3.10.81 \(\int \frac {x^2}{\sqrt {-1+x^4}} \, dx\) [981]

Optimal. Leaf size=126 \[ \frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+x^4}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {312, 228, 1199} \begin {gather*} \frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4-1}}-\frac {\sqrt {2} \sqrt {x^2-1} \sqrt {x^2+1} E\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{\sqrt {x^4-1}}+\frac {x \left (x^2+1\right )}{\sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[-1 + x^4],x]

[Out]

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

Rule 228

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*b, 2]}, Simp[Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2
)/q]/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]))*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; IntegerQ[q]]
 /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 312

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x]
, x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 1199

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[e*x*((q + c*x
^2)/(c*Sqrt[a + c*x^4])), x] - Simp[Sqrt[2]*e*q*Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2)/q]/(Sqrt[-a]*c*Sqrt[a + c*x
^4]))*EllipticE[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; EqQ[c*d + e*q, 0] && IntegerQ[q]] /; FreeQ[{a,
c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]

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\\ &=\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 40, normalized size = 0.32 \begin {gather*} \frac {x^3 \sqrt {1-x^4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^4\right )}{3 \sqrt {-1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[-1 + x^4],x]

[Out]

(x^3*Sqrt[1 - x^4]*Hypergeometric2F1[1/2, 3/4, 7/4, x^4])/(3*Sqrt[-1 + x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 44, normalized size = 0.35

method result size
meijerg \(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, x^{3} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x^{4}\right )}{3 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}}\) \(33\)
default \(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\EllipticF \left (i x , i\right )-\EllipticE \left (i x , i\right )\right )}{\sqrt {x^{4}-1}}\) \(44\)
elliptic \(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\EllipticF \left (i x , i\right )-\EllipticE \left (i x , i\right )\right )}{\sqrt {x^{4}-1}}\) \(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^4-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(x^4 - 1), x)

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Fricas [A]
time = 0.07, size = 29, normalized size = 0.23 \begin {gather*} \frac {x E(\arcsin \left (\frac {1}{x}\right )\,|\,-1) - x F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) + \sqrt {x^{4} - 1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*elliptic_e(arcsin(1/x), -1) - x*elliptic_f(arcsin(1/x), -1) + sqrt(x^4 - 1))/x

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Sympy [C] Result contains complex when optimal does not.
time = 0.31, size = 27, normalized size = 0.21 \begin {gather*} - \frac {i 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}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**4)/(4*gamma(7/4))

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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^4-1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(x^4 - 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{\sqrt {x^4-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^4 - 1)^(1/2),x)

[Out]

int(x^2/(x^4 - 1)^(1/2), x)

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