Optimal. Leaf size=127 \[ -\frac{(n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac{12 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}+\frac{24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (n^4-20 n^2+64\right )} \]
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Rubi [A] time = 0.129511, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ -\frac{(n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac{12 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}+\frac{24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (n^4-20 n^2+64\right )} \]
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 )^3} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac{12 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{c \left (16-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac{12 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}+\frac{24 \int \frac{e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c^2 \left (64-20 n^2+n^4\right )}\\ &=\frac{24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (64-20 n^2+n^4\right )}-\frac{e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac{12 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.216089, size = 97, normalized size = 0.76 \[ \frac{\left (4 n^2 \left (3 a^2 x^2-4\right )-8 a n x \left (3 a^2 x^2-5\right )+24 \left (a^2 x^2-1\right )^2-4 a n^3 x+n^4\right ) e^{n \coth ^{-1}(a x)}}{a c^3 n \left (n^2-16\right ) \left (n^2-4\right ) \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 101, normalized size = 0.8 \begin{align*}{\frac{ \left ( 24\,{x}^{4}{a}^{4}-24\,{x}^{3}{a}^{3}n+12\,{a}^{2}{n}^{2}{x}^{2}-4\,a{n}^{3}x-48\,{a}^{2}{x}^{2}+{n}^{4}+40\,nax-16\,{n}^{2}+24 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{ \left ({a}^{2}{x}^{2}-1 \right ) ^{2}{c}^{3}a \left ({n}^{2}-16 \right ) \left ({n}^{2}-4 \right ) n}} \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66716, size = 373, normalized size = 2.94 \begin{align*} -\frac{{\left (24 \, a^{4} x^{4} + 24 \, a^{3} n x^{3} + n^{4} + 12 \,{\left (a^{2} n^{2} - 4 \, a^{2}\right )} x^{2} - 16 \, n^{2} + 4 \,{\left (a n^{3} - 10 \, a n\right )} x + 24\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{3} n^{5} - 20 \, a c^{3} n^{3} + 64 \, a c^{3} n +{\left (a^{5} c^{3} n^{5} - 20 \, a^{5} c^{3} n^{3} + 64 \, a^{5} c^{3} n\right )} x^{4} - 2 \,{\left (a^{3} c^{3} n^{5} - 20 \, a^{3} c^{3} n^{3} + 64 \, a^{3} c^{3} 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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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