3.3.62 \(\int \frac {x^2}{(1-x^2) \sqrt {1-x^4}} \, dx\) [262]

Optimal. Leaf size=61 \[ \frac {x \left (1+x^2\right )}{2 \sqrt {1-x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {1-x^4}} \]

[Out]

1/2*x*(x^2+1)/(-x^4+1)^(1/2)-1/2*EllipticE(x,I)*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1270, 482, 435} \begin {gather*} \frac {x \left (x^2+1\right )}{2 \sqrt {1-x^4}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} E(\text {ArcSin}(x)|-1)}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/((1 - x^2)*Sqrt[1 - x^4]),x]

[Out]

(x*(1 + x^2))/(2*Sqrt[1 - x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticE[ArcSin[x], -1])/(2*Sqrt[1 - x^4])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 1270

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + c*x^4)^Fr
acPart[p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + (c/e
)*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (1-x^2\right ) \sqrt {1-x^4}} \, dx &=\frac {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {x^2}{\left (1-x^2\right )^{3/2} \sqrt {1+x^2}} \, dx}{\sqrt {1-x^4}}\\ &=\frac {x \left (1+x^2\right )}{2 \sqrt {1-x^4}}-\frac {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx}{2 \sqrt {1-x^4}}\\ &=\frac {x \left (1+x^2\right )}{2 \sqrt {1-x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {1-x^4}}\\ \end {align*}

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Mathematica [A]
time = 10.07, size = 37, normalized size = 0.61 \begin {gather*} \frac {x+x^3-\sqrt {1-x^4} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/((1 - x^2)*Sqrt[1 - x^4]),x]

[Out]

(x + x^3 - Sqrt[1 - x^4]*EllipticE[ArcSin[x], -1])/(2*Sqrt[1 - x^4])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (49 ) = 98\).
time = 0.16, size = 143, normalized size = 2.34

method	result	size
risch	\(\frac {x \left (x^{2}+1\right )}{2 \sqrt {-x^{4}+1}}-\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\)	\(88\)
elliptic	\(-\frac {\left (-x^{2}-1\right ) x}{2 \sqrt {\left (x^{2}-1\right ) \left (-x^{2}-1\right )}}-\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\)	\(96\)
default	\(-\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}-\frac {-x^{3}-x^{2}-x -1}{4 \sqrt {\left (-1+x \right ) \left (-x^{3}-x^{2}-x -1\right )}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}-\frac {-x^{3}+x^{2}-x +1}{4 \sqrt {\left (1+x \right ) \left (-x^{3}+x^{2}-x +1\right )}}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-x^2+1)/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*EllipticF(x,I)*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)-1/4*(-x^3-x^2-x-1)/((-1+x)*(-x^3-x^2-x-1))^(1/
2)+1/2*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(EllipticF(x,I)-EllipticE(x,I))-1/4*(-x^3+x^2-x+1)/((1+x)*(
-x^3+x^2-x+1))^(1/2)

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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.

[In]

integrate(x^2/(-x^2+1)/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-integrate(x^2/(sqrt(-x^4 + 1)*(x^2 - 1)), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^2+1)/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{2}}{x^{2} \sqrt {1 - x^{4}} - \sqrt {1 - x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-x**2+1)/(-x**4+1)**(1/2),x)

[Out]

-Integral(x**2/(x**2*sqrt(1 - x**4) - sqrt(1 - x**4)), x)

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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.

[In]

integrate(x^2/(-x^2+1)/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(-x^2/(sqrt(-x^4 + 1)*(x^2 - 1)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {x^2}{\left (x^2-1\right )\,\sqrt {1-x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^2/((x^2 - 1)*(1 - x^4)^(1/2)),x)

[Out]

-int(x^2/((x^2 - 1)*(1 - x^4)^(1/2)), x)

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