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

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

[Out]

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

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Rubi [A]  time = 0.10, 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} \[ -\frac {\text {Ei}\left (-\frac {\sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/(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 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 {\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*}

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

Antiderivative was successfully verified.

[In]

Integrate[1/(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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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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