Optimal. Leaf size=76 \[ -\frac {\sqrt {1+x^4}}{7 x^7}+\frac {5 \sqrt {1+x^4}}{21 x^3}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{42 \sqrt {1+x^4}} \]
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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 = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {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 )}{42 \sqrt {x^4+1}}-\frac {\sqrt {x^4+1}}{7 x^7}+\frac {5 \sqrt {x^4+1}}{21 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^8 \sqrt {1+x^4}} \, dx &=-\frac {\sqrt {1+x^4}}{7 x^7}-\frac {5}{7} \int \frac {1}{x^4 \sqrt {1+x^4}} \, dx\\ &=-\frac {\sqrt {1+x^4}}{7 x^7}+\frac {5 \sqrt {1+x^4}}{21 x^3}+\frac {5}{21} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=-\frac {\sqrt {1+x^4}}{7 x^7}+\frac {5 \sqrt {1+x^4}}{21 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 )}{42 \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 {7}{4},\frac {1}{2};-\frac {3}{4};-x^4\right )}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 86, normalized size = 1.13
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {7}{4}, \frac {1}{2}\right ], \left [-\frac {3}{4}\right ], -x^{4}\right )}{7 x^{7}}\) | \(17\) |
default | \(-\frac {\sqrt {x^{4}+1}}{7 x^{7}}+\frac {5 \sqrt {x^{4}+1}}{21 x^{3}}+\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{21 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(86\) |
risch | \(\frac {5 x^{8}+2 x^{4}-3}{21 x^{7} \sqrt {x^{4}+1}}+\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{21 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(86\) |
elliptic | \(-\frac {\sqrt {x^{4}+1}}{7 x^{7}}+\frac {5 \sqrt {x^{4}+1}}{21 x^{3}}+\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{21 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(86\) |
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 = 0.07, size = 37, normalized size = 0.49 \begin {gather*} \frac {-5 i \, \sqrt {i} x^{7} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) + {\left (5 \, x^{4} - 3\right )} \sqrt {x^{4} + 1}}{21 \, x^{7}} \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 = 0.47, size = 36, normalized size = 0.47 \begin {gather*} \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{7} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^8\,\sqrt {x^4+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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