\(\int \frac {(F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}})^n}{1-a^2 x^2} \, dx\) [479]

Optimal result

Integrand size = 37, antiderivative size = 77 \[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx=-\frac {F^{-\frac {n \sqrt {1-a x}}{\sqrt {1+a x}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n \operatorname {ExpIntegralEi}\left (\frac {n \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx=-\frac {F^{-\frac {n \sqrt {1-a x}}{\sqrt {1+a x}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n \operatorname {ExpIntegralEi}\left (\frac {n \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2717, 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.

\(\displaystyle \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}\right )^n}{1-a^2 x^2} \, dx\)

\(\Big \downarrow \) 2717

\(\displaystyle F^{-\frac {n \sqrt {1-a x}}{\sqrt {a x+1}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}\right )^n \int \frac {F^{\frac {n \sqrt {1-a x}}{\sqrt {a x+1}}}}{1-a^2 x^2}dx\)

\(\Big \downarrow \) 2729

\(\displaystyle -\frac {F^{-\frac {n \sqrt {1-a x}}{\sqrt {a x+1}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}\right )^n \int \frac {F^{\frac {n \sqrt {1-a x}}{\sqrt {a x+1}}} \sqrt {a x+1}}{\sqrt {1-a x}}d\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}{a}\)

\(\Big \downarrow \) 2609

\(\displaystyle -\frac {F^{-\frac {n \sqrt {1-a x}}{\sqrt {a x+1}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}\right )^n \operatorname {ExpIntegralEi}\left (\frac {n \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2609

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]
 

rule 2717

Int[(u_.)*((a_.)*(F_)^(v_))^(n_), x_Symbol] :> Simp[(a*F^v)^n/F^(n*v)   Int 
[u*F^(n*v), x], x] /; FreeQ[{F, a, n}, x] &&  !IntegerQ[n]
 

rule 2729

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]
 

Maple [F]

\[\int \frac {\left (F^{\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}}\right )^{n}}{-a^{2} x^{2}+1}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx=\int { -\frac {{\left (F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx=- \int \frac {\left (F^{\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}}}\right )^{n}}{a^{2} x^{2} - 1}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx=\int { -\frac {{\left (F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx=\int { -\frac {{\left (F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx=\int -\frac {{\left (F^{\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}}\right )}^n}{a^2\,x^2-1} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx=-\left (\int \frac {f^{\frac {\sqrt {-a x +1}\, n}{\sqrt {a x +1}}}}{a^{2} x^{2}-1}d x \right ) \] Input:

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

Output:

 - int(f**((sqrt( - a*x + 1)*n)/sqrt(a*x + 1))/(a**2*x**2 - 1),x)