Optimal. Leaf size=68 \[ -\frac{c 2^{\frac{n}{2}+2} (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (-\frac{n}{2}-1,2-\frac{n}{2},3-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (4-n)} \]
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Rubi [A] time = 0.0441174, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6140, 69} \[ -\frac{c 2^{\frac{n}{2}+2} (1-a x)^{2-\frac{n}{2}} \, _2F_1\left (-\frac{n}{2}-1,2-\frac{n}{2};3-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (4-n)} \]
Antiderivative was successfully verified.
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Rule 6140
Rule 69
Rubi steps
\begin{align*} \int e^{n \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int (1-a x)^{1-\frac{n}{2}} (1+a x)^{1+\frac{n}{2}} \, dx\\ &=-\frac{2^{2+\frac{n}{2}} c (1-a x)^{2-\frac{n}{2}} \, _2F_1\left (-1-\frac{n}{2},2-\frac{n}{2};3-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (4-n)}\\ \end{align*}
Mathematica [A] time = 0.0190884, size = 65, normalized size = 0.96 \[ \frac{c 2^{\frac{n}{2}+2} (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (-\frac{n}{2}-1,2-\frac{n}{2},3-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (n-4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -{a}^{2}c{x}^{2}+c \right ) \, 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{\left (a^{2} c x^{2} - c\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{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 (-{\left (a^{2} c x^{2} - c\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}, 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*} - c \left (\int a^{2} x^{2} e^{n \operatorname{atanh}{\left (a x \right )}}\, dx + \int - e^{n \operatorname{atanh}{\left (a x \right )}}\, dx\right ) \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 -{\left (a^{2} c x^{2} - c\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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