Optimal. Leaf size=29 \[ -\frac{\text{Ei}\left (-\frac{2 \sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]
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Rubi [A] time = 0.104807, 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, 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.
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Rule 2291
Rule 2178
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*}
Mathematica [A] time = 0.245116, size = 29, normalized size = 1. \[ -\frac{\text{Ei}\left (-\frac{2 \sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1} \left ({F}^{2\,{\frac{\sqrt{-ax+1}}{\sqrt{ax+1}}}} \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac{2 \, \sqrt{-a x + 1}}{\sqrt{a x + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac{2 \, \sqrt{-a x + 1}}{\sqrt{a x + 1}}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac{2 \, \sqrt{-a x + 1}}{\sqrt{a x + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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