Integrand size = 10, antiderivative size = 92 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=-\frac {b \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)}{2 x^2}+\frac {a b^2 \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{2 \left (1+a^2\right )^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5859, 5828, 745, 739, 212} \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\frac {a b^2 \text {arctanh}\left (\frac {a (a+b x)+1}{\sqrt {a^2+1} \sqrt {(a+b x)^2+1}}\right )}{2 \left (a^2+1\right )^{3/2}}-\frac {b \sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) x}-\frac {\text {arcsinh}(a+b x)}{2 x^2} \]
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Rule 212
Rule 739
Rule 745
Rule 5828
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {arcsinh}(a+b x)}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {b \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)}{2 x^2}-\frac {(a b) \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 \left (1+a^2\right )} \\ & = -\frac {b \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)}{2 x^2}+\frac {(a b) \text {Subst}\left (\int \frac {1}{\frac {1}{b^2}+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {\frac {1}{b}+\frac {a (a+b x)}{b}}{\sqrt {1+(a+b x)^2}}\right )}{2 \left (1+a^2\right )} \\ & = -\frac {b \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)}{2 x^2}+\frac {a b^2 \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{2 \left (1+a^2\right )^{3/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=-\frac {\text {arcsinh}(a+b x)+\frac {b x \left (\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}+a b x \log (x)-a b x \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )\right )}{\left (1+a^2\right )^{3/2}}}{2 x^2} \]
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Time = 0.02 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.12
method | result | size |
parts | \(-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2 x^{2}}+\frac {b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2}\) | \(103\) |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(112\) |
default | \(b^{2} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(112\) |
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.57 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\frac {\sqrt {a^{2} + 1} a b^{2} x^{2} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + \sqrt {a^{2} + 1} a + 1\right )} + {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - {\left (a^{2} + 1\right )} b^{2} x^{2} + {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} b x - {\left (a^{4} - {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2} + 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}} \]
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\[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x^{3}}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.59 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\frac {1}{2} \, {\left (\frac {a b \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (a^{2} + 1\right )} x}\right )} b - \frac {\operatorname {arsinh}\left (b x + a\right )}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (78) = 156\).
Time = 0.36 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.16 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=-\frac {1}{2} \, {\left (\frac {a b \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2 \, {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a b + a^{2} {\left | b \right |} + {\left | b \right |}\right )}}{{\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )} {\left (a^{2} + 1\right )}}\right )} b - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}{2 \, x^{2}} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x^3} \,d x \]
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