Integrand size = 23, antiderivative size = 16 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx=\frac {\left (1+x^4\right )^{3/2}}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {531, 460} \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx=\frac {\left (x^4+1\right )^{3/2}}{3 x^3} \]
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Rule 460
Rule 531
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{x^4} \, dx \\ & = \frac {\left (1+x^4\right )^{3/2}}{3 x^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx=\frac {\left (1+x^4\right )^{3/2}}{3 x^3} \]
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Time = 0.92 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(\frac {\left (x^{4}+1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
trager | \(\frac {\left (x^{4}+1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
elliptic | \(\frac {\left (x^{4}+1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
pseudoelliptic | \(\frac {\left (x^{4}+1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {x^{8}+2 x^{4}+1}{3 x^{3} \sqrt {x^{4}+1}}\) | \(23\) |
default | \(\frac {\sqrt {x^{4}+1}\, x}{3}+\frac {\sqrt {x^{4}+1}}{3 x^{3}}\) | \(24\) |
meijerg | \(\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{4}\right )}{3 x^{3}}+x \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{4}\right )\) | \(31\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx=\frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{3 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 65, normalized size of antiderivative = 4.06 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx=\frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{3 \, x^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx=\frac {1}{3} \, \sqrt {x^{4} + 1} x + \frac {\sqrt {\frac {1}{x^{4}} + 1}}{3 \, x} \]
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Time = 0.12 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx=\frac {{\left (x^4+1\right )}^{3/2}}{3\,x^3} \]
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