Integrand size = 12, antiderivative size = 514 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=-\frac {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right )}-\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {3 a b^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {3 a b^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}} \]
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Time = 0.62 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5859, 5828, 5843, 3405, 3403, 2296, 2221, 2611, 2320, 6724, 5680, 2317, 2438} \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\frac {3 a b^2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {3 a b^2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{a^2+1}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{a^2+1}-\frac {3 a b^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}+\frac {3 a b^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (a^2+1\right )}+\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \left (a^2+1\right )^{3/2}}-\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \left (a^2+1\right )^{3/2}}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{a^2+1}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{a^2+1}-\frac {3 b \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2}{2 \left (a^2+1\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3403
Rule 3405
Rule 5680
Rule 5828
Rule 5843
Rule 5859
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)^3}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {\text {arcsinh}(x)^2}{\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)^3}{2 x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {x^2}{\left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2} \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = -\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {(3 b) \text {Subst}\left (\int \frac {x \cosh (x)}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{1+a^2}-\frac {(3 a b) \text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 \left (1+a^2\right )} \\ & = -\frac {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right )}-\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {(3 b) \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 )}{1+a^2}+\frac {(3 b) \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 )}{1+a^2}-\frac {(3 a b) \text {Subst}\left (\int \frac {e^x x^2}{-\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 {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right )}-\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {(3 a b) \text {Subst}\left (\int \frac {e^x x^2}{-\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 {(3 a b) \text {Subst}\left (\int \frac {e^x x^2}{-\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 {\left (3 b^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 )}{1+a^2}-\frac {\left (3 b^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 )}{1+a^2} \\ & = -\frac {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right )}-\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int x \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 (3 a b^2\right ) \text {Subst}\left (\int x \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 (3 b^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 )}{1+a^2}-\frac {\left (3 b^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 )}{1+a^2} \\ & = -\frac {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right )}-\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x) \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 {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x) \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 {\left (3 a b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\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 (3 a b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\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 {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right )}-\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x) \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 {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x) \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 {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {x}{a-\sqrt {1+a^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{\left (1+a^2\right )^{3/2}}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {x}{a+\sqrt {1+a^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{\left (1+a^2\right )^{3/2}} \\ & = -\frac {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right )}-\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x) \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 {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x) \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 {3 a b^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {3 a b^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.02 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\frac {-3 \sqrt {1+a^2} b^2 x^2 \text {arcsinh}(a+b x)^2-3 \sqrt {1+a^2} b x \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2-\sqrt {1+a^2} \text {arcsinh}(a+b x)^3-a^2 \sqrt {1+a^2} \text {arcsinh}(a+b x)^3+6 \sqrt {1+a^2} b^2 x^2 \text {arcsinh}(a+b x) \log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-3 a b^2 x^2 \text {arcsinh}(a+b x)^2 \log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+6 \sqrt {1+a^2} b^2 x^2 \text {arcsinh}(a+b x) \log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )+3 a b^2 x^2 \text {arcsinh}(a+b x)^2 \log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )+6 b^2 x^2 \left (\sqrt {1+a^2}+a \text {arcsinh}(a+b x)\right ) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+6 b^2 x^2 \left (\sqrt {1+a^2}-a \text {arcsinh}(a+b x)\right ) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-6 a b^2 x^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+6 a b^2 x^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2} x^2} \]
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\[\int \frac {\operatorname {arcsinh}\left (b x +a \right )^{3}}{x^{3}}d x\]
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\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^3}{x^3} \,d x \]
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