Optimal. Leaf size=13 \[ \frac {\sqrt {x^4+1}}{x} \]
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Rubi [A] time = 0.01, 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 = {449} \begin {gather*} \frac {\sqrt {x^4+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
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 = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^4+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^4+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 1}{\sqrt {x^{4} + 1} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x^{4}+1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, 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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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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sympy [C] time = 1.51, 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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