Optimal. Leaf size=164 \[ -\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a^2 c (1-n) \sqrt{c-a^2 c x^2}}-\frac{(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.335145, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {6189, 6192, 6195, 131} \[ -\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a^2 c (1-n) \sqrt{c-a^2 c x^2}}-\frac{(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6189
Rule 6192
Rule 6195
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{\int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx}{a^2 c}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{\left (\sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a^2 x^2}} x} \, dx}{a^2 c \sqrt{c-a^2 c x^2}}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\left (\sqrt{1-\frac{1}{a^2 x^2}} x\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{a^2 c \sqrt{c-a^2 c x^2}}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a^2 c (1-n) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.375046, size = 127, normalized size = 0.77 \[ -\frac{e^{n \coth ^{-1}(a x)} \left (2 (n-1) \left (a^2 x^2-1\right ) e^{\coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )+a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-n)\right )}{a^4 c (n-1) (n+1) x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.309, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{x}^{2} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, 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{x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{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{\sqrt{-a^{2} c x^{2} + c} x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{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*} \int \frac{x^{2} e^{n \operatorname{acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \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^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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