3.11.7 \(\int \frac {x^2}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx\) [1007]

Optimal. Leaf size=17 \[ -\frac {1}{2} E\left (\left .\cos ^{-1}(x)\right |2\right )-\frac {1}{2} F\left (\left .\cos ^{-1}(x)\right |2\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {507, 436, 431} \begin {gather*} -\frac {F(\text {ArcCos}(x)|2)}{2}-\frac {1}{2} E(\text {ArcCos}(x)|2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[1 - x^2]*Sqrt[-1 + 2*x^2]),x]

[Out]

-1/2*EllipticE[ArcCos[x], 2] - EllipticF[ArcCos[x], 2]/2

Rule 431

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(-(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a -
b*(c/d)])^(-1))*EllipticF[ArcCos[Rt[-d/c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c
] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]

Rule 436

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

Rule 507

Int[(x_)^(n_)/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/b, Int[Sqrt[a +
b*x^n]/Sqrt[c + d*x^n], x], x] - Dist[a/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c,
d}, x] && NeQ[b*c - a*d, 0] && (EqQ[n, 2] || EqQ[n, 4]) &&  !(EqQ[n, 2] && SimplerSqrtQ[-b/a, -d/c])

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(17)=34\).
time = 0.27, size = 37, normalized size = 2.18 \begin {gather*} \frac {\sqrt {1-2 x^2} \left (-E\left (\left .\sin ^{-1}(x)\right |2\right )+F\left (\left .\sin ^{-1}(x)\right |2\right )\right )}{2 \sqrt {-1+2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 34, normalized size = 2.00

method	result	size
default	\(\frac {\left (\EllipticF \left (x , \sqrt {2}\right )-\EllipticE \left (x , \sqrt {2}\right )\right ) \sqrt {-2 x^{2}+1}}{2 \sqrt {2 x^{2}-1}}\)	\(34\)
elliptic	\(\frac {\sqrt {-\left (2 x^{2}-1\right ) \left (x^{2}-1\right )}\, \sqrt {-2 x^{2}+1}\, \left (\EllipticF \left (x , \sqrt {2}\right )-\EllipticE \left (x , \sqrt {2}\right )\right )}{2 \sqrt {2 x^{2}-1}\, \sqrt {-2 x^{4}+3 x^{2}-1}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(EllipticF(x,2^(1/2))-EllipticE(x,2^(1/2)))*(-2*x^2+1)^(1/2)/(2*x^2-1)^(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)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.53, size = 23, normalized size = 1.35 \begin {gather*} -\frac {\sqrt {2 \, x^{2} - 1} \sqrt {-x^{2} + 1}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2*x^2 - 1)*sqrt(-x^2 + 1)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(sqrt(-(x - 1)*(x + 1))*sqrt(2*x**2 - 1)), 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)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x^2}{\sqrt {1-x^2}\,\sqrt {2\,x^2-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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