Optimal. Leaf size=11 \[ E\left (\left .\sin ^{-1}(x)\right |-1\right )-F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {313, 227, 1195,
435} \begin {gather*} E(\text {ArcSin}(x)|-1)-F(\text {ArcSin}(x)|-1) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 313
Rule 435
Rule 1195
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1-x^4}} \, dx &=-\int \frac {1}{\sqrt {1-x^4}} \, dx+\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=-F\left (\left .\sin ^{-1}(x)\right |-1\right )+\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=E\left (\left .\sin ^{-1}(x)\right |-1\right )-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 = 10.02, size = 20, normalized size = 1.82 \begin {gather*} \frac {1}{3} x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^4\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 38 vs. \(2 (11 ) = 22\).
time = 0.14, size = 39, normalized size = 3.55
method | result | size |
meijerg | \(\frac {x^{3} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x^{4}\right )}{3}\) | \(15\) |
default | \(-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\) | \(39\) |
elliptic | \(-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\) | \(39\) |
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 = 14, normalized size = 1.27 \begin {gather*} -\frac {\sqrt {-x^{4} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 31 vs. \(2 (5) = 10\).
time = 0.31, size = 31, normalized size = 2.82 \begin {gather*} \frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {7}{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 = -0.09 \begin {gather*} \int \frac {x^2}{\sqrt {1-x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________