Optimal. Leaf size=76 \[ \frac {1}{2 x^3 \sqrt {1+x^4}}-\frac {5 \sqrt {1+x^4}}{6 x^3}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {296, 331, 226}
\begin {gather*} -\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{12 \sqrt {x^4+1}}-\frac {5 \sqrt {x^4+1}}{6 x^3}+\frac {1}{2 x^3 \sqrt {x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{2 x^3 \sqrt {1+x^4}}+\frac {5}{2} \int \frac {1}{x^4 \sqrt {1+x^4}} \, dx\\ &=\frac {1}{2 x^3 \sqrt {1+x^4}}-\frac {5 \sqrt {1+x^4}}{6 x^3}-\frac {5}{6} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {1}{2 x^3 \sqrt {1+x^4}}-\frac {5 \sqrt {1+x^4}}{6 x^3}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 22, normalized size = 0.29 \begin {gather*} -\frac {\, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};-x^4\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 84, normalized size = 1.11
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {1}{4}\right ], -x^{4}\right )}{3 x^{3}}\) | \(17\) |
risch | \(-\frac {5 x^{4}+2}{6 x^{3} \sqrt {x^{4}+1}}-\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{6 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(81\) |
default | \(-\frac {x}{2 \sqrt {x^{4}+1}}-\frac {\sqrt {x^{4}+1}}{3 x^{3}}-\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{6 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(84\) |
elliptic | \(-\frac {x}{2 \sqrt {x^{4}+1}}-\frac {\sqrt {x^{4}+1}}{3 x^{3}}-\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{6 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.08, size = 51, normalized size = 0.67 \begin {gather*} -\frac {5 \, \sqrt {i} {\left (-i \, x^{7} - i \, x^{3}\right )} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) + {\left (5 \, x^{4} + 2\right )} \sqrt {x^{4} + 1}}{6 \, {\left (x^{7} + x^{3}\right )}} \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 = 0.42, size = 32, normalized size = 0.42 \begin {gather*} \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{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 [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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\left (x^4+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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