Optimal. Leaf size=37 \[ -\frac {\sqrt {1+x^8}}{x}+\frac {3}{7} x^7 \, _2F_1\left (\frac {1}{2},\frac {7}{8};\frac {15}{8};-x^8\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {331, 371}
\begin {gather*} \frac {3}{7} x^7 \, _2F_1\left (\frac {1}{2},\frac {7}{8};\frac {15}{8};-x^8\right )-\frac {\sqrt {x^8+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 331
Rule 371
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {1+x^8}} \, dx &=-\frac {\sqrt {1+x^8}}{x}+3 \int \frac {x^6}{\sqrt {1+x^8}} \, dx\\ &=-\frac {\sqrt {1+x^8}}{x}+\frac {3}{7} x^7 \, _2F_1\left (\frac {1}{2},\frac {7}{8};\frac {15}{8};-x^8\right )\\ \end {align*}
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Mathematica [A]
time = 10.01, size = 20, normalized size = 0.54 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{8},\frac {1}{2};\frac {7}{8};-x^8\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 17, normalized size = 0.46
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {1}{8}, \frac {1}{2}\right ], \left [\frac {7}{8}\right ], -x^{8}\right )}{x}\) | \(17\) |
risch | \(\frac {3 x^{7} \hypergeom \left (\left [\frac {1}{2}, \frac {7}{8}\right ], \left [\frac {15}{8}\right ], -x^{8}\right )}{7}-\frac {\sqrt {x^{8}+1}}{x}\) | \(30\) |
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 [F]
time = 0.38, size = 19, normalized size = 0.51 \begin {gather*} {\rm integral}\left (\frac {\sqrt {x^{8} + 1}}{x^{10} + x^{2}}, x\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 = 0.33, size = 31, normalized size = 0.84 \begin {gather*} \frac {\Gamma \left (- \frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{8}, \frac {1}{2} \\ \frac {7}{8} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 x \Gamma \left (\frac {7}{8}\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 [B]
time = 1.12, size = 15, normalized size = 0.41 \begin {gather*} -\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{8},\frac {1}{2};\ \frac {7}{8};\ -x^8\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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