Optimal. Leaf size=20 \[ -\frac{2 \sinh ^{-1}\left (\frac{\sqrt{-b x}}{\sqrt{2}}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0053592, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {63, 215} \[ -\frac{2 \sinh ^{-1}\left (\frac{\sqrt{-b x}}{\sqrt{2}}\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 215
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-b x} \sqrt{2-b x}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^2}} \, dx,x,\sqrt{-b x}\right )}{b}\\ &=-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{-b x}}{\sqrt{2}}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0085961, size = 37, normalized size = 1.85 \[ \frac{2 \sqrt{x} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b} \sqrt{-b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.004, size = 64, normalized size = 3.2 \begin{align*}{\sqrt{-bx \left ( -bx+2 \right ) }\ln \left ({({b}^{2}x-b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}-2\,bx} \right ){\frac{1}{\sqrt{-bx}}}{\frac{1}{\sqrt{-bx+2}}}{\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 [A] time = 1.99129, size = 62, normalized size = 3.1 \begin{align*} -\frac{\log \left (-b x + \sqrt{-b x + 2} \sqrt{-b x} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.48838, size = 53, normalized size = 2.65 \begin{align*} \begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{2 i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1284, size = 32, normalized size = 1.6 \begin{align*} \frac{2 \, \log \left ({\left | -\sqrt{-b x + 2} + \sqrt{-b x} \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]