Optimal. Leaf size=72 \[ \frac {5 x}{12 \sqrt {x^4+1}}+\frac {x}{6 \left (x^4+1\right )^{3/2}}+\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {x^4+1}} \]
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Rubi [A] time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {199, 220} \[ \frac {5 x}{12 \sqrt {x^4+1}}+\frac {x}{6 \left (x^4+1\right )^{3/2}}+\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 220
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx &=\frac {x}{6 \left (1+x^4\right )^{3/2}}+\frac {5}{6} \int \frac {1}{\left (1+x^4\right )^{3/2}} \, dx\\ &=\frac {x}{6 \left (1+x^4\right )^{3/2}}+\frac {5 x}{12 \sqrt {1+x^4}}+\frac {5}{12} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {x}{6 \left (1+x^4\right )^{3/2}}+\frac {5 x}{12 \sqrt {1+x^4}}+\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 )}{24 \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 49, normalized size = 0.68 \[ \frac {5}{12} x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-x^4\right )+\frac {5 x}{12 \sqrt {x^4+1}}+\frac {x}{6 \left (x^4+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1}}{x^{12} + 3 \, x^{8} + 3 \, x^{4} + 1}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 82, normalized size = 1.14 \[ \frac {x}{6 \left (x^{4}+1\right )^{\frac {3}{2}}}+\frac {5 x}{12 \sqrt {x^{4}+1}}+\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )}{12 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}} \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 12, normalized size = 0.17 \[ x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {5}{2};\ \frac {5}{4};\ -x^4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.03, size = 27, normalized size = 0.38 \[ \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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