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

Optimal. Leaf size=28 \[ -\frac{\text{Ei}\left (\frac{\sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.0958213, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {2291, 2178} \[ -\frac{\text{Ei}\left (\frac{\sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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{F^{\frac{\sqrt{1-a x}}{\sqrt{1+a x}}}}{1-a^2 x^2} \, dx &=-\frac{\operatorname{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]  time = 0.229814, size = 28, normalized size = 1. \[ -\frac{\text{Ei}\left (\frac{\sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1}{F}^{{\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}}}}\, 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))/(-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)

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

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Fricas [A]  time = 2.45206, size = 59, normalized size = 2.11 \begin{align*} -\frac{{\rm Ei}\left (\frac{\sqrt{-a x + 1} \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))/(-a^2*x^2+1),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{F^{\frac{\sqrt{- a x + 1}}{\sqrt{a x + 1}}}}{a^{2} x^{2} - 1}\, dx \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))/(-a**2*x**2+1),x)

[Out]

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

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