Integrand size = 10, antiderivative size = 131 \[ \int \frac {\text {arccosh}(a+b x)}{x} \, dx=-\frac {1}{2} \text {arccosh}(a+b x)^2+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a+\sqrt {-1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arccosh}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arccosh}(a+b x)}}{a+\sqrt {-1+a^2}}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5996, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {\text {arccosh}(a+b x)}{x} \, dx=\operatorname {PolyLog}\left (2,\frac {e^{\text {arccosh}(a+b x)}}{a-\sqrt {a^2-1}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arccosh}(a+b x)}}{a+\sqrt {a^2-1}}\right )+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a-\sqrt {a^2-1}}\right )+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{\sqrt {a^2-1}+a}\right )-\frac {1}{2} \text {arccosh}(a+b x)^2 \]
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Rule 2221
Rule 2317
Rule 2438
Rule 5681
Rule 5962
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arccosh}(x)}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {x \sinh (x)}{-\frac {a}{b}+\frac {\cosh (x)}{b}} \, dx,x,\text {arccosh}(a+b x)\right )}{b} \\ & = -\frac {1}{2} \text {arccosh}(a+b x)^2+\frac {\text {Subst}\left (\int \frac {e^x x}{-\frac {a}{b}-\frac {\sqrt {-1+a^2}}{b}+\frac {e^x}{b}} \, dx,x,\text {arccosh}(a+b x)\right )}{b}+\frac {\text {Subst}\left (\int \frac {e^x x}{-\frac {a}{b}+\frac {\sqrt {-1+a^2}}{b}+\frac {e^x}{b}} \, dx,x,\text {arccosh}(a+b x)\right )}{b} \\ & = -\frac {1}{2} \text {arccosh}(a+b x)^2+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a+\sqrt {-1+a^2}}\right )-\text {Subst}\left (\int \log \left (1+\frac {e^x}{\left (-\frac {a}{b}-\frac {\sqrt {-1+a^2}}{b}\right ) b}\right ) \, dx,x,\text {arccosh}(a+b x)\right )-\text {Subst}\left (\int \log \left (1+\frac {e^x}{\left (-\frac {a}{b}+\frac {\sqrt {-1+a^2}}{b}\right ) b}\right ) \, dx,x,\text {arccosh}(a+b x)\right ) \\ & = -\frac {1}{2} \text {arccosh}(a+b x)^2+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a+\sqrt {-1+a^2}}\right )-\text {Subst}\left (\int \frac {\log \left (1+\frac {x}{\left (-\frac {a}{b}-\frac {\sqrt {-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\text {arccosh}(a+b x)}\right )-\text {Subst}\left (\int \frac {\log \left (1+\frac {x}{\left (-\frac {a}{b}+\frac {\sqrt {-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\text {arccosh}(a+b x)}\right ) \\ & = -\frac {1}{2} \text {arccosh}(a+b x)^2+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+\text {arccosh}(a+b x) \log \left (1-\frac {e^{\text {arccosh}(a+b x)}}{a+\sqrt {-1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arccosh}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arccosh}(a+b x)}}{a+\sqrt {-1+a^2}}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.17 \[ \int \frac {\text {arccosh}(a+b x)}{x} \, dx=-\frac {1}{2} \text {arccosh}(a+b x)^2+\text {arccosh}(a+b x) \log \left (1+\frac {e^{\text {arccosh}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {-1+a^2}}{b}\right ) b}\right )+\text {arccosh}(a+b x) \log \left (1+\frac {e^{\text {arccosh}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {-1+a^2}}{b}\right ) b}\right )+\operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(a+b x)}}{-a+\sqrt {-1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arccosh}(a+b x)}}{a+\sqrt {-1+a^2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(435\) vs. \(2(177)=354\).
Time = 0.92 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.33
method | result | size |
derivativedivides | \(-\frac {\operatorname {arccosh}\left (b x +a \right )^{2}}{2}+\frac {a \,\operatorname {arccosh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}-\frac {a \,\operatorname {arccosh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}-\frac {\left (a^{2}-1+a \sqrt {a^{2}-1}\right ) \operatorname {arccosh}\left (b x +a \right ) \left (2 \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right ) a^{2}-\ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )-\ln \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )-2 a \sqrt {a^{2}-1}\, \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )\right )}{a^{2}-1}+\operatorname {dilog}\left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )+\operatorname {dilog}\left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )\) | \(436\) |
default | \(-\frac {\operatorname {arccosh}\left (b x +a \right )^{2}}{2}+\frac {a \,\operatorname {arccosh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}-\frac {a \,\operatorname {arccosh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}-\frac {\left (a^{2}-1+a \sqrt {a^{2}-1}\right ) \operatorname {arccosh}\left (b x +a \right ) \left (2 \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right ) a^{2}-\ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )-\ln \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )-2 a \sqrt {a^{2}-1}\, \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )\right )}{a^{2}-1}+\operatorname {dilog}\left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )+\operatorname {dilog}\left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )\) | \(436\) |
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\[ \int \frac {\text {arccosh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (b x + a\right )}{x} \,d x } \]
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\[ \int \frac {\text {arccosh}(a+b x)}{x} \, dx=\int \frac {\operatorname {acosh}{\left (a + b x \right )}}{x}\, dx \]
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\[ \int \frac {\text {arccosh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (b x + a\right )}{x} \,d x } \]
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\[ \int \frac {\text {arccosh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (b x + a\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a+b x)}{x} \, dx=\int \frac {\mathrm {acosh}\left (a+b\,x\right )}{x} \,d x \]
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