Optimal. Leaf size=43 \[ \frac {x^5}{2 \sqrt {1-x^4}}+\frac {5}{6} x \sqrt {1-x^4}-\frac {5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {294, 327, 227}
\begin {gather*} -\frac {5}{6} F(\text {ArcSin}(x)|-1)+\frac {5}{6} \sqrt {1-x^4} x+\frac {x^5}{2 \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {x^8}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {x^5}{2 \sqrt {1-x^4}}-\frac {5}{2} \int \frac {x^4}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^5}{2 \sqrt {1-x^4}}+\frac {5}{6} x \sqrt {1-x^4}-\frac {5}{6} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^5}{2 \sqrt {1-x^4}}+\frac {5}{6} x \sqrt {1-x^4}-\frac {5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 4.32, size = 49, normalized size = 1.14 \begin {gather*} -\frac {x \left (-5+2 x^4+5 \sqrt {1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )\right )}{6 \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 57, normalized size = 1.33
method | result | size |
meijerg | \(\frac {x^{9} \hypergeom \left (\left [\frac {3}{2}, \frac {9}{4}\right ], \left [\frac {13}{4}\right ], x^{4}\right )}{9}\) | \(15\) |
risch | \(-\frac {x \left (2 x^{4}-5\right )}{6 \sqrt {-x^{4}+1}}-\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(52\) |
default | \(\frac {x}{2 \sqrt {-x^{4}+1}}+\frac {x \sqrt {-x^{4}+1}}{3}-\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(57\) |
elliptic | \(\frac {x}{2 \sqrt {-x^{4}+1}}+\frac {x \sqrt {-x^{4}+1}}{3}-\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.07, size = 27, normalized size = 0.63 \begin {gather*} \frac {{\left (2 \, x^{5} - 5 \, x\right )} \sqrt {-x^{4} + 1}}{6 \, {\left (x^{4} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.43, size = 31, normalized size = 0.72 \begin {gather*} \frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^8}{{\left (1-x^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________