Optimal. Leaf size=13 \[ \frac {\sqrt {1+x^4}}{x} \]
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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460}
\begin {gather*} \frac {\sqrt {x^4+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 460
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx &=\frac {\sqrt {1+x^4}}{x}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^4}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 12, normalized size = 0.92
method | result | size |
gosper | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
default | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
trager | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
risch | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
elliptic | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
meijerg | \(\frac {x^{3} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{3}+\frac {\hypergeom \left (\left [-\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{4}\right ], -x^{4}\right )}{x}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 11, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x^{4} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 11, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x^{4} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.98, size = 61, normalized size = 4.69 \begin {gather*} \frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 11, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x^4+1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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