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

Optimal. Leaf size=60 \[ \frac{\tanh ^{-1}\left (\frac{1-2 b x}{\sqrt{1-4 a b}}\right )}{\sqrt{1-4 a b}}-\frac{\tanh ^{-1}\left (\frac{2 b x+1}{\sqrt{1-4 a b}}\right )}{\sqrt{1-4 a b}} \]

[Out]

ArcTanh[(1 - 2*b*x)/Sqrt[1 - 4*a*b]]/Sqrt[1 - 4*a*b] - ArcTanh[(1 + 2*b*x)/Sqrt[1 - 4*a*b]]/Sqrt[1 - 4*a*b]

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Rubi [A]  time = 0.0686734, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1161, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{1-2 b x}{\sqrt{1-4 a b}}\right )}{\sqrt{1-4 a b}}-\frac{\tanh ^{-1}\left (\frac{2 b x+1}{\sqrt{1-4 a b}}\right )}{\sqrt{1-4 a b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[(1 - 2*b*x)/Sqrt[1 - 4*a*b]]/Sqrt[1 - 4*a*b] - ArcTanh[(1 + 2*b*x)/Sqrt[1 - 4*a*b]]/Sqrt[1 - 4*a*b]

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]))

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 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])

Rubi steps

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

Mathematica [B]  time = 0.199963, size = 138, normalized size = 2.3 \[ \frac{\frac{\left (\sqrt{1-4 a b}+1\right ) \tan ^{-1}\left (\frac{b x}{\sqrt{a b-\frac{1}{2} \sqrt{1-4 a b}-\frac{1}{2}}}\right )}{\sqrt{2 a b-\sqrt{1-4 a b}-1}}+\frac{\left (\sqrt{1-4 a b}-1\right ) \tan ^{-1}\left (\frac{\sqrt{2} b x}{\sqrt{2 a b+\sqrt{1-4 a b}-1}}\right )}{\sqrt{2 a b+\sqrt{1-4 a b}-1}}}{\sqrt{2-8 a b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((1 + Sqrt[1 - 4*a*b])*ArcTan[(b*x)/Sqrt[-1/2 + a*b - Sqrt[1 - 4*a*b]/2]])/Sqrt[-1 + 2*a*b - Sqrt[1 - 4*a*b]]
+ ((-1 + Sqrt[1 - 4*a*b])*ArcTan[(Sqrt[2]*b*x)/Sqrt[-1 + 2*a*b + Sqrt[1 - 4*a*b]]])/Sqrt[-1 + 2*a*b + Sqrt[1
- 4*a*b]])/Sqrt[2 - 8*a*b]

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Maple [A]  time = 0.123, size = 52, normalized size = 0.9 \begin{align*}{\arctan \left ({(2\,bx-1){\frac{1}{\sqrt{4\,ab-1}}}} \right ){\frac{1}{\sqrt{4\,ab-1}}}}+{\arctan \left ({(2\,bx+1){\frac{1}{\sqrt{4\,ab-1}}}} \right ){\frac{1}{\sqrt{4\,ab-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3072, size = 389, normalized size = 6.48 \begin{align*} \left [-\frac{\sqrt{-4 \, a b + 1} \log \left (\frac{b^{2} x^{4} -{\left (6 \, a b - 1\right )} x^{2} + a^{2} - 2 \,{\left (b x^{3} - a x\right )} \sqrt{-4 \, a b + 1}}{b^{2} x^{4} +{\left (2 \, a b - 1\right )} x^{2} + a^{2}}\right )}{2 \,{\left (4 \, a b - 1\right )}}, \frac{\sqrt{4 \, a b - 1} \arctan \left (\frac{b x}{\sqrt{4 \, a b - 1}}\right ) + \sqrt{4 \, a b - 1} \arctan \left (\frac{{\left (b^{2} x^{3} +{\left (3 \, a b - 1\right )} x\right )} \sqrt{4 \, a b - 1}}{4 \, a^{2} b - a}\right )}{4 \, a b - 1}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-4*a*b + 1)*log((b^2*x^4 - (6*a*b - 1)*x^2 + a^2 - 2*(b*x^3 - a*x)*sqrt(-4*a*b + 1))/(b^2*x^4 + (2*
a*b - 1)*x^2 + a^2))/(4*a*b - 1), (sqrt(4*a*b - 1)*arctan(b*x/sqrt(4*a*b - 1)) + sqrt(4*a*b - 1)*arctan((b^2*x
^3 + (3*a*b - 1)*x)*sqrt(4*a*b - 1)/(4*a^2*b - a)))/(4*a*b - 1)]

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Sympy [B]  time = 0.379932, size = 117, normalized size = 1.95 \begin{align*} - \frac{\sqrt{- \frac{1}{4 a b - 1}} \log{\left (- \frac{a}{b} + x^{2} + \frac{x \left (- 4 a b \sqrt{- \frac{1}{4 a b - 1}} + \sqrt{- \frac{1}{4 a b - 1}}\right )}{b} \right )}}{2} + \frac{\sqrt{- \frac{1}{4 a b - 1}} \log{\left (- \frac{a}{b} + x^{2} + \frac{x \left (4 a b \sqrt{- \frac{1}{4 a b - 1}} - \sqrt{- \frac{1}{4 a b - 1}}\right )}{b} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26325, size = 69, normalized size = 1.15 \begin{align*} \frac{\arctan \left (\frac{2 \, b x + 1}{\sqrt{4 \, a b - 1}}\right )}{\sqrt{4 \, a b - 1}} + \frac{\arctan \left (\frac{2 \, b x - 1}{\sqrt{4 \, a b - 1}}\right )}{\sqrt{4 \, a b - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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