Integrand size = 12, antiderivative size = 268 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=-\frac {\text {arcsinh}(a+b x)^3}{x}-\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {6 b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {6 b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}} \]
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Time = 0.43 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5859, 5828, 5843, 3403, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=-\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}+\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}+\frac {6 b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}-\frac {6 b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}-\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}+\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\sqrt {a^2+1}}-\frac {\text {arcsinh}(a+b x)^3}{x} \]
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3403
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 )^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {arcsinh}(a+b x)^3}{x}+3 \text {Subst}\left (\int \frac {\text {arcsinh}(x)^2}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {\text {arcsinh}(a+b x)^3}{x}+3 \text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = -\frac {\text {arcsinh}(a+b x)^3}{x}+6 \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 ) \\ & = -\frac {\text {arcsinh}(a+b x)^3}{x}+\frac {6 \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 )}{\sqrt {1+a^2}}-\frac {6 \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 )}{\sqrt {1+a^2}} \\ & = -\frac {\text {arcsinh}(a+b x)^3}{x}-\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {(6 b) \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 )}{\sqrt {1+a^2}}+\frac {(6 b) \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 )}{\sqrt {1+a^2}} \\ & = -\frac {\text {arcsinh}(a+b x)^3}{x}-\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {(6 b) \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 )}{\sqrt {1+a^2}}+\frac {(6 b) \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 )}{\sqrt {1+a^2}} \\ & = -\frac {\text {arcsinh}(a+b x)^3}{x}-\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {(6 b) \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 )}{\sqrt {1+a^2}}-\frac {(6 b) \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 )}{\sqrt {1+a^2}} \\ & = -\frac {\text {arcsinh}(a+b x)^3}{x}-\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {6 b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {6 b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.97 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=-\frac {\sqrt {1+a^2} \text {arcsinh}(a+b x)^3-3 b x \text {arcsinh}(a+b x)^2 \log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+3 b x \text {arcsinh}(a+b x)^2 \log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )+6 b x \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )-6 b x \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-6 b x \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+6 b x \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2} x} \]
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\[\int \frac {\operatorname {arcsinh}\left (b x +a \right )^{3}}{x^{2}}d x\]
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\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^3}{x^2} \,d x \]
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