Optimal. Leaf size=45 \[ \frac {1}{2 x^3 \sqrt {1-x^4}}-\frac {5 \sqrt {1-x^4}}{6 x^3}+\frac {5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 45, 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 = {296, 331, 227}
\begin {gather*} \frac {5}{6} F(\text {ArcSin}(x)|-1)-\frac {5 \sqrt {1-x^4}}{6 x^3}+\frac {1}{2 x^3 \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{2 x^3 \sqrt {1-x^4}}+\frac {5}{2} \int \frac {1}{x^4 \sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^3 \sqrt {1-x^4}}-\frac {5 \sqrt {1-x^4}}{6 x^3}+\frac {5}{6} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x^3 \sqrt {1-x^4}}-\frac {5 \sqrt {1-x^4}}{6 x^3}+\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 = 10.01, size = 20, normalized size = 0.44 \begin {gather*} -\frac {\, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};x^4\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 59, normalized size = 1.31
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {1}{4}\right ], x^{4}\right )}{3 x^{3}}\) | \(15\) |
risch | \(\frac {5 x^{4}-2}{6 x^{3} \sqrt {-x^{4}+1}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(54\) |
default | \(\frac {x}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{3 x^{3}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(59\) |
elliptic | \(\frac {x}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{3 x^{3}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(59\) |
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 = 0.08, size = 47, normalized size = 1.04 \begin {gather*} \frac {5 \, {\left (x^{7} - x^{3}\right )} F(\arcsin \left (x\right )\,|\,-1) - {\left (5 \, x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{6 \, {\left (x^{7} - x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.44, size = 34, normalized size = 0.76 \begin {gather*} \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{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 {1}{x^4\,{\left (1-x^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________