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

Optimal. Leaf size=31 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.028482, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1107, 618, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a - b + 2*a*x^2 + a*x^4),x]

[Out]

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

Rule 1107

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

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

Mathematica [A]  time = 0.0078592, size = 31, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 26, normalized size = 0.8 \begin{align*} -{\frac{1}{2}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58671, size = 215, normalized size = 6.94 \begin{align*} \left [\frac{\sqrt{a b} \log \left (\frac{a x^{4} + 2 \, a x^{2} - 2 \, \sqrt{a b}{\left (x^{2} + 1\right )} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right )}{4 \, a b}, \frac{\sqrt{-a b} \arctan \left (\frac{\sqrt{-a b}}{a x^{2} + a}\right )}{2 \, a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.286649, size = 53, normalized size = 1.71 \begin{align*} \frac{\sqrt{\frac{1}{a b}} \log{\left (- b \sqrt{\frac{1}{a b}} + x^{2} + 1 \right )}}{4} - \frac{\sqrt{\frac{1}{a b}} \log{\left (b \sqrt{\frac{1}{a b}} + x^{2} + 1 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 4.48347, size = 31, normalized size = 1. \begin{align*} \frac{\arctan \left (\frac{a x^{2} + a}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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