Integrand size = 10, antiderivative size = 46 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\sqrt {1+x}}{6 x^{3/2}}+\frac {\sqrt {1+x}}{3 \sqrt {x}}-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2 x^2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 46, 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 = {5875, 12, 47, 37} \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2 x^2}-\frac {\sqrt {x+1}}{6 x^{3/2}}+\frac {\sqrt {x+1}}{3 \sqrt {x}} \]
[In]
[Out]
Rule 12
Rule 37
Rule 47
Rule 5875
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{2} \int \frac {1}{2 x^{5/2} \sqrt {1+x}} \, dx \\ & = -\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{x^{5/2} \sqrt {1+x}} \, dx \\ & = -\frac {\sqrt {1+x}}{6 x^{3/2}}-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{6} \int \frac {1}{x^{3/2} \sqrt {1+x}} \, dx \\ & = -\frac {\sqrt {1+x}}{6 x^{3/2}}+\frac {\sqrt {1+x}}{3 \sqrt {x}}-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2 x^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {\sqrt {x} \sqrt {1+x} (-1+2 x)-3 \text {arcsinh}\left (\sqrt {x}\right )}{6 x^2} \]
[In]
[Out]
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(-\frac {\operatorname {arcsinh}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {1+x}}{6 x^{\frac {3}{2}}}+\frac {\sqrt {1+x}}{3 \sqrt {x}}\) | \(31\) |
default | \(-\frac {\operatorname {arcsinh}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {1+x}}{6 x^{\frac {3}{2}}}+\frac {\sqrt {1+x}}{3 \sqrt {x}}\) | \(31\) |
parts | \(-\frac {\operatorname {arcsinh}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {1+x}}{6 x^{\frac {3}{2}}}+\frac {\sqrt {1+x}}{3 \sqrt {x}}\) | \(31\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {{\left (2 \, x - 1\right )} \sqrt {x + 1} \sqrt {x} - 3 \, \log \left (\sqrt {x + 1} + \sqrt {x}\right )}{6 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\operatorname {asinh}{\left (\sqrt {x} \right )}}{x^{3}}\, dx \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.65 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {\sqrt {x + 1}}{3 \, \sqrt {x}} - \frac {\sqrt {x + 1}}{6 \, x^{\frac {3}{2}}} - \frac {\operatorname {arsinh}\left (\sqrt {x}\right )}{2 \, x^{2}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\log \left (\sqrt {x + 1} + \sqrt {x}\right )}{2 \, x^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left ({\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} - 1\right )}^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\mathrm {asinh}\left (\sqrt {x}\right )}{x^3} \,d x \]
[In]
[Out]