Integrand size = 12, antiderivative size = 178 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=-\frac {\text {arcsinh}(a+b x)^2}{x}-\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}} \]
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Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5859, 5828, 5843, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=-\frac {2 b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}+\frac {2 b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}-\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}+\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\sqrt {a^2+1}}-\frac {\text {arcsinh}(a+b x)^2}{x} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3403
Rule 5828
Rule 5843
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)^2}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {arcsinh}(a+b x)^2}{x}+2 \text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {\text {arcsinh}(a+b x)^2}{x}+2 \text {Subst}\left (\int \frac {x}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = -\frac {\text {arcsinh}(a+b x)^2}{x}+4 \text {Subst}\left (\int \frac {e^x x}{-\frac {1}{b}-\frac {2 a e^x}{b}+\frac {e^{2 x}}{b}} \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = -\frac {\text {arcsinh}(a+b x)^2}{x}+\frac {4 \text {Subst}\left (\int \frac {e^x x}{-\frac {2 a}{b}-\frac {2 \sqrt {1+a^2}}{b}+\frac {2 e^x}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{\sqrt {1+a^2}}-\frac {4 \text {Subst}\left (\int \frac {e^x x}{-\frac {2 a}{b}+\frac {2 \sqrt {1+a^2}}{b}+\frac {2 e^x}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{\sqrt {1+a^2}} \\ & = -\frac {\text {arcsinh}(a+b x)^2}{x}-\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {(2 b) \text {Subst}\left (\int \log \left (1+\frac {2 e^x}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{\sqrt {1+a^2}}+\frac {(2 b) \text {Subst}\left (\int \log \left (1+\frac {2 e^x}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{\sqrt {1+a^2}} \\ & = -\frac {\text {arcsinh}(a+b x)^2}{x}-\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {(2 b) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{\sqrt {1+a^2}}+\frac {(2 b) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{\sqrt {1+a^2}} \\ & = -\frac {\text {arcsinh}(a+b x)^2}{x}-\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=\frac {-\text {arcsinh}(a+b x) \left (\sqrt {1+a^2} \text {arcsinh}(a+b x)+2 b x \left (-\log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+\log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )\right )\right )-2 b x \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 b x \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2} x} \]
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Time = 0.25 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(b \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{b x}+\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \left (\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{\sqrt {a^{2}+1}}+\frac {2 \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {2 \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}\right )\) | \(206\) |
default | \(b \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{b x}+\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \left (\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{\sqrt {a^{2}+1}}+\frac {2 \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {2 \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}\right )\) | \(206\) |
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\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{x^2} \,d x \]
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