3.25 \(\int \frac{1}{\sqrt{b x-b^2 x^2}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\sin ^{-1}(1-2 b x)}{b} \]

[Out]

-(ArcSin[1 - 2*b*x]/b)

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Rubi [A]  time = 0.0076931, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {619, 216} \[ -\frac{\sin ^{-1}(1-2 b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b*x - b^2*x^2],x]

[Out]

-(ArcSin[1 - 2*b*x]/b)

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [B]  time = 0.014204, size = 47, normalized size = 3.92 \[ \frac{2 \sqrt{x} \sqrt{1-b x} \sin ^{-1}\left (\sqrt{b} \sqrt{x}\right )}{\sqrt{b} \sqrt{-b x (b x-1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b*x - b^2*x^2],x]

[Out]

(2*Sqrt[x]*Sqrt[1 - b*x]*ArcSin[Sqrt[b]*Sqrt[x]])/(Sqrt[b]*Sqrt[-(b*x*(-1 + b*x))])

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Maple [B]  time = 0.05, size = 35, normalized size = 2.9 \begin{align*}{\arctan \left ({\sqrt{{b}^{2}} \left ( x-{\frac{1}{2\,b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}+bx}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.23184, size = 55, normalized size = 4.58 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{-b^{2} x^{2} + b x}}{b x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- b^{2} x^{2} + b x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-b**2*x**2 + b*x), x)

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Giac [A]  time = 2.63835, size = 20, normalized size = 1.67 \begin{align*} -\frac{\arcsin \left (-2 \, b x + 1\right ) \mathrm{sgn}\left (b\right )}{{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(-2*b*x + 1)*sgn(b)/abs(b)