Optimal. Leaf size=77 \[ -\frac{F^{-\frac{n \sqrt{1-a x}}{\sqrt{a x+1}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{a x+1}}}\right )^n \text{Ei}\left (\frac{n \sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]
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Rubi [A] time = 0.240574, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {2281, 2291, 2178} \[ -\frac{F^{-\frac{n \sqrt{1-a x}}{\sqrt{a x+1}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{a x+1}}}\right )^n \text{Ei}\left (\frac{n \sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2281
Rule 2291
Rule 2178
Rubi steps
\begin{align*} \int \frac{\left (F^{\frac{\sqrt{1-a x}}{\sqrt{1+a x}}}\right )^n}{1-a^2 x^2} \, dx &=\left (F^{-\frac{n \sqrt{1-a x}}{\sqrt{1+a x}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{1+a x}}}\right )^n\right ) \int \frac{F^{\frac{n \sqrt{1-a x}}{\sqrt{1+a x}}}}{1-a^2 x^2} \, dx\\ &=-\frac{\left (F^{-\frac{n \sqrt{1-a x}}{\sqrt{1+a x}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{1+a x}}}\right )^n\right ) \operatorname{Subst}\left (\int \frac{F^{n x}}{x} \, dx,x,\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}\\ &=-\frac{F^{-\frac{n \sqrt{1-a x}}{\sqrt{1+a x}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{1+a x}}}\right )^n \text{Ei}\left (\frac{n \sqrt{1-a x} \log (F)}{\sqrt{1+a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.317459, size = 77, normalized size = 1. \[ -\frac{F^{-\frac{n \sqrt{1-a x}}{\sqrt{a x+1}}} \left (F^{\frac{\sqrt{1-a x}}{\sqrt{a x+1}}}\right )^n \text{Ei}\left (\frac{n \sqrt{1-a x} \log (F)}{\sqrt{a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1} \left ({F}^{{\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}}} \right ) ^{n}}\, 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{{\left (F^{\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.87811, size = 62, normalized size = 0.81 \begin{align*} -\frac{{\rm Ei}\left (\frac{\sqrt{-a x + 1} n \log \left (F\right )}{\sqrt{a x + 1}}\right )}{a} \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{{\left (F^{\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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