Integrand size = 13, antiderivative size = 23 \[ \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx=\arctan \left (\sqrt [4]{1+x^2}\right )-\text {arctanh}\left (\sqrt [4]{1+x^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 65, 304, 209, 212} \[ \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx=\arctan \left (\sqrt [4]{x^2+1}\right )-\text {arctanh}\left (\sqrt [4]{x^2+1}\right ) \]
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Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^2\right ) \\ & = 2 \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^2}\right ) \\ & = -\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^2}\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^2}\right ) \\ & = \arctan \left (\sqrt [4]{1+x^2}\right )-\text {arctanh}\left (\sqrt [4]{1+x^2}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx=\arctan \left (\sqrt [4]{1+x^2}\right )-\text {arctanh}\left (\sqrt [4]{1+x^2}\right ) \]
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Time = 1.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\left (x^{2}+1\right )^{\frac {1}{4}}-1\right )}{2}-\frac {\ln \left (\left (x^{2}+1\right )^{\frac {1}{4}}+1\right )}{2}+\arctan \left (\left (x^{2}+1\right )^{\frac {1}{4}}\right )\) | \(34\) |
meijerg | \(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {\pi \sqrt {2}\, x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{2}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+2 \ln \left (x \right )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{4 \pi }\) | \(59\) |
trager | \(\frac {\ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{2}+1}-x^{2}+2 \left (x^{2}+1\right )^{\frac {1}{4}}-2}{x^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{2}+1\right )^{\frac {1}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{2}}\right )}{2}\) | \(111\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx=\arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx=\arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx=\arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 5.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx=\mathrm {atan}\left ({\left (x^2+1\right )}^{1/4}\right )-\mathrm {atanh}\left ({\left (x^2+1\right )}^{1/4}\right ) \]
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