operatorname {arccosh}\left (c x \right )}{4}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(131\) |
derivativedivides | \(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {d b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b d \,\operatorname {arccosh}\left (c x \right )}{4}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(133\) |
default | \(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {d b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b d \,\operatorname {arccosh}\left (c x \right )}{4}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(133\) |
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\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=- d \left (\int \left (- \frac {a}{x}\right )\, dx + \int a c^{2} x\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x}\right )\, dx + \int b c^{2} x \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x} \,d x \]
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