Optimal. Leaf size=33 \[ \frac {2-x+\frac {2 x}{5+\frac {6}{x}}}{e^4+\frac {x}{-5+x^2}} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(33)=66\).
time = 0.23, antiderivative size = 90, normalized size of antiderivative = 2.73, number of steps
used = 7, number of rules used = 4, integrand size = 140, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2099, 652,
632, 212} \begin {gather*} -\frac {15 e^4 \left (6+13 e^4\right )-\left (18+99 e^4+118 e^8\right ) x}{e^8 \left (30+89 e^4\right ) \left (-e^4 x^2-x+5 e^4\right )}-\frac {3 x}{5 e^4}+\frac {6408}{25 \left (30+89 e^4\right ) (5 x+6)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 652
Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3}{5 e^4}-\frac {6408}{5 \left (30+89 e^4\right ) (6+5 x)^2}+\frac {5 e^4 \left (18+159 e^4+236 e^8\right )-\left (18+99 e^4+298 e^8+390 e^{12}\right ) x}{e^8 \left (30+89 e^4\right ) \left (5 e^4-x-e^4 x^2\right )^2}+\frac {18+99 e^4+118 e^8}{e^8 \left (30+89 e^4\right ) \left (-5 e^4+x+e^4 x^2\right )}\right ) \, dx\\ &=-\frac {3 x}{5 e^4}+\frac {6408}{25 \left (30+89 e^4\right ) (6+5 x)}+\frac {\int \frac {5 e^4 \left (18+159 e^4+236 e^8\right )-\left (18+99 e^4+298 e^8+390 e^{12}\right ) x}{\left (5 e^4-x-e^4 x^2\right )^2} \, dx}{e^8 \left (30+89 e^4\right )}+\frac {\left (18+99 e^4+118 e^8\right ) \int \frac {1}{-5 e^4+x+e^4 x^2} \, dx}{e^8 \left (30+89 e^4\right )}\\ &=-\frac {3 x}{5 e^4}+\frac {6408}{25 \left (30+89 e^4\right ) (6+5 x)}-\frac {15 e^4 \left (6+13 e^4\right )-\left (18+99 e^4+118 e^8\right ) x}{e^8 \left (30+89 e^4\right ) \left (5 e^4-x-e^4 x^2\right )}+\frac {\left (18+99 e^4+118 e^8\right ) \int \frac {1}{5 e^4-x-e^4 x^2} \, dx}{e^8 \left (30+89 e^4\right )}-\frac {\left (2 \left (18+99 e^4+118 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{1+20 e^8-x^2} \, dx,x,1+2 e^4 x\right )}{e^8 \left (30+89 e^4\right )}\\ &=-\frac {3 x}{5 e^4}+\frac {6408}{25 \left (30+89 e^4\right ) (6+5 x)}-\frac {15 e^4 \left (6+13 e^4\right )-\left (18+99 e^4+118 e^8\right ) x}{e^8 \left (30+89 e^4\right ) \left (5 e^4-x-e^4 x^2\right )}-\frac {2 \left (18+99 e^4+118 e^8\right ) \tanh ^{-1}\left (\frac {1+2 e^4 x}{\sqrt {1+20 e^8}}\right )}{e^8 \left (30+89 e^4\right ) \sqrt {1+20 e^8}}-\frac {\left (2 \left (18+99 e^4+118 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{1+20 e^8-x^2} \, dx,x,-1-2 e^4 x\right )}{e^8 \left (30+89 e^4\right )}\\ &=-\frac {3 x}{5 e^4}+\frac {6408}{25 \left (30+89 e^4\right ) (6+5 x)}-\frac {15 e^4 \left (6+13 e^4\right )-\left (18+99 e^4+118 e^8\right ) x}{e^8 \left (30+89 e^4\right ) \left (5 e^4-x-e^4 x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(33)=66\).
time = 0.07, size = 88, normalized size = 2.67 \begin {gather*} \frac {-15 x (6+5 x)+e^4 \left (450+147 x-280 x^2-75 x^3\right )+e^8 \left (-360+450 x+447 x^2-90 x^3-75 x^4\right )}{25 e^8 \left (x (6+5 x)+e^4 \left (-30-25 x+6 x^2+5 x^3\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.43, size = 196, normalized size = 5.94
method | result | size |
norman | \(\frac {-3 x^{4}-\frac {24 x \,{\mathrm e}^{-4}}{5}+\frac {{\mathrm e}^{-4} \left (-20+111 \,{\mathrm e}^{4}\right ) x^{2}}{5}-36}{\left (5 x +6\right ) \left (x^{2} {\mathrm e}^{4}-5 \,{\mathrm e}^{4}+x \right )}\) | \(52\) |
gosper | \(-\frac {\left (15 x^{4} {\mathrm e}^{4}-111 x^{2} {\mathrm e}^{4}+20 x^{2}+180 \,{\mathrm e}^{4}+24 x \right ) {\mathrm e}^{-4}}{5 \left (5 x^{3} {\mathrm e}^{4}+6 x^{2} {\mathrm e}^{4}-25 x \,{\mathrm e}^{4}+5 x^{2}-30 \,{\mathrm e}^{4}+6 x \right )}\) | \(68\) |
risch | \(-\frac {3 x \,{\mathrm e}^{-4}}{5}+\frac {\left (\frac {{\mathrm e}^{-4} \left (72 \,{\mathrm e}^{8}-190 \,{\mathrm e}^{4}-75\right ) x^{2}}{25}+\frac {3 \left (49 \,{\mathrm e}^{4}-30\right ) {\mathrm e}^{-4} x}{25}-\frac {72 \,{\mathrm e}^{4}}{5}+18\right ) {\mathrm e}^{-4}}{5 x^{3} {\mathrm e}^{4}+6 x^{2} {\mathrm e}^{4}-25 x \,{\mathrm e}^{4}+5 x^{2}-30 \,{\mathrm e}^{4}+6 x}\) | \(76\) |
