Optimal. Leaf size=81 \[ \frac {64 c^2 \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-6}{2}} \, _2F_1\left (6,3-\frac {n}{2};4-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (6-n)} \]
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Rubi [A] time = 0.14, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6191, 6195, 131} \[ \frac {64 c^2 \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-6}{2}} \, _2F_1\left (6,3-\frac {n}{2};4-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (6-n)} \]
Antiderivative was successfully verified.
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Rule 131
Rule 6191
Rule 6195
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=\left (a^4 c^2\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^2 x^4 \, dx\\ &=-\left (\left (a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{2+\frac {n}{2}}}{x^6} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {64 c^2 \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-6+n)} \, _2F_1\left (6,3-\frac {n}{2};4-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (6-n)}\\ \end {align*}
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Mathematica [B] time = 1.42, size = 179, normalized size = 2.21 \[ \frac {c^2 e^{n \coth ^{-1}(a x)} \left (24 a^5 x^5+6 a^4 n x^4+2 a^3 n^2 x^3-80 a^3 x^3+a^2 n^3 x^2-28 a^2 n x^2+\left (n^4-20 n^2+64\right ) \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \coth ^{-1}(a x)}\right )+n \left (n^3-2 n^2-16 n+32\right ) e^{2 \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \coth ^{-1}(a x)}\right )+a n^4 x-22 a n^2 x+120 a x-n^3+22 n\right )}{120 a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (a^{2} c x^{2} - c\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{2}\, dx \]
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 \[ \int {\left (a^{2} c x^{2} - c\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \]
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 \[ c^{2} \left (\int \left (- 2 a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx + \int a^{4} x^{4} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int e^{n \operatorname {acoth}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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