Integrand size = 12, antiderivative size = 235 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=-\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1+a^2}+\frac {a b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}} \]
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Time = 0.35 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5859, 5828, 5843, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=\frac {a b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {a b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}+\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\left (a^2+1\right )^{3/2}}-\frac {b \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) x}+\frac {b^2 \log (x)}{a^2+1}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2} \]
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3403
Rule 3405
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 )^3} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\text {Subst}\left (\int \frac {x}{\left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2} \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = -\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\frac {b \text {Subst}\left (\int \frac {\cosh (x)}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{1+a^2}-\frac {(a b) \text {Subst}\left (\int \frac {x}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{1+a^2} \\ & = -\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2}-\frac {(2 a b) \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 )}{1+a^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+x} \, dx,x,\frac {a}{b}+x\right )}{1+a^2} \\ & = -\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\frac {b^2 \log (x)}{1+a^2}-\frac {(2 a b) \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 )}{\left (1+a^2\right )^{3/2}}+\frac {(2 a b) \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 )}{\left (1+a^2\right )^{3/2}} \\ & = -\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1+a^2}+\frac {\left (a b^2\right ) \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 )}{\left (1+a^2\right )^{3/2}}-\frac {\left (a b^2\right ) \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 )}{\left (1+a^2\right )^{3/2}} \\ & = -\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1+a^2}+\frac {\left (a b^2\right ) \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 )}{\left (1+a^2\right )^{3/2}}-\frac {\left (a b^2\right ) \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 )}{\left (1+a^2\right )^{3/2}} \\ & = -\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1+a^2}+\frac {a b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.19 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=-\frac {2 \sqrt {1+a^2} b x \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)+\sqrt {1+a^2} \text {arcsinh}(a+b x)^2+a^2 \sqrt {1+a^2} \text {arcsinh}(a+b x)^2+2 a b^2 x^2 \text {arcsinh}(a+b x) \log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-2 a b^2 x^2 \text {arcsinh}(a+b x) \log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )-2 \sqrt {1+a^2} b^2 x^2 \log (x)-2 a b^2 x^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 a b^2 x^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2} x^2} \]
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Time = 0.46 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.59
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (-2 \left (b x +a \right )^{2}+4 a \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )+a^{2} \operatorname {arcsinh}\left (b x +a \right )-2 a \sqrt {1+\left (b x +a \right )^{2}}+2 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}-2 a^{2}\right )}{2 b^{2} x^{2} \left (a^{2}+1\right )}-\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {a \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {2 \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{a^{2}+1}+\frac {\ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{a^{2}+1}\right )\) | \(374\) |
default | \(b^{2} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (-2 \left (b x +a \right )^{2}+4 a \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )+a^{2} \operatorname {arcsinh}\left (b x +a \right )-2 a \sqrt {1+\left (b x +a \right )^{2}}+2 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}-2 a^{2}\right )}{2 b^{2} x^{2} \left (a^{2}+1\right )}-\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {a \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {2 \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{a^{2}+1}+\frac {\ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{a^{2}+1}\right )\) | \(374\) |
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\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{x^3} \,d x \]
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