Integrand size = 25, antiderivative size = 15 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=e^x \sqrt {1-x^2} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2326} \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=e^x \sqrt {1-x^2} \]
[In]
[Out]
Rule 2326
Rubi steps \begin{align*} \text {integral}& = e^x \sqrt {1-x^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=e^x \sqrt {1-x^2} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33
method | result | size |
gosper | \(-\frac {{\mathrm e}^{x} \left (-1+x \right ) \left (1+x \right )}{\sqrt {-x^{2}+1}}\) | \(20\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=\sqrt {-x^{2} + 1} e^{x} \]
[In]
[Out]
\[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=- \int \left (- \frac {e^{x}}{\sqrt {1 - x^{2}}}\right )\, dx - \int \frac {x e^{x}}{\sqrt {1 - x^{2}}}\, dx - \int \frac {x^{2} e^{x}}{\sqrt {1 - x^{2}}}\, dx \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.40 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=-\frac {{\left (x^{2} - 1\right )} e^{x}}{\sqrt {x + 1} \sqrt {-x + 1}} \]
[In]
[Out]
\[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=\int { -\frac {{\left (x^{2} + x - 1\right )} e^{x}}{\sqrt {-x^{2} + 1}} \,d x } \]
[In]
[Out]
Time = 0.52 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx={\mathrm {e}}^x\,\sqrt {1-x^2} \]
[In]
[Out]