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

Optimal. Leaf size=32 \[ -\frac{\tanh ^{-1}\left (\frac{a-b x}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]

[Out]

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

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Rubi [A]  time = 0.0254946, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, 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 = 0.0102485, size = 41, normalized size = 1.28 \[ -\frac{\tan ^{-1}\left (\frac{b x-a}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}} \]

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 = 0.136, size = 31, normalized size = 1. \begin{align*}{{\it Artanh} \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)*arctanh(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 [B]  time = 2.09927, size = 149, normalized size = 4.66 \begin{align*} \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}}} \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)

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Sympy [B]  time = 0.323216, size = 102, normalized size = 3.19 \begin{align*} - \frac{\sqrt{\frac{1}{a^{2} + b^{2}}} \log{\left (x + \frac{- a^{2} \sqrt{\frac{1}{a^{2} + b^{2}}} - a - b^{2} \sqrt{\frac{1}{a^{2} + b^{2}}}}{b} \right )}}{2} + \frac{\sqrt{\frac{1}{a^{2} + b^{2}}} \log{\left (x + \frac{a^{2} \sqrt{\frac{1}{a^{2} + b^{2}}} - a + b^{2} \sqrt{\frac{1}{a^{2} + b^{2}}}}{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**2 + b**2))*log(x + (-a**2*sqrt(1/(a**2 + b**2)) - a - b**2*sqrt(1/(a**2 + b**2)))/b)/2 + sqrt(1/(a
**2 + b**2))*log(x + (a**2*sqrt(1/(a**2 + b**2)) - a + b**2*sqrt(1/(a**2 + b**2)))/b)/2

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Giac [A]  time = 1.37932, size = 74, normalized size = 2.31 \begin{align*} -\frac{\log \left (\frac{{\left | 2 \, b x - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b x - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{2 \, \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]

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