Optimal. Leaf size=1039 \[ \text{result too large to display} \]
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Rubi [A] time = 0.770349, antiderivative size = 1039, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {6160, 6150, 100, 159, 128, 45, 37, 69, 94, 90, 79} \[ -\frac{(a x+1)^{\frac{n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1}{2} (-n-3)}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}+\frac{(n+4) (a x+1)^{\frac{n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1}{2} (-n-3)}}{a^3 (n+3) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}-\frac{3 (n+4) (a x+1)^{\frac{n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1}{2} (-n-1)}}{a^5 (n+3) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}+\frac{n (a x+1)^{\frac{n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (2-n) (n+4) (a x+1)^{\frac{n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1}{2} (-n-1)}}{a^6 \left (9-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 (n+4) \left (-n^2+2 n+1\right ) (a x+1)^{\frac{n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1}{2} (-n-1)}}{a^6 (3-n) (n+1) (n+3) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 n (a x+1)^{\frac{n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 n (a x+1)^{\frac{n+1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{2^{\frac{n+3}{2}} n \left (1-a^2 x^2\right )^{5/2} \, _2F_1\left (\frac{1}{2} (-n-1),\frac{1}{2} (-n-1);\frac{1-n}{2};\frac{1}{2} (1-a x)\right ) (1-a x)^{\frac{1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{2 n (a x+1)^{\frac{n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1-n}{2}}}{a^6 \left (1-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 n (a x+1)^{\frac{n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1-n}{2}}}{a^6 \left (1-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (n+4) \left (-n^2+2 n+1\right ) (a x+1)^{\frac{n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{1-n}{2}}}{a^6 \left (n^4-10 n^2+9\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{2 n (a x+1)^{\frac{n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac{3-n}{2}}}{a^6 (n+1) \left (n^2-4 n+3\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 100
Rule 159
Rule 128
Rule 45
Rule 37
Rule 69
Rule 94
Rule 90
Rule 79
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \, dx &=\frac{\left (1-a^2 x^2\right )^{5/2} \int \frac{e^{n \tanh ^{-1}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (1-a^2 x^2\right )^{5/2} \int x^5 (1-a x)^{-\frac{5}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{\left (1-a^2 x^2\right )^{5/2} \int x^3 (1-a x)^{-\frac{5}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} (-4-a n x) \, dx}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{\left (n \left (1-a^2 x^2\right )^{5/2}\right ) \int x^3 (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{\left ((4+n) \left (1-a^2 x^2\right )^{5/2}\right ) \int x^3 (1-a x)^{-\frac{5}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{(4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^3 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{\left (n \left (1-a^2 x^2\right )^{5/2}\right ) \int \left (-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{-\frac{5}{2}+\frac{n}{2}}}{a^3}+\frac{3 (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{-\frac{3}{2}+\frac{n}{2}}}{a^3}-\frac{3 (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{-\frac{1}{2}+\frac{n}{2}}}{a^3}+\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2}+\frac{n}{2}}}{a^3}\right ) \, dx}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (3 (4+n) \left (1-a^2 x^2\right )^{5/2}\right ) \int x^2 (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{a^3 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{(4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^3 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{3 (4+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^5 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}+\frac{\left (n \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (n \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2}+\frac{n}{2}} \, dx}{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (3 n \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{-\frac{3}{2}+\frac{n}{2}} \, dx}{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{\left (3 n \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{-\frac{1}{2}+\frac{n}{2}} \, dx}{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (3 (4+n) \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} (-1+a (1-n) x) \, dx}{a^5 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{(4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^3 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}+\frac{n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (2-n) (4+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 \left (9-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (4+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^5 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}-\frac{3 n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1+n}{2}} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{2^{\frac{3+n}{2}} n (1-a x)^{\frac{1}{2} (-1-n)} \left (1-a^2 x^2\right )^{5/2} \, _2F_1\left (\frac{1}{2} (-1-n),\frac{1}{2} (-1-n);\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{\left (2 n \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{a^5 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (3 n \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{-\frac{3}{2}+\frac{n}{2}} \, dx}{a^5 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{\left (3 (4+n) \left (1+2 n-n^2\right ) \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \, dx}{a^5 (3-n) (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{(4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^3 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}+\frac{n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (2-n) (4+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 \left (9-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (4+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^5 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}-\frac{2 n (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 \left (1-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 (4+n) \left (1+2 n-n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (3-n) (1+n) (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 n (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 \left (1-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1+n}{2}} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{2^{\frac{3+n}{2}} n (1-a x)^{\frac{1}{2} (-1-n)} \left (1-a^2 x^2\right )^{5/2} \, _2F_1\left (\frac{1}{2} (-1-n),\frac{1}{2} (-1-n);\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (2 n \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{\frac{1-n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \, dx}{a^5 (1-n) (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{\left (3 (4+n) \left (1+2 n-n^2\right ) \left (1-a^2 x^2\right )^{5/2}\right ) \int (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \, dx}{a^5 (3-n) (1+n) (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{(4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^3 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}+\frac{n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (2-n) (4+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 \left (9-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (4+n) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^5 (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}-\frac{2 n (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 \left (1-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{2 n (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1-n) (3-n) (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 (4+n) \left (1+2 n-n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 (3-n) (1+n) (3+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 n (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 \left (1-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{3 (4+n) \left (1+2 n-n^2\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{5/2}}{a^6 \left (9-10 n^2+n^4\right ) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}+\frac{3 n (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1+n}{2}} \left (1-a^2 x^2\right )^{5/2}}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{2^{\frac{3+n}{2}} n (1-a x)^{\frac{1}{2} (-1-n)} \left (1-a^2 x^2\right )^{5/2} \, _2F_1\left (\frac{1}{2} (-1-n),\frac{1}{2} (-1-n);\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{a^6 (1+n) \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ \end{align*}
Mathematica [A] time = 6.31678, size = 227, normalized size = 0.22 \[ \frac{\left (a^2 x^2-1\right )^2 \left (-\frac{4 \left (a^2 x^2-1\right ) \left (\frac{2 n e^{(n+1) \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-e^{2 \tanh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-(n+1) e^{n \tanh ^{-1}(a x)}\right )}{n+1}-\frac{e^{n \tanh ^{-1}(a x)} \left (3 \left (n^2-1\right ) \sqrt{1-a^2 x^2} \cosh \left (3 \tanh ^{-1}(a x)\right )-2 a n^3 x-2 a \left (n^2-1\right ) n x \cosh \left (2 \tanh ^{-1}(a x)\right )+10 a n x+n^2-9\right )}{n^4-10 n^2+9}-\frac{8 (a n x-1) e^{n \tanh ^{-1}(a x)}}{n^2-1}\right )}{4 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.116, 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 ) ^{-{\frac{5}{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 \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{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^{6} x^{6} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right ) \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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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