3.1.25 \(\int \frac {1}{\sqrt {b x-b^2 x^2}} \, dx\) [25]

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

[Out]

arcsin(2*b*x-1)/b

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {633, 222} \begin {gather*} -\frac {\text {ArcSin}(1-2 b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b*x - b^2*x^2],x]

[Out]

-(ArcSin[1 - 2*b*x]/b)

Rule 222

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]

Rule 633

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {b x-b^2 x^2}} \, dx &=-\frac {\text {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] Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(12)=24\).
time = 0.04, size = 57, normalized size = 4.75 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {-1+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {-1+b x}\right )}{\sqrt {b} \sqrt {-b x (-1+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b*x - b^2*x^2],x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(34\) vs. \(2(11)=22\).
time = 0.38, size = 35, normalized size = 2.92

method	result	size
default	\(\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x -\frac {1}{2 b}\right )}{\sqrt {-b^{2} x^{2}+b x}}\right )}{\sqrt {b^{2}}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 21, normalized size = 1.75 \begin {gather*} -\frac {\arcsin \left (-\frac {2 \, b^{2} x - b}{b}\right )}{b} \end {gather*}

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]

-arcsin(-(2*b^2*x - b)/b)/b

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (11) = 22\).
time = 1.27, size = 27, normalized size = 2.25 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {-b^{2} x^{2} + b x}}{b x}\right )}{b} \end {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- b^{2} x^{2} + b x}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (11) = 22\).
time = 2.73, size = 41, normalized size = 3.42 \begin {gather*} \frac {1}{4} \, \sqrt {-b^{2} x^{2} + b x} {\left (2 \, x - \frac {1}{b}\right )} - \frac {\arcsin \left (-2 \, b x + 1\right ) \mathrm {sgn}\left (b\right )}{8 \, {\left | b \right |}} \end {gather*}

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]

1/4*sqrt(-b^2*x^2 + b*x)*(2*x - 1/b) - 1/8*arcsin(-2*b*x + 1)*sgn(b)/abs(b)

________________________________________________________________________________________

Mupad [B]
time = 0.31, size = 42, normalized size = 3.50 \begin {gather*} \frac {\ln \left (\frac {\frac {b}{2}-b^2\,x}{\sqrt {-b^2}}+\sqrt {b\,x-b^2\,x^2}\right )}{\sqrt {-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x - b^2*x^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________