Optimal. Leaf size=123 \[ -\frac{2 a (a x+1)^{n/2} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n+2}{2},\frac{a x+1}{1-a x}\right )}{c}+\frac{a (n+1) (a x+1)^{n/2} (1-a x)^{-n/2}}{c n}-\frac{(a x+1)^{n/2} (1-a x)^{-n/2}}{c x} \]
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Rubi [A] time = 0.143568, antiderivative size = 137, normalized size of antiderivative = 1.11, number of steps used = 5, 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} \[ -\frac{2 a n (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{a x+1}\right )}{c (2-n)}+\frac{a (n+1) (a x+1)^{n/2} (1-a x)^{-n/2}}{c n}-\frac{(a x+1)^{n/2} (1-a x)^{-n/2}}{c x} \]
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^2 \left (c-a^2 c x^2\right )} \, dx &=\frac{\int \frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}}}{x^2} \, dx}{c}\\ &=-\frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}-\frac{\int \frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}} \left (-a n-a^2 x\right )}{x} \, dx}{c}\\ &=\frac{a (1+n) (1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}+\frac{\int \frac{a^2 n^2 (1-a x)^{-n/2} (1+a x)^{-1+\frac{n}{2}}}{x} \, dx}{a c n}\\ &=\frac{a (1+n) (1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}+\frac{(a n) \int \frac{(1-a x)^{-n/2} (1+a x)^{-1+\frac{n}{2}}}{x} \, dx}{c}\\ &=\frac{a (1+n) (1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}-\frac{2 a n (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-n)}\\ \end{align*}
Mathematica [A] time = 0.044348, size = 103, normalized size = 0.84 \[ \frac{(1-a x)^{-n/2} (a x+1)^{\frac{n}{2}-1} \left ((n-2) (a x+1) (n (a x-1)+a x)-2 a n^2 x (a x-1) \text{Hypergeometric2F1}\left (1,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )\right )}{c (n-2) n x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{2} \left ( -{a}^{2}c{x}^{2}+c \right ) }}\, 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 )} x^{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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{4} - c x^{2}}, 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^{2} x^{4} - x^{2}}\, dx}{c} \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 )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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