Optimal. Leaf size=21 \[ 3 \left (\frac {1}{x^6}+e^{\left (-2+x^2\right )^4} \left (-4+x^2\right )\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.70, antiderivative size = 99, normalized size of antiderivative = 4.71, number of steps
used = 9, number of rules used = 7, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {1607, 6820,
6847, 2258, 2239, 2240, 2250} \begin {gather*} \frac {3 \left (2-x^2\right )^5 \text {Gamma}\left (\frac {5}{4},-\left (2-x^2\right )^4\right )}{\left (-\left (2-x^2\right )^4\right )^{5/4}}+\frac {3 \left (2-x^2\right ) \text {Gamma}\left (\frac {1}{4},-\left (2-x^2\right )^4\right )}{4 \sqrt [4]{-\left (2-x^2\right )^4}}+\frac {3}{x^6}-6 e^{\left (x^2-2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 1607
Rule 2239
Rule 2240
Rule 2250
Rule 2258
Rule 6820
Rule 6847
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{x^7 \left (-4+x^2\right )} \, dx\\ &=\int \left (-\frac {18}{x^7}+6 e^{\left (-2+x^2\right )^4} x \left (129-224 x^2+144 x^4-40 x^6+4 x^8\right )\right ) \, dx\\ &=\frac {3}{x^6}+6 \int e^{\left (-2+x^2\right )^4} x \left (129-224 x^2+144 x^4-40 x^6+4 x^8\right ) \, dx\\ &=\frac {3}{x^6}+3 \text {Subst}\left (\int e^{(-2+x)^4} \left (129-224 x+144 x^2-40 x^3+4 x^4\right ) \, dx,x,x^2\right )\\ &=\frac {3}{x^6}+3 \text {Subst}\left (\int \left (e^{(-2+x)^4}-8 e^{(-2+x)^4} (-2+x)^3+4 e^{(-2+x)^4} (-2+x)^4\right ) \, dx,x,x^2\right )\\ &=\frac {3}{x^6}+3 \text {Subst}\left (\int e^{(-2+x)^4} \, dx,x,x^2\right )+12 \text {Subst}\left (\int e^{(-2+x)^4} (-2+x)^4 \, dx,x,x^2\right )-24 \text {Subst}\left (\int e^{(-2+x)^4} (-2+x)^3 \, dx,x,x^2\right )\\ &=-6 e^{\left (-2+x^2\right )^4}+\frac {3}{x^6}+\frac {3 \left (2-x^2\right ) \Gamma \left (\frac {1}{4},-\left (2-x^2\right )^4\right )}{4 \sqrt [4]{-\left (2-x^2\right )^4}}+\frac {3 \left (2-x^2\right )^5 \Gamma \left (\frac {5}{4},-\left (2-x^2\right )^4\right )}{\left (-\left (2-x^2\right )^4\right )^{5/4}}\\ \end {aligned} \end {gather*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.43, size = 95, normalized size = 4.52 \begin {gather*} \frac {3}{x^6}+3 \left (-2 e^{\left (-2+x^2\right )^4}-\frac {\left (-2+x^2\right ) \Gamma \left (\frac {1}{4},-\left (-2+x^2\right )^4\right )}{4 \sqrt [4]{-\left (-2+x^2\right )^4}}+\frac {\left (2-x^2\right )^5 \Gamma \left (\frac {5}{4},-\left (2-x^2\right )^4\right )}{\left (-\left (2-x^2\right )^4\right )^{5/4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 23, normalized size = 1.10
method | result | size |
risch | \(\frac {3}{x^{6}}+\left (3 x^{2}-12\right ) {\mathrm e}^{\left (x^{2}-2\right )^{4}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (23) = 46\).
time = 0.36, size = 61, normalized size = 2.90 \begin {gather*} 3 \, {\left (x^{2} e^{16} - 4 \, e^{16}\right )} e^{\left (x^{8} - 8 \, x^{6} + 24 \, x^{4} - 32 \, x^{2}\right )} - \frac {9 \, {\left (x^{2} + 2\right )}}{16 \, x^{4}} + \frac {3 \, {\left (3 \, x^{4} + 6 \, x^{2} + 16\right )}}{16 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 38, normalized size = 1.81 \begin {gather*} \frac {3 \, {\left (x^{6} e^{\left (x^{8} - 8 \, x^{6} + 24 \, x^{4} - 32 \, x^{2} + \log \left (x^{2} - 4\right ) + 16\right )} + 1\right )}}{x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 32, normalized size = 1.52 \begin {gather*} \left (3 x^{2} - 12\right ) e^{x^{8} - 8 x^{6} + 24 x^{4} - 32 x^{2} + 16} + \frac {3}{x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs.
\(2 (23) = 46\).
time = 0.61, size = 194, normalized size = 9.24 \begin {gather*} \frac {3 \, {\left ({\left (x^{2} - 4\right )}^{4} e^{\left ({\left (x^{2} - 4\right )}^{4} + 8 \, {\left (x^{2} - 4\right )}^{3} + 24 \, {\left (x^{2} - 4\right )}^{2} + 32 \, x^{2} - 112\right )} + 12 \, {\left (x^{2} - 4\right )}^{3} e^{\left ({\left (x^{2} - 4\right )}^{4} + 8 \, {\left (x^{2} - 4\right )}^{3} + 24 \, {\left (x^{2} - 4\right )}^{2} + 32 \, x^{2} - 112\right )} + 48 \, {\left (x^{2} - 4\right )}^{2} e^{\left ({\left (x^{2} - 4\right )}^{4} + 8 \, {\left (x^{2} - 4\right )}^{3} + 24 \, {\left (x^{2} - 4\right )}^{2} + 32 \, x^{2} - 112\right )} + 64 \, {\left (x^{2} - 4\right )} e^{\left ({\left (x^{2} - 4\right )}^{4} + 8 \, {\left (x^{2} - 4\right )}^{3} + 24 \, {\left (x^{2} - 4\right )}^{2} + 32 \, x^{2} - 112\right )} + 1\right )}}{{\left (x^{2} - 4\right )}^{3} + 12 \, {\left (x^{2} - 4\right )}^{2} + 48 \, x^{2} - 128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 61, normalized size = 2.90 \begin {gather*} \frac {3}{x^6}-12\,{\mathrm {e}}^{x^8}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{-8\,x^6}\,{\mathrm {e}}^{24\,x^4}\,{\mathrm {e}}^{-32\,x^2}+3\,x^2\,{\mathrm {e}}^{x^8}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{-8\,x^6}\,{\mathrm {e}}^{24\,x^4}\,{\mathrm {e}}^{-32\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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