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

Optimal. Leaf size=66 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b+4}+4 x}{\sqrt {4-b}}\right )}{\sqrt {4-b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b+4}-4 x}{\sqrt {4-b}}\right )}{\sqrt {4-b}} \]

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Rubi [A]  time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1161, 618, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b+4}+4 x}{\sqrt {4-b}}\right )}{\sqrt {4-b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b+4}-4 x}{\sqrt {4-b}}\right )}{\sqrt {4-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2)/(1 - b*x^2 + 4*x^4),x]

[Out]

-(ArcTan[(Sqrt[4 + b] - 4*x)/Sqrt[4 - b]]/Sqrt[4 - b]) + ArcTan[(Sqrt[4 + b] + 4*x)/Sqrt[4 - b]]/Sqrt[4 - b]

Rule 204

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1161

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rubi steps

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

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Mathematica [B]  time = 0.06, size = 134, normalized size = 2.03 \begin {gather*} \frac {\frac {\left (\sqrt {b^2-16}+b+4\right ) \tan ^{-1}\left (\frac {2 \sqrt {2} x}{\sqrt {-\sqrt {b^2-16}-b}}\right )}{\sqrt {-\sqrt {b^2-16}-b}}+\frac {\left (\sqrt {b^2-16}-b-4\right ) \tan ^{-1}\left (\frac {2 \sqrt {2} x}{\sqrt {\sqrt {b^2-16}-b}}\right )}{\sqrt {\sqrt {b^2-16}-b}}}{\sqrt {2} \sqrt {b^2-16}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((4 + b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[-b - Sqrt[-16 + b^2]]])/Sqrt[-b - Sqrt[-16 + b^2]] + ((-
4 - b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[-b + Sqrt[-16 + b^2]]])/Sqrt[-b + Sqrt[-16 + b^2]])/(Sqrt[2
]*Sqrt[-16 + b^2])

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + 2*x^2)/(1 - b*x^2 + 4*x^4),x]

[Out]

IntegrateAlgebraic[(1 + 2*x^2)/(1 - b*x^2 + 4*x^4), x]

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fricas [A]  time = 0.80, size = 120, normalized size = 1.82 \begin {gather*} \left [\frac {\log \left (\frac {4 \, x^{4} + {\left (b - 8\right )} x^{2} - 2 \, {\left (2 \, x^{3} - x\right )} \sqrt {b - 4} + 1}{4 \, x^{4} - b x^{2} + 1}\right )}{2 \, \sqrt {b - 4}}, \frac {\sqrt {-b + 4} \arctan \left (\frac {{\left (4 \, x^{3} - {\left (b - 2\right )} x\right )} \sqrt {-b + 4}}{b - 4}\right ) + \sqrt {-b + 4} \arctan \left (\frac {2 \, \sqrt {-b + 4} x}{b - 4}\right )}{b - 4}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((4*x^4 + (b - 8)*x^2 - 2*(2*x^3 - x)*sqrt(b - 4) + 1)/(4*x^4 - b*x^2 + 1))/sqrt(b - 4), (sqrt(-b + 4)
*arctan((4*x^3 - (b - 2)*x)*sqrt(-b + 4)/(b - 4)) + sqrt(-b + 4)*arctan(2*sqrt(-b + 4)*x/(b - 4)))/(b - 4)]

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giac [A]  time = 0.31, size = 80, normalized size = 1.21 \begin {gather*} \frac {{\left (b + 8\right )} \sqrt {-b + 4} \arctan \left (\frac {x}{\sqrt {-\frac {1}{8} \, b + \frac {1}{8} \, \sqrt {b^{2} - 16}}}\right )}{b^{2} + 4 \, b - 32} - \frac {{\left (b + 8\right )} \sqrt {-b + 4} \arctan \left (\frac {x}{\sqrt {-\frac {1}{8} \, b - \frac {1}{8} \, \sqrt {b^{2} - 16}}}\right )}{b^{2} + 4 \, b - 32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b + 8)*sqrt(-b + 4)*arctan(x/sqrt(-1/8*b + 1/8*sqrt(b^2 - 16)))/(b^2 + 4*b - 32) - (b + 8)*sqrt(-b + 4)*arcta
n(x/sqrt(-1/8*b - 1/8*sqrt(b^2 - 16)))/(b^2 + 4*b - 32)

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maple [B]  time = 0.03, size = 277, normalized size = 4.20 \begin {gather*} \frac {b \arctan \left (\frac {4 x}{\sqrt {-2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {\left (b -4\right ) \left (b +4\right )}\, \sqrt {-2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}-\frac {b \arctan \left (\frac {4 x}{\sqrt {-2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {\left (b -4\right ) \left (b +4\right )}\, \sqrt {-2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}+\frac {4 \arctan \left (\frac {4 x}{\sqrt {-2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {\left (b -4\right ) \left (b +4\right )}\, \sqrt {-2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}+\frac {\arctan \left (\frac {4 x}{\sqrt {-2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {-2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}-\frac {4 \arctan \left (\frac {4 x}{\sqrt {-2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {\left (b -4\right ) \left (b +4\right )}\, \sqrt {-2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}+\frac {\arctan \left (\frac {4 x}{\sqrt {-2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {-2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} + 1}{4 \, x^{4} - b x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.41, size = 24, normalized size = 0.36 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {b-4}}{2\,x^2-1}\right )}{\sqrt {b-4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-atanh((x*(b - 4)^(1/2))/(2*x^2 - 1))/(b - 4)^(1/2)

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sympy [A]  time = 0.39, size = 83, normalized size = 1.26 \begin {gather*} \frac {\sqrt {\frac {1}{b - 4}} \log {\left (x^{2} + x \left (- \frac {b \sqrt {\frac {1}{b - 4}}}{2} + 2 \sqrt {\frac {1}{b - 4}}\right ) - \frac {1}{2} \right )}}{2} - \frac {\sqrt {\frac {1}{b - 4}} \log {\left (x^{2} + x \left (\frac {b \sqrt {\frac {1}{b - 4}}}{2} - 2 \sqrt {\frac {1}{b - 4}}\right ) - \frac {1}{2} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(1/(b - 4))*log(x**2 + x*(-b*sqrt(1/(b - 4))/2 + 2*sqrt(1/(b - 4))) - 1/2)/2 - sqrt(1/(b - 4))*log(x**2 +
x*(b*sqrt(1/(b - 4))/2 - 2*sqrt(1/(b - 4))) - 1/2)/2

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