Optimal. Leaf size=30 \[ e^{\frac {5 e^x}{5+10 \left (\frac {4}{5 x}-3 x\right )^2}}-x \]
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Rubi [F] time = 16.80, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-1024+13760 x^2-75025 x^4+193500 x^6-202500 x^8+e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} \left (1600 x+800 x^2-5375 x^4-22500 x^5+11250 x^6\right )}{1024-13760 x^2+75025 x^4-193500 x^6+202500 x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {25 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x \left (64+32 x-215 x^3-900 x^4+450 x^5\right )}{\left (32-215 x^2+450 x^4\right )^2}\right ) \, dx\\ &=-x+25 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x \left (64+32 x-215 x^3-900 x^4+450 x^5\right )}{\left (32-215 x^2+450 x^4\right )^2} \, dx\\ &=-x+25 \int \left (\frac {2 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x \left (64-215 x^2\right )}{\left (32-215 x^2+450 x^4\right )^2}+\frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} (-2+x) x}{32-215 x^2+450 x^4}\right ) \, dx\\ &=-x+25 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} (-2+x) x}{32-215 x^2+450 x^4} \, dx+50 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x \left (64-215 x^2\right )}{\left (32-215 x^2+450 x^4\right )^2} \, dx\\ &=-x+25 \int \left (-\frac {2 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{32-215 x^2+450 x^4}+\frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x^2}{32-215 x^2+450 x^4}\right ) \, dx+50 \int \left (\frac {64 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (32-215 x^2+450 x^4\right )^2}-\frac {215 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x^3}{\left (32-215 x^2+450 x^4\right )^2}\right ) \, dx\\ &=-x+25 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x^2}{32-215 x^2+450 x^4} \, dx-50 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{32-215 x^2+450 x^4} \, dx+3200 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (32-215 x^2+450 x^4\right )^2} \, dx-10750 \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x^3}{\left (32-215 x^2+450 x^4\right )^2} \, dx\\ &=-x+25 \int \left (\frac {\left (1-\frac {43 i}{\sqrt {455}}\right ) e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}}}{-215-5 i \sqrt {455}+900 x^2}+\frac {\left (1+\frac {43 i}{\sqrt {455}}\right ) e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}}}{-215+5 i \sqrt {455}+900 x^2}\right ) \, dx-50 \int \left (\frac {36 i \sqrt {\frac {5}{91}} e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{215+5 i \sqrt {455}-900 x^2}+\frac {36 i \sqrt {\frac {5}{91}} e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{-215+5 i \sqrt {455}+900 x^2}\right ) \, dx+3200 \int \left (-\frac {6480 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{91 \left (215+5 i \sqrt {455}-900 x^2\right )^2}+\frac {1296 i e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{91 \sqrt {455} \left (215+5 i \sqrt {455}-900 x^2\right )}-\frac {6480 e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{91 \left (-215+5 i \sqrt {455}+900 x^2\right )^2}+\frac {1296 i e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{91 \sqrt {455} \left (-215+5 i \sqrt {455}+900 x^2\right )}\right ) \, dx-10750 \int \left (-\frac {36 \left (215+5 i \sqrt {455}\right ) e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{455 \left (215+5 i \sqrt {455}-900 x^2\right )^2}+\frac {1548 i e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{455 \sqrt {455} \left (215+5 i \sqrt {455}-900 x^2\right )}+\frac {36 \left (-215+5 i \sqrt {455}\right ) e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{455 \left (-215+5 i \sqrt {455}+900 x^2\right )^2}+\frac {1548 i e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{455 \sqrt {455} \left (-215+5 i \sqrt {455}+900 x^2\right )}\right ) \, dx\\ &=-x-\frac {20736000}{91} \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (215+5 i \sqrt {455}-900 x^2\right )^2} \, dx-\frac {20736000}{91} \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (-215+5 i \sqrt {455}+900 x^2\right )^2} \, dx-\left (1800 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{215+5 i \sqrt {455}-900 x^2} \, dx-\left (1800 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{-215+5 i \sqrt {455}+900 x^2} \, dx-\frac {1}{91} \left (665640 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{215+5 i \sqrt {455}-900 x^2} \, dx-\frac {1}{91} \left (665640 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{-215+5 i \sqrt {455}+900 x^2} \, dx+\frac {1}{91} \left (829440 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{215+5 i \sqrt {455}-900 x^2} \, dx+\frac {1}{91} \left (829440 i \sqrt {\frac {5}{91}}\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{-215+5 i \sqrt {455}+900 x^2} \, dx+\frac {1}{91} \left (387000 \left (43-i \sqrt {455}\right )\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (-215+5 i \sqrt {455}+900 x^2\right )^2} \, dx+\frac {1}{91} \left (387000 \left (43+i \sqrt {455}\right )\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}} x}{\left (215+5 i \sqrt {455}-900 x^2\right )^2} \, dx+\frac {1}{91} \left (5 \left (455-43 i \sqrt {455}\right )\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}}}{-215-5 i \sqrt {455}+900 x^2} \, dx+\frac {1}{91} \left (5 \left (455+43 i \sqrt {455}\right )\right ) \int \frac {e^{x+\frac {25 e^x x^2}{32-215 x^2+450 x^4}}}{-215+5 i \sqrt {455}+900 x^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 1.86, size = 28, normalized size = 0.93 \begin {gather*} e^{\frac {25 e^x x^2}{32-215 x^2+450 x^4}}-x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 50, normalized size = 1.67 \begin {gather*} -{\left (x e^{x} - e^{\left (\frac {450 \, x^{5} - 215 \, x^{3} + 25 \, x^{2} e^{x} + 32 \, x}{450 \, x^{4} - 215 \, x^{2} + 32}\right )}\right )} e^{\left (-x\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*} \int -\frac {202500 \, x^{8} - 193500 \, x^{6} + 75025 \, x^{4} - 13760 \, x^{2} - 25 \, {\left (450 \, x^{6} - 900 \, x^{5} - 215 \, x^{4} + 32 \, x^{2} + 64 \, x\right )} e^{\left (\frac {25 \, x^{2} e^{x}}{450 \, x^{4} - 215 \, x^{2} + 32} + x\right )} + 1024}{202500 \, x^{8} - 193500 \, x^{6} + 75025 \, x^{4} - 13760 \, x^{2} + 1024}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.77, size = 27, normalized size = 0.90
method | result | size |
risch | \(-x +{\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{x}}{450 x^{4}-215 x^{2}+32}}\) | \(27\) |
norman | \(\frac {-32 x +215 x^{3}-450 x^{5}-215 x^{2} {\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{x}}{450 x^{4}-215 x^{2}+32}}+450 x^{4} {\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{x}}{450 x^{4}-215 x^{2}+32}}+32 \,{\mathrm e}^{\frac {25 x^{2} {\mathrm e}^{x}}{450 x^{4}-215 x^{2}+32}}}{450 x^{4}-215 x^{2}+32}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 26, normalized size = 0.87 \begin {gather*} -x + e^{\left (\frac {25 \, x^{2} e^{x}}{450 \, x^{4} - 215 \, x^{2} + 32}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 26, normalized size = 0.87 \begin {gather*} {\mathrm {e}}^{\frac {25\,x^2\,{\mathrm {e}}^x}{450\,x^4-215\,x^2+32}}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 22, normalized size = 0.73 \begin {gather*} - x + e^{\frac {25 x^{2} e^{x}}{450 x^{4} - 215 x^{2} + 32}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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