Optimal. Leaf size=79 \[ -\frac{(n-2 a x) e^{n \tanh ^{-1}(a x)}}{a^3 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}-\frac{\left (2-n^2\right ) e^{n \tanh ^{-1}(a x)}}{a^3 c^2 n \left (4-n^2\right )} \]
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Rubi [A] time = 0.127441, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6147, 6137} \[ -\frac{(n-2 a x) e^{n \tanh ^{-1}(a x)}}{a^3 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}-\frac{\left (2-n^2\right ) e^{n \tanh ^{-1}(a x)}}{a^3 c^2 n \left (4-n^2\right )} \]
Antiderivative was successfully verified.
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Rule 6147
Rule 6137
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac{e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a^3 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}-\frac{\left (2-n^2\right ) \int \frac{e^{n \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{a^2 c \left (4-n^2\right )}\\ &=-\frac{e^{n \tanh ^{-1}(a x)} \left (2-n^2\right )}{a^3 c^2 n \left (4-n^2\right )}-\frac{e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a^3 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.052017, size = 65, normalized size = 0.82 \[ -\frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n}{2}-1} \left (-a^2 \left (n^2-2\right ) x^2+2 a n x-2\right )}{a^3 c^2 n \left (n^2-4\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.028, size = 62, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ({a}^{2}{n}^{2}{x}^{2}-2\,{a}^{2}{x}^{2}-2\,nax+2 \right ) }{ \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}{a}^{3}n \left ({n}^{2}-4 \right ) }} \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{x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17578, size = 178, normalized size = 2.25 \begin{align*} -\frac{{\left (2 \, a n x -{\left (a^{2} n^{2} - 2 \, a^{2}\right )} x^{2} - 2\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{3} c^{2} n^{3} - 4 \, a^{3} c^{2} n -{\left (a^{5} c^{2} n^{3} - 4 \, a^{5} c^{2} n\right )} x^{2}} \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{x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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