Integrand size = 21, antiderivative size = 16 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx=-\frac {2 \text {arcsinh}\left (\sqrt {1-b x}\right )}{b} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {65, 221} \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx=-\frac {2 \text {arcsinh}\left (\sqrt {1-b x}\right )}{b} \]
[In]
[Out]
Rule 65
Rule 221
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {1-b x}\right )}{b} \\ & = -\frac {2 \sinh ^{-1}\left (\sqrt {1-b x}\right )}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(16)=32\).
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.75 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx=\frac {\log \left (\sqrt {1-b x}-\sqrt {2-b x}\right )}{b}-\frac {\log \left (b \sqrt {1-b x}+b \sqrt {2-b x}\right )}{b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(14)=28\).
Time = 0.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.38
method | result | size |
default | \(\frac {\sqrt {\left (-b x +1\right ) \left (-b x +2\right )}\, \ln \left (\frac {-\frac {3}{2} b +b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}-3 b x +2}\right )}{\sqrt {-b x +1}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\) | \(70\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx=-\frac {\log \left (-2 \, b x + 2 \, \sqrt {-b x + 2} \sqrt {-b x + 1} + 3\right )}{b} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx=\int \frac {1}{\sqrt {- b x + 1} \sqrt {- b x + 2}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx=\frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 3 \, b x + 2} b - 3 \, b\right )}{b} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx=\frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x + 1}\right )}{b} \]
[In]
[Out]
Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.81 \[ \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {2-b\,x}\right )}{\left (\sqrt {1-b\,x}-1\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
[In]
[Out]