Optimal. Leaf size=27 \[ -e^{x/4}+x+x \left (2+\left (e^x-9 \log (3)\right )^2\right )^2 \]
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Rubi [B] time = 0.13, antiderivative size = 153, normalized size of antiderivative = 5.67, number of steps used = 13, number of rules used = 4, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12, 2194, 2176, 2187} \begin {gather*} -e^{x/4}-\frac {e^{4 x}}{4}+\frac {1}{4} e^{4 x} (4 x+1)+e^{2 x} \left (2 x \left (2+243 \log ^2(3)\right )+2+243 \log ^2(3)\right )-e^{2 x} \left (2+243 \log ^2(3)\right )+36 e^x \log (3) \left (2+81 \log ^2(3)\right )+x \left (5+6561 \log ^4(3)+324 \log ^2(3)\right )-36 e^x \left (x \log (3) \left (2+81 \log ^2(3)\right )+81 \log ^3(3)+\log (9)\right )+12 e^{3 x} \log (3)-12 e^{3 x} (3 x+1) \log (3) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2187
Rule 2194
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx\\ &=x \left (5+324 \log ^2(3)+6561 \log ^4(3)\right )-\frac {1}{4} \int e^{x/4} \, dx+\frac {1}{4} \int e^{4 x} (4+16 x) \, dx+\frac {1}{4} \int e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right ) \, dx+\frac {1}{4} \int e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right ) \, dx+\frac {1}{4} \log (3) \int e^{3 x} (-144-432 x) \, dx\\ &=-e^{x/4}+\frac {1}{4} e^{4 x} (1+4 x)-12 e^{3 x} (1+3 x) \log (3)+x \left (5+324 \log ^2(3)+6561 \log ^4(3)\right )+\frac {1}{4} \int e^{2 x} \left (8 \left (2+243 \log ^2(3)\right )+16 x \left (2+243 \log ^2(3)\right )\right ) \, dx+\frac {1}{4} \int e^x \left (-144 x \log (3) \left (2+81 \log ^2(3)\right )-144 \left (81 \log ^3(3)+\log (9)\right )\right ) \, dx+(36 \log (3)) \int e^{3 x} \, dx-\int e^{4 x} \, dx\\ &=-e^{x/4}-\frac {e^{4 x}}{4}+\frac {1}{4} e^{4 x} (1+4 x)+12 e^{3 x} \log (3)-12 e^{3 x} (1+3 x) \log (3)+x \left (5+324 \log ^2(3)+6561 \log ^4(3)\right )+e^{2 x} \left (2+243 \log ^2(3)+2 x \left (2+243 \log ^2(3)\right )\right )-36 e^x \left (81 \log ^3(3)+x \log (3) \left (2+81 \log ^2(3)\right )+\log (9)\right )+\left (36 \log (3) \left (2+81 \log ^2(3)\right )\right ) \int e^x \, dx-\left (2 \left (2+243 \log ^2(3)\right )\right ) \int e^{2 x} \, dx\\ &=-e^{x/4}-\frac {e^{4 x}}{4}+\frac {1}{4} e^{4 x} (1+4 x)+12 e^{3 x} \log (3)-12 e^{3 x} (1+3 x) \log (3)+36 e^x \log (3) \left (2+81 \log ^2(3)\right )-e^{2 x} \left (2+243 \log ^2(3)\right )+x \left (5+324 \log ^2(3)+6561 \log ^4(3)\right )+e^{2 x} \left (2+243 \log ^2(3)+2 x \left (2+243 \log ^2(3)\right )\right )-36 e^x \left (81 \log ^3(3)+x \log (3) \left (2+81 \log ^2(3)\right )+\log (9)\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.05, size = 76, normalized size = 2.81 \begin {gather*} -e^{x/4}+5 x+e^{4 x} x-36 e^{3 x} x \log (3)+324 x \log ^2(3)+6561 x \log ^4(3)-36 e^x x \log (3) \left (2+81 \log ^2(3)\right )+2 e^{2 x} x \left (2+243 \log ^2(3)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 73, normalized size = 2.70 \begin {gather*} 6561 \, x \log \relax (3)^{4} - 36 \, x e^{\left (3 \, x\right )} \log \relax (3) + 324 \, x \log \relax (3)^{2} + x e^{\left (4 \, x\right )} + 2 \, {\left (243 \, x \log \relax (3)^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} - 36 \, {\left (81 \, x \log \relax (3)^{3} + 2 \, x \log \relax (3)\right )} e^{x} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 73, normalized size = 2.70 \begin {gather*} 6561 \, x \log \relax (3)^{4} - 36 \, x e^{\left (3 \, x\right )} \log \relax (3) + 324 \, x \log \relax (3)^{2} + x e^{\left (4 \, x\right )} + 2 \, {\left (243 \, x \log \relax (3)^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} - 36 \, {\left (81 \, x \log \relax (3)^{3} + 2 \, x \log \relax (3)\right )} e^{x} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 70, normalized size = 2.59
method | result | size |
risch | \(x \,{\mathrm e}^{4 x}-36 \ln \relax (3) {\mathrm e}^{3 x} x +2 \left (243 \ln \relax (3)^{2}+2\right ) x \,{\mathrm e}^{2 x}-36 \ln \relax (3) \left (81 \ln \relax (3)^{2}+2\right ) x \,{\mathrm e}^{x}-{\mathrm e}^{\frac {x}{4}}+6561 x \ln \relax (3)^{4}+324 x \ln \relax (3)^{2}+5 x\) | \(70\) |
default | \(5 x +324 x \ln \relax (3)^{2}+6561 x \ln \relax (3)^{4}+x \,{\mathrm e}^{4 x}-72 x \ln \relax (3) {\mathrm e}^{x}-2916 \,{\mathrm e}^{x} \ln \relax (3)^{3} x +4 x \,{\mathrm e}^{2 x}+486 x \ln \relax (3)^{2} {\mathrm e}^{2 x}-36 \ln \relax (3) {\mathrm e}^{3 x} x -{\mathrm e}^{\frac {x}{4}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 70, normalized size = 2.59 \begin {gather*} 6561 \, x \log \relax (3)^{4} + 2 \, {\left (243 \, \log \relax (3)^{2} + 2\right )} x e^{\left (2 \, x\right )} - 36 \, {\left (81 \, \log \relax (3)^{3} + 2 \, \log \relax (3)\right )} x e^{x} - 36 \, x e^{\left (3 \, x\right )} \log \relax (3) + 324 \, x \log \relax (3)^{2} + x e^{\left (4 \, x\right )} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.07, size = 68, normalized size = 2.52 \begin {gather*} x\,{\mathrm {e}}^{4\,x}-{\mathrm {e}}^{x/4}+x\,\left (324\,{\ln \relax (3)}^2+6561\,{\ln \relax (3)}^4+5\right )-36\,x\,{\mathrm {e}}^{3\,x}\,\ln \relax (3)+\frac {x\,{\mathrm {e}}^{2\,x}\,\left (1944\,{\ln \relax (3)}^2+16\right )}{4}-36\,x\,{\mathrm {e}}^x\,\ln \relax (3)\,\left (81\,{\ln \relax (3)}^2+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.32, size = 76, normalized size = 2.81 \begin {gather*} x e^{4 x} - 36 x e^{3 x} \log {\relax (3 )} + x \left (5 + 324 \log {\relax (3 )}^{2} + 6561 \log {\relax (3 )}^{4}\right ) + \left (4 x + 486 x \log {\relax (3 )}^{2}\right ) e^{2 x} + \left (- 2916 x \log {\relax (3 )}^{3} - 72 x \log {\relax (3 )}\right ) e^{x} - e^{\frac {x}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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