Optimal. Leaf size=97 \[ \frac{2 n (n-a x) e^{n \coth ^{-1}(a x)}}{a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{(3-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.147321, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6187, 6184} \[ \frac{2 n (n-a x) e^{n \coth ^{-1}(a x)}}{a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{(3-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6187
Rule 6184
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{e^{n \coth ^{-1}(a x)} (3-a n x)}{a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{(2 n) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{a c \left (9-n^2\right )}\\ &=\frac{e^{n \coth ^{-1}(a x)} (3-a n x)}{a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 e^{n \coth ^{-1}(a x)} n (n-a x)}{a^2 c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.586381, size = 108, normalized size = 1.11 \[ \frac{e^{n \coth ^{-1}(a x)} \left (-a \left (n^2-1\right ) n x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )+a n^3 x+6 \left (n^2-1\right ) \cosh \left (2 \coth ^{-1}(a x)\right )-9 a n x+2 n^2+6\right )}{4 a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.041, size = 86, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,{a}^{3}n{x}^{3}-2\,{n}^{2}{x}^{2}{a}^{2}+a{n}^{3}x-3\,nax-{n}^{2}+3 \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{{a}^{2} \left ({n}^{4}-10\,{n}^{2}+9 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \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{x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45122, size = 346, normalized size = 3.57 \begin{align*} -\frac{{\left (2 \, a^{3} n x^{3} + 2 \, a^{2} n^{2} x^{2} + n^{2} +{\left (a n^{3} - 3 \, a n\right )} x - 3\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{2} c^{3} n^{4} - 10 \, a^{2} c^{3} n^{2} + 9 \, a^{2} c^{3} +{\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{4} - 2 \,{\left (a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3}\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{x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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