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

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

[Out]

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

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Rubi [A]  time = 0.173049, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{\text{ExpIntegralEi}\left (-\frac{\sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]

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)

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Rubi in Sympy [A]  time = 12.1013, size = 26, normalized size = 0.9 \[ - \frac{\operatorname{Ei}{\left (- \frac{\sqrt{- a x + 1} \log{\left (F \right )}}{\sqrt{a x + 1}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(F**((-a*x+1)**(1/2)/(a*x+1)**(1/2)))/(-a**2*x**2+1),x)

[Out]

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

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Mathematica [A]  time = 0.0946574, size = 29, normalized size = 1. \[ -\frac{\text{ExpIntegralEi}\left (-\frac{\sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((a^2*x^2 - 1)*F^(sqrt(-a*x + 1)/sqrt(a*x + 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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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((a^2*x^2 - 1)*F^(sqrt(-a*x + 1)/sqrt(a*x + 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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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