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

Optimal. Leaf size=35 \[ -\frac{\tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

[Out]

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

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Rubi [A]  time = 0.0273843, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {618, 206} \[ -\frac{\tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*a*x + b*x^2)^(-1),x]

[Out]

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

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

Mathematica [A]  time = 0.0093187, size = 34, normalized size = 0.97 \[ \frac{\tan ^{-1}\left (\frac{a+b x}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*a*x + b*x^2)^(-1),x]

[Out]

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

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Maple [A]  time = 0.142, size = 35, normalized size = 1. \begin{align*}{\arctan \left ({\frac{2\,bx+2\,a}{2}{\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.17575, size = 261, normalized size = 7.46 \begin{align*} \left [\frac{\log \left (\frac{b^{2} x^{2} + 2 \, a b x + 2 \, a^{2} - b^{2} - 2 \, \sqrt{a^{2} - b^{2}}{\left (b x + a\right )}}{b x^{2} + 2 \, a x + b}\right )}{2 \, \sqrt{a^{2} - b^{2}}}, -\frac{\sqrt{-a^{2} + b^{2}} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b x + a\right )}}{a^{2} - b^{2}}\right )}{a^{2} - b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.234572, size = 100, normalized size = 2.86 \begin{align*} \frac{\sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}} \log{\left (x + \frac{- a^{2} \sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}} + a + b^{2} \sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}}}{b} \right )}}{2} - \frac{\sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}} \log{\left (x + \frac{a^{2} \sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}} + a - b^{2} \sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}}}{b} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.31692, size = 41, normalized size = 1.17 \begin{align*} \frac{\arctan \left (\frac{b x + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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