Optimal. Leaf size=331 \[ -\frac{c^2 n \left (10-n^2\right ) (a x+1)^{\frac{n-4}{2}} (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-4}{2},\frac{n-2}{2},\frac{a x+1}{1-a x}\right )}{3 a (4-n)}+\frac{c^2 2^{\frac{n}{2}-1} n (1-a x)^{3-\frac{n}{2}} \text{Hypergeometric2F1}\left (\frac{4-n}{2},3-\frac{n}{2},4-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a \left (n^2-10 n+24\right )}-\frac{c^2 \left (n^2+5 n+14\right ) (a x+1)^{\frac{n-4}{2}} (1-a x)^{3-\frac{n}{2}}}{6 a^2 x}-\frac{c^2 (n+10) (a x+1)^{\frac{n-4}{2}} (1-a x)^{3-\frac{n}{2}}}{6 a^3 x^2}-\frac{c^2 (a x+1)^{\frac{n-4}{2}} (1-a x)^{3-\frac{n}{2}}}{3 a^4 x^3}-\frac{4 c^2 (a x+1)^{\frac{n-4}{2}} (1-a x)^{3-\frac{n}{2}}}{a (4-n)} \]
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Rubi [C] time = 0.133327, antiderivative size = 71, normalized size of antiderivative = 0.21, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6157, 6150, 136} \[ \frac{c^2 2^{3-\frac{n}{2}} (a x+1)^{\frac{n+6}{2}} F_1\left (\frac{n+6}{2};\frac{n-4}{2},4;\frac{n+8}{2};\frac{1}{2} (a x+1),a x+1\right )}{a (n+6)} \]
Warning: Unable to verify antiderivative.
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Rule 6157
Rule 6150
Rule 136
Rubi steps
\begin{align*} \int e^{n \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^2 \, dx &=\frac{c^2 \int \frac{e^{n \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac{c^2 \int \frac{(1-a x)^{2-\frac{n}{2}} (1+a x)^{2+\frac{n}{2}}}{x^4} \, dx}{a^4}\\ &=\frac{2^{3-\frac{n}{2}} c^2 (1+a x)^{\frac{6+n}{2}} F_1\left (\frac{6+n}{2};\frac{1}{2} (-4+n),4;\frac{8+n}{2};\frac{1}{2} (1+a x),1+a x\right )}{a (6+n)}\\ \end{align*}
Mathematica [A] time = 0.800958, size = 229, normalized size = 0.69 \[ -\frac{c^2 e^{n \tanh ^{-1}(a x)} \left (a^3 \left (n^2-10\right ) n x^3 e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \tanh ^{-1}(a x)}\right )+a^3 \left (n^3+2 n^2-10 n-20\right ) x^3 \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \tanh ^{-1}(a x)}\right )-24 a^3 x^3 e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (2,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )+a^2 n^3 x^2-a^3 n^2 x^3+2 a^2 n^2 x^2-2 a^3 n x^3-12 a^2 n x^2-24 a^2 x^2+a n^2 x+2 a n x+2 n+4\right )}{6 a^4 (n+2) x^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \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 (c - \frac{c}{a^{2} x^{2}}\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 (\frac{{\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}}{a^{4} x^{4}}, 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*} \frac{c^{2} \left (\int a^{4} e^{n \operatorname{atanh}{\left (a x \right )}}\, dx + \int \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{x^{4}}\, dx + \int - \frac{2 a^{2} e^{n \operatorname{atanh}{\left (a x \right )}}}{x^{2}}\, dx\right )}{a^{4}} \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 (c - \frac{c}{a^{2} x^{2}}\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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