Integrand size = 10, antiderivative size = 62 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^4} \, dx=-\frac {\sqrt {1+x}}{15 x^{5/2}}+\frac {4 \sqrt {1+x}}{45 x^{3/2}}-\frac {8 \sqrt {1+x}}{45 \sqrt {x}}-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5875, 12, 47, 37} \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^4} \, dx=-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{3 x^3}+\frac {4 \sqrt {x+1}}{45 x^{3/2}}-\frac {\sqrt {x+1}}{15 x^{5/2}}-\frac {8 \sqrt {x+1}}{45 \sqrt {x}} \]
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Rule 12
Rule 37
Rule 47
Rule 5875
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}\left (\sqrt {x}\right )}{3 x^3}+\frac {1}{3} \int \frac {1}{2 x^{7/2} \sqrt {1+x}} \, dx \\ & = -\frac {\text {arcsinh}\left (\sqrt {x}\right )}{3 x^3}+\frac {1}{6} \int \frac {1}{x^{7/2} \sqrt {1+x}} \, dx \\ & = -\frac {\sqrt {1+x}}{15 x^{5/2}}-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{3 x^3}-\frac {2}{15} \int \frac {1}{x^{5/2} \sqrt {1+x}} \, dx \\ & = -\frac {\sqrt {1+x}}{15 x^{5/2}}+\frac {4 \sqrt {1+x}}{45 x^{3/2}}-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{3 x^3}+\frac {4}{45} \int \frac {1}{x^{3/2} \sqrt {1+x}} \, dx \\ & = -\frac {\sqrt {1+x}}{15 x^{5/2}}+\frac {4 \sqrt {1+x}}{45 x^{3/2}}-\frac {8 \sqrt {1+x}}{45 \sqrt {x}}-\frac {\text {arcsinh}\left (\sqrt {x}\right )}{3 x^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.63 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^4} \, dx=\frac {\sqrt {x} \sqrt {1+x} \left (-3+4 x-8 x^2\right )-15 \text {arcsinh}\left (\sqrt {x}\right )}{45 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(-\frac {\operatorname {arcsinh}\left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {1+x}}{15 x^{\frac {5}{2}}}+\frac {4 \sqrt {1+x}}{45 x^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{45 \sqrt {x}}\) | \(41\) |
default | \(-\frac {\operatorname {arcsinh}\left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {1+x}}{15 x^{\frac {5}{2}}}+\frac {4 \sqrt {1+x}}{45 x^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{45 \sqrt {x}}\) | \(41\) |
parts | \(-\frac {\operatorname {arcsinh}\left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {1+x}}{15 x^{\frac {5}{2}}}+\frac {4 \sqrt {1+x}}{45 x^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{45 \sqrt {x}}\) | \(41\) |
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.60 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^4} \, dx=-\frac {{\left (8 \, x^{2} - 4 \, x + 3\right )} \sqrt {x + 1} \sqrt {x} + 15 \, \log \left (\sqrt {x + 1} + \sqrt {x}\right )}{45 \, x^{3}} \]
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\[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^4} \, dx=\int \frac {\operatorname {asinh}{\left (\sqrt {x} \right )}}{x^{4}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.65 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^4} \, dx=-\frac {8 \, \sqrt {x + 1}}{45 \, \sqrt {x}} + \frac {4 \, \sqrt {x + 1}}{45 \, x^{\frac {3}{2}}} - \frac {\sqrt {x + 1}}{15 \, x^{\frac {5}{2}}} - \frac {\operatorname {arsinh}\left (\sqrt {x}\right )}{3 \, x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08 \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^4} \, dx=-\frac {\log \left (\sqrt {x + 1} + \sqrt {x}\right )}{3 \, x^{3}} + \frac {16 \, {\left (10 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{4} - 5 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} + 1\right )}}{45 \, {\left ({\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} - 1\right )}^{5}} \]
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Timed out. \[ \int \frac {\text {arcsinh}\left (\sqrt {x}\right )}{x^4} \, dx=\int \frac {\mathrm {asinh}\left (\sqrt {x}\right )}{x^4} \,d x \]
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