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

Optimal. Leaf size=69 \[ -\frac{\log \left (a x^4+2 a x^2+a+b\right )}{4 (a+b)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a+b)}+\frac{\log (x)}{a+b} \]

[Out]

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

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Rubi [A]  time = 0.067098, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1114, 705, 29, 634, 618, 204, 628} \[ -\frac{\log \left (a x^4+2 a x^2+a+b\right )}{4 (a+b)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a+b)}+\frac{\log (x)}{a+b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1114

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

Rule 705

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 634

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

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 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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [C]  time = 0.0514506, size = 105, normalized size = 1.52 \[ \frac{i \left (\sqrt{a}+i \sqrt{b}\right ) \log \left (\sqrt{a} \left (x^2+1\right )-i \sqrt{b}\right )+\left (-\sqrt{b}-i \sqrt{a}\right ) \log \left (\sqrt{a} \left (x^2+1\right )+i \sqrt{b}\right )+4 \sqrt{b} \log (x)}{4 \sqrt{b} (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[b]*Log[x] + I*(Sqrt[a] + I*Sqrt[b])*Log[(-I)*Sqrt[b] + Sqrt[a]*(1 + x^2)] + ((-I)*Sqrt[a] - Sqrt[b])*L
og[I*Sqrt[b] + Sqrt[a]*(1 + x^2)])/(4*Sqrt[b]*(a + b))

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Maple [A]  time = 0.049, size = 63, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{a+b}}-{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a+b \right ) }{4\,a+4\,b}}-{\frac{a}{2\,a+2\,b}\arctan \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(1/x/(a*x^4+2*a*x^2+a+b),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54784, size = 347, normalized size = 5.03 \begin{align*} \left [\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{a x^{4} + 2 \, a x^{2} - 2 \,{\left (b x^{2} + b\right )} \sqrt{-\frac{a}{b}} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right ) - \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 4 \, \log \left (x\right )}{4 \,{\left (a + b\right )}}, \frac{2 \, \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a x^{2} + a}\right ) - \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 4 \, \log \left (x\right )}{4 \,{\left (a + b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [B]  time = 2.26912, size = 194, normalized size = 2.81 \begin{align*} \left (- \frac{1}{4 \left (a + b\right )} - \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (- \frac{1}{4 \left (a + b\right )} - \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) + a - 4 b^{2} \left (- \frac{1}{4 \left (a + b\right )} - \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) - b}{a} \right )} + \left (- \frac{1}{4 \left (a + b\right )} + \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (- \frac{1}{4 \left (a + b\right )} + \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) + a - 4 b^{2} \left (- \frac{1}{4 \left (a + b\right )} + \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) - b}{a} \right )} + \frac{\log{\left (x \right )}}{a + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 3.48728, size = 82, normalized size = 1.19 \begin{align*} -\frac{a \arctan \left (\frac{a x^{2} + a}{\sqrt{a b}}\right )}{2 \, \sqrt{a b}{\left (a + b\right )}} - \frac{\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \,{\left (a + b\right )}} + \frac{\log \left (x^{2}\right )}{2 \,{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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