Optimal. Leaf size=28 \[ -\frac{\text{Ei}\left (\frac{\sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]
[Out]
________________________________________________________________________________________
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]
[Out]
Rule 2291
Rule 2178
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]