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

Optimal. Leaf size=77 \[ -\frac{F^{-\frac{n \sqrt{1-a x}}{\sqrt{a x+1}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{a x+1}}}\right )^n \text{Ei}\left (\frac{n \sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]

[Out]

-(((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])))

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Rubi [A]  time = 0.240574, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {2281, 2291, 2178} \[ -\frac{F^{-\frac{n \sqrt{1-a x}}{\sqrt{a x+1}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{a x+1}}}\right )^n \text{Ei}\left (\frac{n \sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((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])))

Rule 2281

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

Rule 2291

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
]

Rule 2178

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

Rubi steps

\begin{align*} \int \frac{\left (F^{\frac{\sqrt{1-a x}}{\sqrt{1+a x}}}\right )^n}{1-a^2 x^2} \, dx &=\left (F^{-\frac{n \sqrt{1-a x}}{\sqrt{1+a x}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{1+a x}}}\right )^n\right ) \int \frac{F^{\frac{n \sqrt{1-a x}}{\sqrt{1+a x}}}}{1-a^2 x^2} \, dx\\ &=-\frac{\left (F^{-\frac{n \sqrt{1-a x}}{\sqrt{1+a x}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{1+a x}}}\right )^n\right ) \operatorname{Subst}\left (\int \frac{F^{n x}}{x} \, dx,x,\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}\\ &=-\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 \text{Ei}\left (\frac{n \sqrt{1-a x} \log (F)}{\sqrt{1+a x}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.317459, size = 77, normalized size = 1. \[ -\frac{F^{-\frac{n \sqrt{1-a x}}{\sqrt{a x+1}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{a x+1}}}\right )^n \text{Ei}\left (\frac{n \sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((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])))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1} \left ({F}^{{\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}}} \right ) ^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (F^{\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.87811, size = 62, normalized size = 0.81 \begin{align*} -\frac{{\rm Ei}\left (\frac{\sqrt{-a x + 1} n \log \left (F\right )}{\sqrt{a x + 1}}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Ei(sqrt(-a*x + 1)*n*log(F)/sqrt(a*x + 1))/a

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (F^{\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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