Optimal. Leaf size=72 \[ \frac{2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{(n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]
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Rubi [A] time = 0.0778262, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ \frac{2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{(n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6185
Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}+\frac{2 \int \frac{e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c \left (4-n^2\right )}\\ &=\frac{2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.161797, size = 55, normalized size = 0.76 \[ -\frac{\left (2 a^2 x^2-2 a n x+n^2-2\right ) e^{n \coth ^{-1}(a x)}}{a c^2 n \left (n^2-4\right ) \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 55, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( 2\,{a}^{2}{x}^{2}-2\,nax+{n}^{2}-2 \right ) }{ \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}an \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{\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 = 1.73817, size = 165, normalized size = 2.29 \begin{align*} -\frac{{\left (2 \, a^{2} x^{2} + 2 \, a n x + n^{2} - 2\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{2} n^{3} - 4 \, a c^{2} n -{\left (a^{3} c^{2} n^{3} - 4 \, a^{3} 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{\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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