Optimal. Leaf size=11 \[ E\left (\left .\sin ^{-1}(x)\right |-1\right )-\text{EllipticF}\left (\sin ^{-1}(x),-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0120484, antiderivative size = 11, normalized size of antiderivative = 1., 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 = {307, 221, 1181, 424} \[ E\left (\left .\sin ^{-1}(x)\right |-1\right )-F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 307
Rule 221
Rule 1181
Rule 424
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] time = 0.0024427, size = 20, normalized size = 1.82 \[ \frac{1}{3} x^3 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};x^4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 39, normalized size = 3.6 \begin{align*} -{({\it EllipticF} \left ( x,i \right ) -{\it EllipticE} \left ( x,i \right ) )\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 1} x^{2}}{x^{4} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.715487, size = 31, normalized size = 2.82 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]