Integrand size = 10, antiderivative size = 57 \[ \int \frac {\text {arcsinh}(a+b x)}{x^2} \, dx=-\frac {\text {arcsinh}(a+b x)}{x}-\frac {b \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{\sqrt {1+a^2}} \]
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Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5859, 5828, 739, 212} \[ \int \frac {\text {arcsinh}(a+b x)}{x^2} \, dx=-\frac {b \text {arctanh}\left (\frac {a (a+b x)+1}{\sqrt {a^2+1} \sqrt {(a+b x)^2+1}}\right )}{\sqrt {a^2+1}}-\frac {\text {arcsinh}(a+b x)}{x} \]
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Rule 212
Rule 739
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 )^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {arcsinh}(a+b x)}{x}+\text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {\text {arcsinh}(a+b x)}{x}-\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 ) \\ & = -\frac {\text {arcsinh}(a+b x)}{x}-\frac {b \text {arctanh}\left (\frac {b \left (\frac {1}{b}+\frac {a (a+b x)}{b}\right )}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{\sqrt {1+a^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a+b x)}{x^2} \, dx=-\frac {\text {arcsinh}(a+b x)}{x}-\frac {b \text {arctanh}\left (\frac {1+a^2+a b x}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{\sqrt {1+a^2}} \]
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Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19
method | result | size |
parts | \(-\frac {\operatorname {arcsinh}\left (b x +a \right )}{x}-\frac {b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\sqrt {a^{2}+1}}\) | \(68\) |
derivativedivides | \(b \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{b x}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\sqrt {a^{2}+1}}\right )\) | \(75\) |
default | \(b \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{b x}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\sqrt {a^{2}+1}}\right )\) | \(75\) |
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Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (51) = 102\).
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.93 \[ \int \frac {\text {arcsinh}(a+b x)}{x^2} \, dx=\frac {\sqrt {a^{2} + 1} b x \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) + {\left (a^{2} + 1\right )} x \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left (a^{2} - {\left (a^{2} + 1\right )} x + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{{\left (a^{2} + 1\right )} x} \]
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\[ \int \frac {\text {arcsinh}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (51) = 102\).
Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.95 \[ \int \frac {\text {arcsinh}(a+b x)}{x^2} \, dx=-\frac {b \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{\sqrt {a^{2} + 1}} - \frac {\operatorname {arsinh}\left (b x + a\right )}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (51) = 102\).
Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.93 \[ \int \frac {\text {arcsinh}(a+b x)}{x^2} \, dx=\frac {b \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1}} - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}{x} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{x^2} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x^2} \,d x \]
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