Integrand size = 10, antiderivative size = 131 \[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=-\frac {1}{2} \text {arcsinh}(a+b x)^2+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \]
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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 = {5859, 5827, 5680, 2221, 2317, 2438} \[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{2} \text {arcsinh}(a+b x)^2 \]
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Rule 2221
Rule 2317
Rule 2438
Rule 5680
Rule 5827
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {x \cosh (x)}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = -\frac {1}{2} \text {arcsinh}(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 {arcsinh}(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 {arcsinh}(a+b x)\right )}{b} \\ & = -\frac {1}{2} \text {arcsinh}(a+b x)^2+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(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 {arcsinh}(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 {arcsinh}(a+b x)\right ) \\ & = -\frac {1}{2} \text {arcsinh}(a+b x)^2+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(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 {arcsinh}(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 {arcsinh}(a+b x)}\right ) \\ & = -\frac {1}{2} \text {arcsinh}(a+b x)^2+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.17 \[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=-\frac {1}{2} \text {arcsinh}(a+b x)^2+\text {arcsinh}(a+b x) \log \left (1+\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+\text {arcsinh}(a+b x) \log \left (1+\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+\operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(387\) vs. \(2(153)=306\).
Time = 0.66 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.96
method | result | size |
derivativedivides | \(-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{2}+\frac {\left (a^{2}+1+\sqrt {a^{2}+1}\, a \right ) \operatorname {arcsinh}\left (b x +a \right ) \left (2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) a^{2}+\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) \sqrt {a^{2}+1}\, a +\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{a^{2}+1}+\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )+\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )+\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}\) | \(388\) |
default | \(-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{2}+\frac {\left (a^{2}+1+\sqrt {a^{2}+1}\, a \right ) \operatorname {arcsinh}\left (b x +a \right ) \left (2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) a^{2}+\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) \sqrt {a^{2}+1}\, a +\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{a^{2}+1}+\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )+\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )+\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}\) | \(388\) |
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\[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )}{x} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x}\, dx \]
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\[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )}{x} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x} \,d x \]
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