default | \(-\frac {3 \,{\mathrm e}^{4} {\mathrm e}^{-8} x}{5}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4} {\mathrm e}^{8}+2 \textit {\_Z}^{3} {\mathrm e}^{4}+\left (-10 \,{\mathrm e}^{8}+1\right ) \textit {\_Z}^{2}-10 \textit {\_Z} \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}\right )}{\sum }\frac {\left (8100000+\left (39807720 \,{\mathrm e}^{4}+153912951 \,{\mathrm e}^{12}+111620952 \,{\mathrm e}^{8}+83186342 \,{\mathrm e}^{16}+486000 \,{\mathrm e}^{-4}+6998400\right ) \textit {\_R}^{2}+30 \left (-162000-1792800 \,{\mathrm e}^{4}-13497384 \,{\mathrm e}^{12}-7401240 \,{\mathrm e}^{8}-9164597 \,{\mathrm e}^{16}\right ) \textit {\_R} +88020000 \,{\mathrm e}^{4}+632095800 \,{\mathrm e}^{12}+355644000 \,{\mathrm e}^{8}+415931710 \,{\mathrm e}^{16}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} {\mathrm e}^{8}+3 \textit {\_R}^{2} {\mathrm e}^{4}-10 \textit {\_R} \,{\mathrm e}^{8}-5 \,{\mathrm e}^{4}+\textit {\_R}}}{2 \left (7921 \,{\mathrm e}^{8}+5340 \,{\mathrm e}^{4}+900\right )^{2}}-\frac {-4568199120 \,{\mathrm e}^{8}-1539842400 \,{\mathrm e}^{4}-4517441352 \,{\mathrm e}^{12}-173016000}{25 \left (7921 \,{\mathrm e}^{8}+5340 \,{\mathrm e}^{4}+900\right )^{2} \left (5 x +6\right )}\) | \(196\) |
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. 79 vs.
\(2 (31) = 62\).
time = 0.29, size = 79, normalized size = 2.39 \begin {gather*} -\frac {3}{5} \, x e^{\left (-4\right )} + \frac {x^{2} {\left (72 \, e^{8} - 190 \, e^{4} - 75\right )} + 3 \, x {\left (49 \, e^{4} - 30\right )} - 360 \, e^{8} + 450 \, e^{4}}{25 \, {\left (5 \, x^{3} e^{12} + x^{2} {\left (6 \, e^{12} + 5 \, e^{8}\right )} - x {\left (25 \, e^{12} - 6 \, e^{8}\right )} - 30 \, e^{12}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (31) = 62\).
time = 0.36, size = 86, normalized size = 2.61 \begin {gather*} -\frac {75 \, x^{2} + 3 \, {\left (25 \, x^{4} + 30 \, x^{3} - 149 \, x^{2} - 150 \, x + 120\right )} e^{8} + {\left (75 \, x^{3} + 280 \, x^{2} - 147 \, x - 450\right )} e^{4} + 90 \, x}{25 \, {\left ({\left (5 \, x^{3} + 6 \, x^{2} - 25 \, x - 30\right )} e^{12} + {\left (5 \, x^{2} + 6 \, x\right )} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (20) = 40\).
time = 2.33, size = 82, normalized size = 2.48 \begin {gather*} - \frac {3 x}{5 e^{4}} - \frac {x^{2} \left (- 72 e^{8} + 75 + 190 e^{4}\right ) + x \left (90 - 147 e^{4}\right ) - 450 e^{4} + 360 e^{8}}{125 x^{3} e^{12} + x^{2} \cdot \left (125 e^{8} + 150 e^{12}\right ) + x \left (- 625 e^{12} + 150 e^{8}\right ) - 750 e^{12}} \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. 80 vs.
\(2 (31) = 62\).
time = 0.43, size = 80, normalized size = 2.42 \begin {gather*} -\frac {3}{5} \, x e^{\left (-4\right )} + \frac {{\left (72 \, x^{2} e^{8} - 190 \, x^{2} e^{4} - 75 \, x^{2} + 147 \, x e^{4} - 90 \, x - 360 \, e^{8} + 450 \, e^{4}\right )} e^{\left (-8\right )}}{25 \, {\left (5 \, x^{3} e^{4} + 6 \, x^{2} e^{4} + 5 \, x^{2} - 25 \, x e^{4} + 6 \, x - 30 \, e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.45, size = 80, normalized size = 2.42 \begin {gather*} -\frac {3\,x\,{\mathrm {e}}^{-4}}{5}-\frac {\frac {{\mathrm {e}}^{-4}\,\left (190\,{\mathrm {e}}^4-72\,{\mathrm {e}}^8+75\right )\,x^2}{5}-\frac {3\,{\mathrm {e}}^{-4}\,\left (49\,{\mathrm {e}}^4-30\right )\,x}{5}+72\,{\mathrm {e}}^4-90}{25\,{\mathrm {e}}^8\,x^3+\left (25\,{\mathrm {e}}^4+30\,{\mathrm {e}}^8\right )\,x^2+\left (30\,{\mathrm {e}}^4-125\,{\mathrm {e}}^8\right )\,x-150\,{\mathrm {e}}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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