Integrand size = 36, antiderivative size = 29 \[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \] Output:
-Ei(-2*(-a*x+1)^(1/2)*ln(F)/(a*x+1)^(1/2))/a
Time = 0.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \] Input:
Integrate[1/(F^((2*Sqrt[1 - a*x])/Sqrt[1 + a*x])*(1 - a^2*x^2)),x]
Output:
-(ExpIntegralEi[(-2*Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)
Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2729, 2609}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {a x+1}}}}{1-a^2 x^2} \, dx\) |
\(\Big \downarrow \) 2729 |
\(\displaystyle -\frac {\int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {a x+1}}} \sqrt {a x+1}}{\sqrt {1-a x}}d\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}{a}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle -\frac {\operatorname {ExpIntegralEi}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a}\) |
Input:
Int[1/(F^((2*Sqrt[1 - a*x])/Sqrt[1 + a*x])*(1 - a^2*x^2)),x]
Output:
-(ExpIntegralEi[(-2*Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[((a_.) + (b_.)*(F_)^(((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_. )*(x_)]))^(n_.)/((A_) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d *g))) Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
\[\int \frac {F^{-\frac {2 \sqrt {-a x +1}}{\sqrt {a x +1}}}}{-a^{2} x^{2}+1}d x\]
Input:
int(1/(F^(2*(-a*x+1)^(1/2)/(a*x+1)^(1/2)))/(-a^2*x^2+1),x)
Output:
int(1/(F^(2*(-a*x+1)^(1/2)/(a*x+1)^(1/2)))/(-a^2*x^2+1),x)
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}} \,d x } \] Input:
integrate(1/(F^(2*(-a*x+1)^(1/2)/(a*x+1)^(1/2)))/(-a^2*x^2+1),x, algorithm ="fricas")
Output:
integral(-1/((a^2*x^2 - 1)*F^(2*sqrt(-a*x + 1)/sqrt(a*x + 1))), x)
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=- \int \frac {1}{F^{\frac {2 \sqrt {- a x + 1}}{\sqrt {a x + 1}}} a^{2} x^{2} - F^{\frac {2 \sqrt {- a x + 1}}{\sqrt {a x + 1}}}}\, dx \] Input:
integrate(1/(F**(2*(-a*x+1)**(1/2)/(a*x+1)**(1/2)))/(-a**2*x**2+1),x)
Output:
-Integral(1/(F**(2*sqrt(-a*x + 1)/sqrt(a*x + 1))*a**2*x**2 - F**(2*sqrt(-a *x + 1)/sqrt(a*x + 1))), x)
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}} \,d x } \] Input:
integrate(1/(F^(2*(-a*x+1)^(1/2)/(a*x+1)^(1/2)))/(-a^2*x^2+1),x, algorithm ="maxima")
Output:
-integrate(1/((a^2*x^2 - 1)*F^(2*sqrt(-a*x + 1)/sqrt(a*x + 1))), x)
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}} \,d x } \] Input:
integrate(1/(F^(2*(-a*x+1)^(1/2)/(a*x+1)^(1/2)))/(-a^2*x^2+1),x, algorithm ="giac")
Output:
integrate(-1/((a^2*x^2 - 1)*F^(2*sqrt(-a*x + 1)/sqrt(a*x + 1))), x)
Timed out. \[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int -\frac {1}{F^{\frac {2\,\sqrt {1-a\,x}}{\sqrt {a\,x+1}}}\,\left (a^2\,x^2-1\right )} \,d x \] Input:
int(-1/(F^((2*(1 - a*x)^(1/2))/(a*x + 1)^(1/2))*(a^2*x^2 - 1)),x)
Output:
int(-1/(F^((2*(1 - a*x)^(1/2))/(a*x + 1)^(1/2))*(a^2*x^2 - 1)), x)
\[ \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\left (\int \frac {1}{f^{\frac {2 \sqrt {-a x +1}}{\sqrt {a x +1}}} a^{2} x^{2}-f^{\frac {2 \sqrt {-a x +1}}{\sqrt {a x +1}}}}d x \right ) \] Input:
int(1/(F^(2*(-a*x+1)^(1/2)/(a*x+1)^(1/2)))/(-a^2*x^2+1),x)
Output:
- int(1/(f**((2*sqrt( - a*x + 1))/sqrt(a*x + 1))*a**2*x**2 - f**((2*sqrt( - a*x + 1))/sqrt(a*x + 1))),x)