Optimal. Leaf size=110 \[ \frac{a^2 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{2 a b \text{Ei}\left (\frac{c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^2 \text{Ei}\left (\frac{2 c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e} \]
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Rubi [A] time = 0.303162, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.064, Rules used = {2291, 2183, 2178} \[ \frac{a^2 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{2 a b \text{Ei}\left (\frac{c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^2 \text{Ei}\left (\frac{2 c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e} \]
Antiderivative was successfully verified.
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Rule 2291
Rule 2183
Rule 2178
Rubi steps
\begin{align*} \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b F^{c x}\right )^2}{x} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x}+\frac{2 a b F^{c x}}{x}+\frac{b^2 F^{2 c x}}{x}\right ) \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}\\ &=\frac{a^2 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{(2 a b) \operatorname{Subst}\left (\int \frac{F^{c x}}{x} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^2 \operatorname{Subst}\left (\int \frac{F^{2 c x}}{x} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}\\ &=\frac{2 a b \text{Ei}\left (\frac{c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^2 \text{Ei}\left (\frac{2 c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e}+\frac{a^2 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}\\ \end{align*}
Mathematica [F] time = 1.2677, size = 0, normalized size = 0. \[ \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{e}^{2}{x}^{2}+{d}^{2}} \left ( a+b{F}^{{c\sqrt{ex+d}{\frac{1}{\sqrt{-efx+df}}}}} \right ) ^{2}}\, 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*} \frac{1}{2} \, a^{2}{\left (\frac{\log \left (e x + d\right )}{d e} - \frac{\log \left (e x - d\right )}{d e}\right )} - b^{2} \int \frac{F^{\frac{2 \, \sqrt{e x + d} c}{\sqrt{-e x + d} \sqrt{f}}}}{e^{2} x^{2} - d^{2}}\,{d x} - 2 \, a b \int \frac{F^{\frac{\sqrt{e x + d} c}{\sqrt{-e x + d} \sqrt{f}}}}{e^{2} x^{2} - d^{2}}\,{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{a^{2} + \frac{2 \, a b}{F^{\frac{\sqrt{-e f x + d f} \sqrt{e x + d} c}{e f x - d f}}} + \frac{b^{2}}{F^{\frac{2 \, \sqrt{-e f x + d f} \sqrt{e x + d} c}{e f x - d f}}}}{e^{2} x^{2} - d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a^{2}}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac{F^{\frac{2 c \sqrt{d + e x}}{\sqrt{d f - e f x}}} b^{2}}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac{2 F^{\frac{c \sqrt{d + e x}}{\sqrt{d f - e f x}}} a b}{- d^{2} + e^{2} x^{2}}\, dx \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{e x + d} c}{\sqrt{-e f x + d f}}} b + a\right )}^{2}}{e^{2} x^{2} - d^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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