Optimal. Leaf size=35 \[ \frac {x^3}{2 \sqrt {1-x^4}}-\frac {3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {294, 313, 227,
1195, 435} \begin {gather*} \frac {3}{2} F(\text {ArcSin}(x)|-1)-\frac {3}{2} E(\text {ArcSin}(x)|-1)+\frac {x^3}{2 \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 294
Rule 313
Rule 435
Rule 1195
Rubi steps
\begin {align*} \int \frac {x^6}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {x^3}{2 \sqrt {1-x^4}}-\frac {3}{2} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^3}{2 \sqrt {1-x^4}}+\frac {3}{2} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {3}{2} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^3}{2 \sqrt {1-x^4}}+\frac {3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {3}{2} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^3}{2 \sqrt {1-x^4}}-\frac {3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{2} 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 = 4.15, size = 31, normalized size = 0.89 \begin {gather*} x^3 \left (-\frac {1}{\sqrt {1-x^4}}+\, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};x^4\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 54, normalized size = 1.54
method | result | size |
meijerg | \(\frac {x^{7} \hypergeom \left (\left [\frac {3}{2}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x^{4}\right )}{7}\) | \(15\) |
default | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(54\) |
risch | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(54\) |
elliptic | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(54\) |
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 = 27, normalized size = 0.77 \begin {gather*} \frac {{\left (2 \, x^{4} - 3\right )} \sqrt {-x^{4} + 1}}{2 \, {\left (x^{5} - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 31, normalized size = 0.89 \begin {gather*} \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \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 = -0.03 \begin {gather*} \int \frac {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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