Optimal. Leaf size=30 \[ x+25 \left (-3+\log \left (-x+e^{x^2+\left (2-x^2\right )^2} x\right )\right )^2 \]
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Rubi [F] time = 9.39, 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 {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {100 e^{3 x^2} x \left (-3+2 x^2\right ) \left (-3+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right )}{e^{3 x^2}-e^{4+x^4}}+\frac {-150+x+900 x^2-600 x^4+50 \log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )-300 x^2 \log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )+200 x^4 \log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )}{x}\right ) \, dx\\ &=-\left (100 \int \frac {e^{3 x^2} x \left (-3+2 x^2\right ) \left (-3+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right )}{e^{3 x^2}-e^{4+x^4}} \, dx\right )+\int \frac {-150+x+900 x^2-600 x^4+50 \log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )-300 x^2 \log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )+200 x^4 \log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )}{x} \, dx\\ &=-\left (100 \int \left (-\frac {3 e^{3 x^2} x \left (-3+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right )}{e^{3 x^2}-e^{4+x^4}}+\frac {2 e^{3 x^2} x^3 \left (-3+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right )}{e^{3 x^2}-e^{4+x^4}}\right ) \, dx\right )+\int \left (\frac {-150+x+900 x^2-600 x^4}{x}+\frac {50 \left (1-6 x^2+4 x^4\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{x}\right ) \, dx\\ &=50 \int \frac {\left (1-6 x^2+4 x^4\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{x} \, dx-200 \int \frac {e^{3 x^2} x^3 \left (-3+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right )}{e^{3 x^2}-e^{4+x^4}} \, dx+300 \int \frac {e^{3 x^2} x \left (-3+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right )}{e^{3 x^2}-e^{4+x^4}} \, dx+\int \frac {-150+x+900 x^2-600 x^4}{x} \, dx\\ &=50 \int \left (\frac {\log \left (-x+e^{4-3 x^2+x^4} x\right )}{x}-6 x \log \left (-x+e^{4-3 x^2+x^4} x\right )+4 x^3 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \, dx-200 \int \left (-\frac {3 e^{3 x^2} x^3}{e^{3 x^2}-e^{4+x^4}}+\frac {e^{3 x^2} x^3 \log \left (-x+e^{4-3 x^2+x^4} x\right )}{e^{3 x^2}-e^{4+x^4}}\right ) \, dx+300 \int \left (-\frac {3 e^{3 x^2} x}{e^{3 x^2}-e^{4+x^4}}+\frac {e^{3 x^2} x \log \left (-x+e^{4-3 x^2+x^4} x\right )}{e^{3 x^2}-e^{4+x^4}}\right ) \, dx+\int \left (1-\frac {150}{x}+900 x-600 x^3\right ) \, dx\\ &=x+450 x^2-150 x^4-150 \log (x)+50 \int \frac {\log \left (-x+e^{4-3 x^2+x^4} x\right )}{x} \, dx+200 \int x^3 \log \left (-x+e^{4-3 x^2+x^4} x\right ) \, dx-200 \int \frac {e^{3 x^2} x^3 \log \left (-x+e^{4-3 x^2+x^4} x\right )}{e^{3 x^2}-e^{4+x^4}} \, dx-300 \int x \log \left (-x+e^{4-3 x^2+x^4} x\right ) \, dx+300 \int \frac {e^{3 x^2} x \log \left (-x+e^{4-3 x^2+x^4} x\right )}{e^{3 x^2}-e^{4+x^4}} \, dx+600 \int \frac {e^{3 x^2} x^3}{e^{3 x^2}-e^{4+x^4}} \, dx-900 \int \frac {e^{3 x^2} x}{e^{3 x^2}-e^{4+x^4}} \, dx\\ &=x+450 x^2-150 x^4-150 \log (x)-150 x^2 \log \left (-x+e^{4-3 x^2+x^4} x\right )+50 x^4 \log \left (-x+e^{4-3 x^2+x^4} x\right )-50 \int \frac {x^3 \left (e^{3 x^2}+e^{4+x^4} \left (-1+6 x^2-4 x^4\right )\right )}{e^{3 x^2}-e^{4+x^4}} \, dx+50 \int \frac {\log \left (-x+e^{4-3 x^2+x^4} x\right )}{x} \, dx+150 \int \frac {x \left (e^{3 x^2}+e^{4+x^4} \left (-1+6 x^2-4 x^4\right )\right )}{e^{3 x^2}-e^{4+x^4}} \, dx+200 \int \frac {\left (e^{3 x^2}+e^{4+x^4} \left (-1+6 x^2-4 x^4\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{2 \left (e^{3 x^2}-e^{4+x^4}\right ) x} \, dx-300 \int \frac {\left (e^{3 x^2}+e^{4+x^4} \left (-1+6 x^2-4 x^4\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{2 \left (e^{3 x^2}-e^{4+x^4}\right ) x} \, dx+300 \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-450 \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-\left (100 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )+\left (150 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )\\ &=x+450 x^2-150 x^4-150 \log (x)-150 x^2 \log \left (-x+e^{4-3 x^2+x^4} x\right )+50 x^4 \log \left (-x+e^{4-3 x^2+x^4} x\right )-25 \operatorname {Subst}\left (\int \frac {x \left (e^{3 x}+e^{4+x^2} \left (-1+6 x-4 x^2\right )\right )}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )+50 \int \frac {\log \left (-x+e^{4-3 x^2+x^4} x\right )}{x} \, dx+75 \operatorname {Subst}\left (\int \frac {e^{3 x}+e^{4+x^2} \left (-1+6 x-4 x^2\right )}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )+100 \int \frac {\left (e^{3 x^2}+e^{4+x^4} \left (-1+6 x^2-4 x^4\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{\left (e^{3 x^2}-e^{4+x^4}\right ) x} \, dx-150 \int \frac {\left (e^{3 x^2}+e^{4+x^4} \left (-1+6 x^2-4 x^4\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{\left (e^{3 x^2}-e^{4+x^4}\right ) x} \, dx+300 \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-450 \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-\left (100 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )+\left (150 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )\\ &=x+450 x^2-150 x^4-150 \log (x)-150 x^2 \log \left (-x+e^{4-3 x^2+x^4} x\right )+50 x^4 \log \left (-x+e^{4-3 x^2+x^4} x\right )-25 \operatorname {Subst}\left (\int \left (-\frac {2 e^{3 x} x^2 (-3+2 x)}{e^{3 x}-e^{4+x^2}}+x \left (1-6 x+4 x^2\right )\right ) \, dx,x,x^2\right )+50 \int \frac {\log \left (-x+e^{4-3 x^2+x^4} x\right )}{x} \, dx+75 \operatorname {Subst}\left (\int \left (1-6 x+4 x^2-\frac {2 e^{3 x} x (-3+2 x)}{e^{3 x}-e^{4+x^2}}\right ) \, dx,x,x^2\right )+100 \int \left (-\frac {2 e^{3 x^2} x \left (-3+2 x^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{e^{3 x^2}-e^{4+x^4}}+\frac {\left (1-6 x^2+4 x^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{x}\right ) \, dx-150 \int \left (-\frac {2 e^{3 x^2} x \left (-3+2 x^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{e^{3 x^2}-e^{4+x^4}}+\frac {\left (1-6 x^2+4 x^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{x}\right ) \, dx+300 \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-450 \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-\left (100 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )+\left (150 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )\\ &=x+525 x^2-375 x^4+100 x^6-150 \log (x)-150 x^2 \log \left (-x+e^{4-3 x^2+x^4} x\right )+50 x^4 \log \left (-x+e^{4-3 x^2+x^4} x\right )-25 \operatorname {Subst}\left (\int x \left (1-6 x+4 x^2\right ) \, dx,x,x^2\right )+50 \int \frac {\log \left (-x+e^{4-3 x^2+x^4} x\right )}{x} \, dx+50 \operatorname {Subst}\left (\int \frac {e^{3 x} x^2 (-3+2 x)}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )+100 \int \frac {\left (1-6 x^2+4 x^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{x} \, dx-150 \int \frac {\left (1-6 x^2+4 x^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{x} \, dx-150 \operatorname {Subst}\left (\int \frac {e^{3 x} x (-3+2 x)}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-200 \int \frac {e^{3 x^2} x \left (-3+2 x^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{e^{3 x^2}-e^{4+x^4}} \, dx+300 \int \frac {e^{3 x^2} x \left (-3+2 x^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )}{e^{3 x^2}-e^{4+x^4}} \, dx+300 \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-450 \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )-\left (100 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )+\left (150 \log \left (-x+e^{4-3 x^2+x^4} x\right )\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{4+x^2}} \, dx,x,x^2\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.19, size = 174, normalized size = 5.80 \begin {gather*} x+450 x^2-225 x^4+25 \log ^2\left (\left (e^{3 x^2}-e^{4+x^4}\right ) x\right )-150 x^2 \log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )+50 \log \left (e^{3 x^2}-e^{4+x^4}\right ) \left (-3+3 x^2-\log \left (\left (e^{3 x^2}-e^{4+x^4}\right ) x\right )+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right )+50 \log (x) \left (-3+3 x^2-\log \left (\left (e^{3 x^2}-e^{4+x^4}\right ) x\right )+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 44, normalized size = 1.47 \begin {gather*} 25 \, \log \left (x e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - x\right )^{2} + x - 150 \, \log \left (x e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - 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 {{\left (600 \, x^{4} - 900 \, x^{2} - x + 150\right )} e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - 50 \, {\left ({\left (4 \, x^{4} - 6 \, x^{2} + 1\right )} e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - 1\right )} \log \left (x e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - x\right ) + x - 150}{x e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 45, normalized size = 1.50
method | result | size |
norman | \(x +25 \ln \left (x \,{\mathrm e}^{x^{4}-3 x^{2}+4}-x \right )^{2}-150 \ln \relax (x )-150 \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\) | \(45\) |
risch | \(25 \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )^{2}+\left (50 \ln \relax (x )-200\right ) \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )+25 \ln \relax (x )^{2}-200+25 i \pi \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{2}-25 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )+25 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{2}+x -150 \ln \relax (x )-25 i \pi \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{3}+100 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )+100 i \pi \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{3}-100 i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{2}+25 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{2}+50 \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )-100 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{2}-25 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{3}+25 i \pi \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )^{2}-25 i \pi \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )\) | \(547\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 76, normalized size = 2.53 \begin {gather*} 225 \, x^{4} + 450 \, x^{2} - 150 \, {\left (x^{2} + 1\right )} \log \relax (x) + 25 \, \log \relax (x)^{2} - 50 \, {\left (3 \, x^{2} - \log \relax (x) + 3\right )} \log \left (e^{\left (x^{4} + 4\right )} - e^{\left (3 \, x^{2}\right )}\right ) + 25 \, \log \left (e^{\left (x^{4} + 4\right )} - e^{\left (3 \, x^{2}\right )}\right )^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 47, normalized size = 1.57 \begin {gather*} 25\,{\ln \left (x\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-3\,x^2}-x\right )}^2+x-150\,\ln \left ({\mathrm {e}}^{x^4}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-3\,x^2}-1\right )-150\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 42, normalized size = 1.40 \begin {gather*} x - 150 \log {\relax (x )} + 25 \log {\left (x e^{x^{4} - 3 x^{2} + 4} - x \right )}^{2} - 150 \log {\left (e^{x^{4} - 3 x^{2} + 4} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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