Optimal. Leaf size=76 \[ -\frac {5 \sqrt {x^4+1}}{6 x^3}+\frac {1}{2 x^3 \sqrt {x^4+1}}-\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {x^4+1}} \]
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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 = {290, 325, 220} \[ -\frac {5 \sqrt {x^4+1}}{6 x^3}+\frac {1}{2 x^3 \sqrt {x^4+1}}-\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
Rule 325
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] time = 0.00, size = 22, normalized size = 0.29 \[ -\frac {\, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};-x^4\right )}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1}}{x^{12} + 2 \, x^{8} + x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 84, normalized size = 1.11 \[ -\frac {x}{2 \sqrt {x^{4}+1}}-\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )}{6 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {\sqrt {x^{4}+1}}{3 x^{3}} \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,{\left (x^4+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.39, size = 32, normalized size = 0.42 \[ \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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