Integrand size = 11, antiderivative size = 18 \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2054, 212} \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \]
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Rule 212
Rule 2054
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right ) \\ & = \frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(18)=36\).
Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.39 \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {2 \sqrt {x} \sqrt {1+x^3} \log \left (x^{3/2}+\sqrt {1+x^3}\right )}{3 \sqrt {x+x^4}} \]
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Time = 0.82 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.39
method | result | size |
meijerg | \(\frac {2 \,\operatorname {arcsinh}\left (x^{\frac {3}{2}}\right )}{3}\) | \(7\) |
pseudoelliptic | \(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x}}{x^{2}}\right )}{3}\) | \(15\) |
default | \(\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{3}\) | \(21\) |
trager | \(\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{3}\) | \(21\) |
elliptic | \(-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(290\) |
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Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {1}{3} \, \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) \]
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\[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\int \frac {x}{\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]
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\[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\int { \frac {x}{\sqrt {x^{4} + x}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\frac {1}{3} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{3} \, \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {x}{\sqrt {x+x^4}} \, dx=\int \frac {x}{\sqrt {x^4+x}} \,d x \]
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