Optimal. Leaf size=92 \[ -\frac {9 \sqrt {x^4+1}}{14 x^7}+\frac {1}{2 x^7 \sqrt {x^4+1}}+\frac {15 \sqrt {x^4+1}}{14 x^3}+\frac {15 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{28 \sqrt {x^4+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {290, 325, 220} \[ \frac {15 \sqrt {x^4+1}}{14 x^3}-\frac {9 \sqrt {x^4+1}}{14 x^7}+\frac {1}{2 x^7 \sqrt {x^4+1}}+\frac {15 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{28 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{2 x^7 \sqrt {1+x^4}}+\frac {9}{2} \int \frac {1}{x^8 \sqrt {1+x^4}} \, dx\\ &=\frac {1}{2 x^7 \sqrt {1+x^4}}-\frac {9 \sqrt {1+x^4}}{14 x^7}-\frac {45}{14} \int \frac {1}{x^4 \sqrt {1+x^4}} \, dx\\ &=\frac {1}{2 x^7 \sqrt {1+x^4}}-\frac {9 \sqrt {1+x^4}}{14 x^7}+\frac {15 \sqrt {1+x^4}}{14 x^3}+\frac {15}{14} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {1}{2 x^7 \sqrt {1+x^4}}-\frac {9 \sqrt {1+x^4}}{14 x^7}+\frac {15 \sqrt {1+x^4}}{14 x^3}+\frac {15 \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 )}{28 \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 22, normalized size = 0.24 \[ -\frac {\, _2F_1\left (-\frac {7}{4},\frac {3}{2};-\frac {3}{4};-x^4\right )}{7 x^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1}}{x^{16} + 2 \, x^{12} + x^{8}}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{2}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 96, normalized size = 1.04 \[ \frac {x}{2 \sqrt {x^{4}+1}}+\frac {15 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )}{14 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {4 \sqrt {x^{4}+1}}{7 x^{3}}-\frac {\sqrt {x^{4}+1}}{7 x^{7}} \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{2}} x^{8}}\,{d x} \]
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 \[ \int \frac {1}{x^8\,{\left (x^4+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.06, size = 36, normalized size = 0.39 \[ \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {3}{2} \\ - \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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