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

Optimal. Leaf size=29 \[ -\frac {\text {Ei}\left (\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2291, 2178} \[ -\frac {\text {Ei}\left (\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ExpIntegralEi[(2*Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)

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

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
]

Rubi steps

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

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Mathematica [A]  time = 0.25, size = 29, normalized size = 1.00 \[ -\frac {\text {Ei}\left (\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ExpIntegralEi[(2*Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)

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fricas [A]  time = 0.61, size = 25, normalized size = 0.86 \[ -\frac {{\rm Ei}\left (\frac {2 \, \sqrt {-a x + 1} \log \relax (F)}{\sqrt {a x + 1}}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Ei(2*sqrt(-a*x + 1)*log(F)/sqrt(a*x + 1))/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}}{a^{2} x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{\frac {2 \sqrt {-a x +1}}{\sqrt {a x +1}}}}{-a^{2} x^{2}+1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}}{a^{2} x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int -\frac {F^{\frac {2\,\sqrt {1-a\,x}}{\sqrt {a\,x+1}}}}{a^2\,x^2-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {F^{\frac {2 \sqrt {- a x + 1}}{\sqrt {a x + 1}}}}{a^{2} x^{2} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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