Integrand size = 36, antiderivative size = 29 \[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2329, 2209} \[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]
[In]
[Out]
Rule 2209
Rule 2329
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {F^{-2 x}}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \\ & = -\frac {\text {Ei}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \]
[In]
[Out]
\[\int \frac {F^{-\frac {2 \sqrt {-a x +1}}{\sqrt {a x +1}}}}{-a^{2} x^{2}+1}d x\]
[In]
[Out]
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}} \,d x } \]
[In]
[Out]
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=- \int \frac {1}{F^{\frac {2 \sqrt {- a x + 1}}{\sqrt {a x + 1}}} a^{2} x^{2} - F^{\frac {2 \sqrt {- a x + 1}}{\sqrt {a x + 1}}}}\, dx \]
[In]
[Out]
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}} \,d x } \]
[In]
[Out]
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int -\frac {1}{F^{\frac {2\,\sqrt {1-a\,x}}{\sqrt {a\,x+1}}}\,\left (a^2\,x^2-1\right )} \,d x \]
[In]
[Out]