Optimal. Leaf size=130 \[ \frac {1}{10} x^6 \sqrt {1+x^8}-\frac {3 x^2 \sqrt {1+x^8}}{10 \left (1+x^4\right )}+\frac {3 \left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{10 \sqrt {1+x^8}}-\frac {3 \left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{20 \sqrt {1+x^8}} \]
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Rubi [A]
time = 0.04, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {281, 327, 311,
226, 1210} \begin {gather*} -\frac {3 \left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \text {ArcTan}\left (x^2\right )|\frac {1}{2}\right )}{20 \sqrt {x^8+1}}+\frac {3 \left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} E\left (2 \text {ArcTan}\left (x^2\right )|\frac {1}{2}\right )}{10 \sqrt {x^8+1}}+\frac {1}{10} \sqrt {x^8+1} x^6-\frac {3 \sqrt {x^8+1} x^2}{10 \left (x^4+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 281
Rule 311
Rule 327
Rule 1210
Rubi steps
\begin {align*} \int \frac {x^{13}}{\sqrt {1+x^8}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^6}{\sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=\frac {1}{10} x^6 \sqrt {1+x^8}-\frac {3}{10} \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=\frac {1}{10} x^6 \sqrt {1+x^8}-\frac {3}{10} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^2\right )+\frac {3}{10} \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=\frac {1}{10} x^6 \sqrt {1+x^8}-\frac {3 x^2 \sqrt {1+x^8}}{10 \left (1+x^4\right )}+\frac {3 \left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{10 \sqrt {1+x^8}}-\frac {3 \left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{20 \sqrt {1+x^8}}\\ \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 = 34, normalized size = 0.26 \begin {gather*} \frac {1}{10} x^6 \left (\sqrt {1+x^8}-\, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-x^8\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.20, size = 17, normalized size = 0.13
method | result | size |
meijerg | \(\frac {x^{14} \hypergeom \left (\left [\frac {1}{2}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -x^{8}\right )}{14}\) | \(17\) |
risch | \(\frac {x^{6} \sqrt {x^{8}+1}}{10}-\frac {x^{6} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{8}\right )}{10}\) | \(30\) |
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 [C] Result contains complex when optimal does not.
time = 0.09, size = 55, normalized size = 0.42 \begin {gather*} \frac {-3 i \, \sqrt {i} x^{2} E(\arcsin \left (\frac {\sqrt {i}}{x^{2}}\right )\,|\,-1) + 3 i \, \sqrt {i} x^{2} F(\arcsin \left (\frac {\sqrt {i}}{x^{2}}\right )\,|\,-1) + \sqrt {x^{8} + 1} {\left (x^{8} - 3\right )}}{10 \, x^{2}} \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 = 0.67, size = 29, normalized size = 0.22 \begin {gather*} \frac {x^{14} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {11}{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 {x^{13}}{\sqrt {x^8+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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