Optimal. Leaf size=190 \[ -\frac{2 (a x+1)^{n/2} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n+2}{2},\frac{a x+1}{1-a x}\right )}{c^2 n}-\frac{\left (-n^2-n+4\right ) (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{c^2 n \left (4-n^2\right )}+\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2 (n+2)}+\frac{(n+4) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{c^2 n (n+2)} \]
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Rubi [A] time = 0.202266, antiderivative size = 200, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6150, 129, 155, 12, 131} \[ -\frac{2 (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{a x+1}\right )}{c^2 (2-n)}-\frac{\left (-n^2-n+4\right ) (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{c^2 n \left (4-n^2\right )}+\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{c^2 (n+2)}+\frac{(n+4) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{c^2 n (n+2)} \]
Warning: Unable to verify antiderivative.
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Rule 6150
Rule 129
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{(1-a x)^{-2-\frac{n}{2}} (1+a x)^{-2+\frac{n}{2}}}{x} \, dx}{c^2}\\ &=\frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 (2+n)}-\frac{\int \frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{-2+\frac{n}{2}} \left (-a (2+n)-2 a^2 x\right )}{x} \, dx}{a c^2 (2+n)}\\ &=\frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 (2+n)}+\frac{(4+n) (1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 n (2+n)}+\frac{\int \frac{(1-a x)^{-n/2} (1+a x)^{-2+\frac{n}{2}} \left (a^2 n (2+n)+a^3 (4+n) x\right )}{x} \, dx}{a^2 c^2 n (2+n)}\\ &=\frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 (2+n)}-\frac{\left (4-n-n^2\right ) (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 n \left (4-n^2\right )}+\frac{(4+n) (1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 n (2+n)}+\frac{\int \frac{a^3 (2-n) n (2+n) (1-a x)^{-n/2} (1+a x)^{-1+\frac{n}{2}}}{x} \, dx}{a^3 c^2 n \left (4-n^2\right )}\\ &=\frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 (2+n)}-\frac{\left (4-n-n^2\right ) (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 n \left (4-n^2\right )}+\frac{(4+n) (1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 n (2+n)}+\frac{\int \frac{(1-a x)^{-n/2} (1+a x)^{-1+\frac{n}{2}}}{x} \, dx}{c^2}\\ &=\frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 (2+n)}-\frac{\left (4-n-n^2\right ) (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 n \left (4-n^2\right )}+\frac{(4+n) (1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)}}{c^2 n (2+n)}-\frac{2 (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{1+a x}\right )}{c^2 (2-n)}\\ \end{align*}
Mathematica [A] time = 0.069729, size = 121, normalized size = 0.64 \[ -\frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n}{2}-1} \left (-2 (n+2) n (a x-1)^2 \text{Hypergeometric2F1}\left (1,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )+n^2 \left (a^2 x^2-a x-1\right )+a^2 n x^2-4 a^2 x^2+n+4\right )}{c^2 n \left (n^2-4\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.183, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{x \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 \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2} x}\,{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 (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{5} - 2 \, a^{2} c^{2} x^{3} + c^{2} x}, 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{\int \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{4} x^{5} - 2 a^{2} x^{3} + x}\, dx}{c^{2}} \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 (a^{2} c x^{2} - c\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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