Optimal. Leaf size=12 \[ -\frac{\sin ^{-1}(1-2 b x)}{b} \]
[Out]
________________________________________________________________________________________
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]
[Out]
Rule 619
Rule 216
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]