Optimal. Leaf size=150 \[ \frac{2 \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2} \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n+2}{2},\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right )}{a c}-\frac{(n+1) \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2}}{a c n}+\frac{x \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2}}{c} \]
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Rubi [A] time = 0.123679, antiderivative size = 164, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6194, 129, 155, 12, 131} \[ \frac{2 n \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a c (2-n)}-\frac{(n+1) \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2}}{a c n}+\frac{x \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2}}{c} \]
Warning: Unable to verify antiderivative.
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Rule 6194
Rule 129
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-1+\frac{n}{2}}}{x^2} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{n/2} x}{c}+\frac{\operatorname{Subst}\left (\int \frac{\left (-\frac{n}{a}-\frac{x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{-1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-1+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{(1+n) \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{n/2}}{a c n}+\frac{\left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{n/2} x}{c}-\frac{a \operatorname{Subst}\left (\int \frac{n^2 \left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{-1+\frac{n}{2}}}{a^2 x} \, dx,x,\frac{1}{x}\right )}{c n}\\ &=-\frac{(1+n) \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{n/2}}{a c n}+\frac{\left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{n/2} x}{c}-\frac{n \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{-1+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=-\frac{(1+n) \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{n/2}}{a c n}+\frac{\left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{n/2} x}{c}+\frac{2 n \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a c (2-n)}\\ \end{align*}
Mathematica [A] time = 0.252392, size = 94, normalized size = 0.63 \[ \frac{e^{n \coth ^{-1}(a x)} \left (n^2 e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \coth ^{-1}(a x)}\right )+(n+2) \left (n \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \coth ^{-1}(a x)}\right )+a n x-1\right )\right )}{a c n (n+2)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) ^{-1}}\, 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}}{c - \frac{c}{a^{2} 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{a^{2} x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, 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{a^{2} \int \frac{x^{2} e^{n \operatorname{acoth}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, 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}}{c - \frac{c}{a^{2} x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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