Optimal. Leaf size=71 \[ \frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {296, 331, 313,
227, 1195, 435} \begin {gather*} \frac {21}{10} F(\text {ArcSin}(x)|-1)-\frac {21}{10} E(\text {ArcSin}(x)|-1)-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {7 \sqrt {1-x^4}}{10 x^5}+\frac {1}{2 x^5 \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 296
Rule 313
Rule 331
Rule 435
Rule 1195
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{2 x^5 \sqrt {1-x^4}}+\frac {7}{2} \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}+\frac {21}{10} \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {21}{10} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}+\frac {21}{10} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {21}{10} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \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 = 20, normalized size = 0.28 \begin {gather*} -\frac {\, _2F_1\left (-\frac {5}{4},\frac {3}{2};-\frac {1}{4};x^4\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 82, normalized size = 1.15
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {5}{4}, \frac {3}{2}\right ], \left [-\frac {1}{4}\right ], x^{4}\right )}{5 x^{5}}\) | \(15\) |
risch | \(\frac {21 x^{8}-14 x^{4}-2}{10 x^{5} \sqrt {-x^{4}+1}}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\) | \(66\) |
default | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}-\frac {8 \sqrt {-x^{4}+1}}{5 x}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\) | \(82\) |
elliptic | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}-\frac {8 \sqrt {-x^{4}+1}}{5 x}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\) | \(82\) |
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 [A]
time = 0.08, size = 66, normalized size = 0.93 \begin {gather*} -\frac {21 \, {\left (x^{9} - x^{5}\right )} E(\arcsin \left (x\right )\,|\,-1) - 21 \, {\left (x^{9} - x^{5}\right )} F(\arcsin \left (x\right )\,|\,-1) + {\left (21 \, x^{8} - 14 \, x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{10 \, {\left (x^{9} - x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.50, size = 37, normalized size = 0.52 \begin {gather*} \frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{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 {1}{x^6\,{\left (1-x^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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