Optimal. Leaf size=16 \[ \frac {\left (1+x^4\right )^{3/2}}{3 x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {531, 460}
\begin {gather*} \frac {\left (x^4+1\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 460
Rule 531
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+x^4}}{x^4} \, dx &=\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*}
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Mathematica [A]
time = 0.10, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^4\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 24, normalized size = 1.50
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 | \(\frac {\left (x^{4}+1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
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 | \(\frac {\hypergeom \left (\left [-\frac {3}{4}, -\frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{4}\right )}{3 x^{3}}+x \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{4}\right )\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{3 \, x^{3}} \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 = 1.17, size = 65, normalized size = 4.06 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 23, normalized size = 1.44 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} + 1} x + \frac {\sqrt {\frac {1}{x^{4}} + 1}}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^4+1\right )}^{3/2}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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