Optimal. Leaf size=70 \[ -\frac{c^2 2^{\frac{n}{2}+3} (1-a x)^{3-\frac{n}{2}} \text{Hypergeometric2F1}\left (-\frac{n}{2}-2,3-\frac{n}{2},4-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (6-n)} \]
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Rubi [A] time = 0.0536608, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6140, 69} \[ -\frac{c^2 2^{\frac{n}{2}+3} (1-a x)^{3-\frac{n}{2}} \, _2F_1\left (-\frac{n}{2}-2,3-\frac{n}{2};4-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (6-n)} \]
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 )^2 \, dx &=c^2 \int (1-a x)^{2-\frac{n}{2}} (1+a x)^{2+\frac{n}{2}} \, dx\\ &=-\frac{2^{3+\frac{n}{2}} c^2 (1-a x)^{3-\frac{n}{2}} \, _2F_1\left (-2-\frac{n}{2},3-\frac{n}{2};4-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (6-n)}\\ \end{align*}
Mathematica [A] time = 0.0205565, size = 67, normalized size = 0.96 \[ \frac{c^2 2^{\frac{n}{2}+3} (1-a x)^{3-\frac{n}{2}} \text{Hypergeometric2F1}\left (-\frac{n}{2}-2,3-\frac{n}{2},4-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (n-6)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.144, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -{a}^{2}c{x}^{2}+c \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*} \int{\left (a^{2} c x^{2} - c\right )}^{2} \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^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\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^{2} \left (\int - 2 a^{2} x^{2} e^{n \operatorname{atanh}{\left (a x \right )}}\, dx + \int a^{4} x^{4} 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 )}^{2} \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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