\(\int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx\) [560]

Optimal result

Integrand size = 35, antiderivative size = 28 \[ \int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (\frac {\sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \]

[Out]

-Ei(ln(F)*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2329, 2209} \[ \int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (\frac {\sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]

[In]

Int[F^(Sqrt[1 - a*x]/Sqrt[1 + a*x])/(1 - a^2*x^2),x]

[Out]

-(ExpIntegralEi[(Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2329

Int[((a_.) + (b_.)*(F_)^(((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.)*(x_)]))^(n_.)/((A_) + (C_.)*(x_)^
2), x_Symbol] :> Dist[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
]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {F^x}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \\ & = -\frac {\text {Ei}\left (\frac {\sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (\frac {\sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \]

[In]

Integrate[F^(Sqrt[1 - a*x]/Sqrt[1 + a*x])/(1 - a^2*x^2),x]

[Out]

-(ExpIntegralEi[(Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)

Maple [F]

\[\int \frac {F^{\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}}}{-a^{2} x^{2}+1}d x\]

[In]

int(F^((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x)

[Out]

int(F^((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x)

Fricas [F]

\[ \int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}}{a^{2} x^{2} - 1} \,d x } \]

[In]

integrate(F^((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="fricas")

[Out]

integral(-F^(sqrt(-a*x + 1)/sqrt(a*x + 1))/(a^2*x^2 - 1), x)

Sympy [F]

\[ \int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=- \int \frac {F^{\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}}}}{a^{2} x^{2} - 1}\, dx \]

[In]

integrate(F**((-a*x+1)**(1/2)/(a*x+1)**(1/2))/(-a**2*x**2+1),x)

[Out]

-Integral(F**(sqrt(-a*x + 1)/sqrt(a*x + 1))/(a**2*x**2 - 1), x)

Maxima [F]

\[ \int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}}{a^{2} x^{2} - 1} \,d x } \]

[In]

integrate(F^((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="maxima")

[Out]

-integrate(F^(sqrt(-a*x + 1)/sqrt(a*x + 1))/(a^2*x^2 - 1), x)

Giac [F]

\[ \int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int { -\frac {F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}}{a^{2} x^{2} - 1} \,d x } \]

[In]

integrate(F^((-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="giac")

[Out]

integrate(-F^(sqrt(-a*x + 1)/sqrt(a*x + 1))/(a^2*x^2 - 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx=\int -\frac {F^{\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}}}{a^2\,x^2-1} \,d x \]

[In]

int(-F^((1 - a*x)^(1/2)/(a*x + 1)^(1/2))/(a^2*x^2 - 1),x)

[Out]

int(-F^((1 - a*x)^(1/2)/(a*x + 1)^(1/2))/(a^2*x^2 - 1), x)