Optimal. Leaf size=81 \[ -\frac {256 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-8+n)} \, _2F_1\left (8,4-\frac {n}{2};5-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \]
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Rubi [A]
time = 0.10, 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 = {6326, 6330,
133} \begin {gather*} -\frac {256 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-8}{2}} \, _2F_1\left (8,4-\frac {n}{2};5-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 133
Rule 6326
Rule 6330
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=-\left (\left (a^6 c^3\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^3 x^6 \, dx\right )\\ &=\left (a^6 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{3+\frac {n}{2}}}{x^8} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {256 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-8+n)} \, _2F_1\left (8,4-\frac {n}{2};5-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(267\) vs. \(2(81)=162\).
time = 1.52, size = 267, normalized size = 3.30 \begin {gather*} -\frac {c^3 e^{n \coth ^{-1}(a x)} \left (-912 n+58 n^3-n^5-5040 a x+912 a n^2 x-58 a n^4 x+a n^6 x+1368 a^2 n x^2-64 a^2 n^3 x^2+a^2 n^5 x^2+5040 a^3 x^3-152 a^3 n^2 x^3+2 a^3 n^4 x^3-576 a^4 n x^4+6 a^4 n^3 x^4-3024 a^5 x^5+24 a^5 n^2 x^5+120 a^6 n x^6+720 a^7 x^7+e^{2 \coth ^{-1}(a x)} n \left (-1152+576 n+104 n^2-52 n^3-2 n^4+n^5\right ) \, _2F_1\left (1,1+\frac {n}{2};2+\frac {n}{2};e^{2 \coth ^{-1}(a x)}\right )+\left (-2304+784 n^2-56 n^4+n^6\right ) \, _2F_1\left (1,\frac {n}{2};1+\frac {n}{2};e^{2 \coth ^{-1}(a x)}\right )\right )}{5040 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{3}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - c^{3} \left (\int 3 a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- 3 a^{4} x^{4} e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx + \int a^{6} x^{6} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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