Optimal. Leaf size=174 \[ \frac {2 \left (n^2+2\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{3 a^3 (2-n)}+\frac {1}{3} x^3 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}+\frac {n x^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}}{6 a} \]
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Rubi [A] time = 0.09, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6171, 129, 151, 12, 131} \[ \frac {2 \left (n^2+2\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{3 a^3 (2-n)}+\frac {1}{3} x^3 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}+\frac {n x^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}}{6 a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 129
Rule 131
Rule 151
Rule 6171
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3+\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (-\frac {n}{a}-\frac {x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{6 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3-\frac {1}{6} \operatorname {Subst}\left (\int \frac {\left (2+n^2\right ) \left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{a^2 x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{6 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3-\frac {\left (2+n^2\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{6 a^2}\\ &=\frac {n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{6 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3+\frac {2 \left (2+n^2\right ) \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 (2,1-\frac {n}{2};2-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{3 a^3 (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 118, normalized size = 0.68 \[ \frac {e^{n \coth ^{-1}(a x)} \left ((n+2) \left (2 a^3 x^3+n \left (a^2 x^2-1\right )+\left (n^2+2\right ) \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \coth ^{-1}(a x)}\right )+a n^2 x\right )+n \left (n^2+2\right ) e^{2 \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \coth ^{-1}(a x)}\right )\right )}{6 a^3 (n+2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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