3.1018 \(\int \frac {x^2}{\sqrt {-1+x^4} (1+x^4)} \, dx\)

Optimal. Leaf size=49 \[ -\frac {1}{4} \tan ^{-1}\left (\frac {x^2+1}{x \sqrt {x^4-1}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {1-x^2}{x \sqrt {x^4-1}}\right ) \]

[Out]

-1/4*arctan((x^2+1)/x/(x^4-1)^(1/2))-1/4*arctanh((-x^2+1)/x/(x^4-1)^(1/2))

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Rubi [C]  time = 0.12, antiderivative size = 47, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {490, 1211, 222, 1699, 206, 203} \[ \left (\frac {1}{8}+\frac {i}{8}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {x^4-1}}\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {x^4-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[-1 + x^4]*(1 + x^4)),x]

[Out]

(-1/8 - I/8)*ArcTan[((1 + I)*x)/Sqrt[-1 + x^4]] + (1/8 + I/8)*ArcTanh[((1 + I)*x)/Sqrt[-1 + x^4]]

Rule 203

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

Rule 206

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

Rule 222

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]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]), x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1211

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

Rule 1699

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}{\sqrt {-1+x^4} \left (1+x^4\right )} \, dx &=-\left (\frac {1}{2} \int \frac {1}{\left (i-x^2\right ) \sqrt {-1+x^4}} \, dx\right )+\frac {1}{2} \int \frac {1}{\left (i+x^2\right ) \sqrt {-1+x^4}} \, dx\\ &=-\left (\frac {1}{4} i \int \frac {i-x^2}{\left (i+x^2\right ) \sqrt {-1+x^4}} \, dx\right )+\frac {1}{4} i \int \frac {i+x^2}{\left (i-x^2\right ) \sqrt {-1+x^4}} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{i-2 x^2} \, dx,x,\frac {x}{\sqrt {-1+x^4}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{i+2 x^2} \, dx,x,\frac {x}{\sqrt {-1+x^4}}\right )\\ &=\left (-\frac {1}{8}-\frac {i}{8}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )+\left (\frac {1}{8}+\frac {i}{8}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 46, normalized size = 0.94 \[ \frac {x^3 \sqrt {1-x^4} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};x^4,-x^4\right )}{3 \sqrt {x^4-1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^3*Sqrt[1 - x^4]*AppellF1[3/4, 1/2, 1, 7/4, x^4, -x^4])/(3*Sqrt[-1 + x^4])

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fricas [A]  time = 0.66, size = 51, normalized size = 1.04 \[ \frac {1}{4} \, \arctan \left (\frac {\sqrt {x^{4} - 1} x}{x^{2} + 1}\right ) + \frac {1}{8} \, \log \left (\frac {x^{4} + 2 \, x^{2} + 2 \, \sqrt {x^{4} - 1} x - 1}{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arctan(sqrt(x^4 - 1)*x/(x^2 + 1)) + 1/8*log((x^4 + 2*x^2 + 2*sqrt(x^4 - 1)*x - 1)/(x^4 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (x^{4} + 1\right )} \sqrt {x^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.03, size = 88, normalized size = 1.80 \[ -\frac {\arctan \left (-\frac {\sqrt {x^{4}-1}}{x}+1\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{8}+\frac {\ln \left (\frac {\frac {\sqrt {x^{4}-1}}{x}+\frac {x^{4}-1}{2 x^{2}}+1}{-\frac {\sqrt {x^{4}-1}}{x}+\frac {x^{4}-1}{2 x^{2}}+1}\right )}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*arctan((x^4-1)^(1/2)/x+1)-1/8*arctan(-(x^4-1)^(1/2)/x+1)+1/16*ln((1/2*(x^4-1)/x^2+(x^4-1)^(1/2)/x+1)/(1/2*
(x^4-1)/x^2-(x^4-1)^(1/2)/x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (x^{4} + 1\right )} \sqrt {x^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{\sqrt {x^4-1}\,\left (x^4+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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