Integrand size = 10, antiderivative size = 50 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^3} \, dx=\frac {\sqrt {1+\frac {a^2}{x^2}}}{4 a x}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 50, 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 = {5870, 6419, 342, 327, 221} \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^3} \, dx=\frac {\sqrt {\frac {a^2}{x^2}+1}}{4 a x}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]
[In]
[Out]
Rule 221
Rule 327
Rule 342
Rule 5870
Rule 6419
Rubi steps \begin{align*} \text {integral}& = \int \frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x^3} \, dx \\ & = -\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2}-\frac {1}{2} a \int \frac {1}{\sqrt {1+\frac {a^2}{x^2}} x^4} \, dx \\ & = -\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2}+\frac {1}{2} a \text {Subst}\left (\int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt {1+\frac {a^2}{x^2}}}{4 a x}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx,x,\frac {1}{x}\right )}{4 a} \\ & = \frac {\sqrt {1+\frac {a^2}{x^2}}}{4 a x}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^3} \, dx=\frac {a \sqrt {1+\frac {a^2}{x^2}} x-\left (2 a^2+x^2\right ) \text {arcsinh}\left (\frac {a}{x}\right )}{4 a^2 x^2} \]
[In]
[Out]
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(-\frac {\frac {a^{2} \operatorname {arcsinh}\left (\frac {a}{x}\right )}{2 x^{2}}-\frac {a \sqrt {\frac {a^{2}}{x^{2}}+1}}{4 x}+\frac {\operatorname {arcsinh}\left (\frac {a}{x}\right )}{4}}{a^{2}}\) | \(46\) |
default | \(-\frac {\frac {a^{2} \operatorname {arcsinh}\left (\frac {a}{x}\right )}{2 x^{2}}-\frac {a \sqrt {\frac {a^{2}}{x^{2}}+1}}{4 x}+\frac {\operatorname {arcsinh}\left (\frac {a}{x}\right )}{4}}{a^{2}}\) | \(46\) |
parts | \(-\frac {\operatorname {arcsinh}\left (\frac {a}{x}\right )}{2 x^{2}}+\frac {\sqrt {a^{2}+x^{2}}\, \left (-\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}+x^{2}}}{x}\right ) x^{2}+\sqrt {a^{2}}\, \sqrt {a^{2}+x^{2}}\right )}{4 a \sqrt {\frac {a^{2}+x^{2}}{x^{2}}}\, x^{3} \sqrt {a^{2}}}\) | \(94\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.16 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^3} \, dx=\frac {a x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} - {\left (2 \, a^{2} + x^{2}\right )} \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right )}{4 \, a^{2} x^{2}} \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^3} \, dx=\int \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{x^{3}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (42) = 84\).
Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.94 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^3} \, dx=\frac {1}{8} \, a {\left (\frac {2 \, x \sqrt {\frac {a^{2}}{x^{2}} + 1}}{a^{2} x^{2} {\left (\frac {a^{2}}{x^{2}} + 1\right )} - a^{4}} - \frac {\log \left (x \sqrt {\frac {a^{2}}{x^{2}} + 1} + a\right )}{a^{3}} + \frac {\log \left (x \sqrt {\frac {a^{2}}{x^{2}} + 1} - a\right )}{a^{3}}\right )} - \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{2 \, x^{2}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.68 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^3} \, dx=-\frac {a {\left (\frac {\log \left (a + \sqrt {a^{2} + x^{2}}\right )}{a^{3}} - \frac {\log \left (-a + \sqrt {a^{2} + x^{2}}\right )}{a^{3}} - \frac {2 \, \sqrt {a^{2} + x^{2}}}{a^{2} x^{2}}\right )}}{8 \, \mathrm {sgn}\left (x\right )} - \frac {\log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right )}{2 \, x^{2}} \]
[In]
[Out]
Time = 2.65 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^3} \, dx=\frac {\sqrt {\frac {a^2}{x^2}+1}}{4\,a\,x}-\frac {\mathrm {asinh}\left (\frac {a}{x}\right )\,\left (\frac {x}{4\,a^2}+\frac {1}{2\,x}\right )}{x} \]
[In]
[Out]