3.3.65 \(\int \frac {x^2}{(1+x^2) \sqrt {-1-x^4}} \, dx\) [265]

Optimal. Leaf size=74 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1-x^4}}\right )}{2 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {-1-x^4}} \]

[Out]

-1/4*arctanh(x*2^(1/2)/(-x^4-1)^(1/2))*2^(1/2)+1/4*(x^2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arctan(x))*Ellipti
cF(sin(2*arctan(x)),1/2*2^(1/2))*((x^4+1)/(x^2+1)^2)^(1/2)/(-x^4-1)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1332, 226, 1713, 212} \begin {gather*} \frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{4 \sqrt {-x^4-1}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-x^4-1}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/((1 + x^2)*Sqrt[-1 - x^4]),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 226

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

Rule 1332

Int[(x_)^2/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[d/(2*d*e), Int[1/Sqrt[a + c*x^
4], x], x] - Dist[d/(2*d*e), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] &
& NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]

Rule 1713

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx &=\frac {1}{2} \int \frac {1}{\sqrt {-1-x^4}} \, dx-\frac {1}{2} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {-1-x^4}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {-1-x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1-x^4}}\right )}{2 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {-1-x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.08, size = 60, normalized size = 0.81 \begin {gather*} \frac {\sqrt [4]{-1} \sqrt {1+x^4} \left (-F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right )}{\sqrt {-1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(1/4)*Sqrt[1 + x^4]*(-EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] + EllipticPi[-I, I*ArcSinh[(-1)^(1/4)*x], -
1]))/Sqrt[-1 - x^4]

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Maple [C] Result contains complex when optimal does not.
time = 0.19, size = 168, normalized size = 2.27

method	result	size
default	\(\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) x , i\right )}{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) \sqrt {-x^{4}-1}}-\frac {i \sqrt {-i}\, \sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-x^{4}-1}}-\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-i}\, \sqrt {-x^{4}-1}}\)	\(168\)
elliptic	\(\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) x , i\right )}{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) \sqrt {-x^{4}-1}}-\frac {i \sqrt {-i}\, \sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-x^{4}-1}}-\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-i}\, \sqrt {-x^{4}-1}}\)	\(168\)

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/(1/2*2^(1/2)-1/2*I*2^(1/2))*(1+I*x^2)^(1/2)*(1-I*x^2)^(1/2)/(-x^4-1)^(1/2)*EllipticF((1/2*2^(1/2)-1/2*I*2^(1
/2))*x,I)-1/2*I*(-I)^(1/2)*(1+I*x^2)^(1/2)*(1-I*x^2)^(1/2)/(-x^4-1)^(1/2)*EllipticPi((-I)^(1/2)*x,-I,(-1)^(1/4
)/(-I)^(1/2))-1/2/(-I)^(1/2)*(1+I*x^2)^(1/2)*(1-I*x^2)^(1/2)/(-x^4-1)^(1/2)*EllipticPi((-I)^(1/2)*x,-I,(-1)^(1
/4)/(-I)^(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 [C] Result contains complex when optimal does not.
time = 0.10, size = 74, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, \sqrt {i} {\rm ellipticF}\left (\sqrt {i} x, -1\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} x + \sqrt {-x^{4} - 1}}{x^{2} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} x - \sqrt {-x^{4} - 1}}{x^{2} + 1}\right ) \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]

-1/2*sqrt(I)*ellipticF(sqrt(I)*x, -1) - 1/8*sqrt(2)*log((sqrt(2)*x + sqrt(-x^4 - 1))/(x^2 + 1)) + 1/8*sqrt(2)*
log(-(sqrt(2)*x - sqrt(-x^4 - 1))/(x^2 + 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (x^{2} + 1\right ) \sqrt {- x^{4} - 1}}\, 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 + 1)*sqrt(-x**4 - 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)/(-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 = -0.01 \begin {gather*} \int \frac {x^2}{\left (x^2+1\right )\,\sqrt {-x^4-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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