Optimal. Leaf size=19 \[ 5 \left (e^x+e^{2+x+\frac {x}{\log (2)}}\right )^4 \]
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Rubi [B] time = 0.13, antiderivative size = 75, normalized size of antiderivative = 3.95, number of steps used = 11, number of rules used = 3, integrand size = 116, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {12, 2194, 2227} \begin {gather*} 5 e^{4 x}+5 e^{4 \left (\frac {x (1+\log (2))}{\log (2)}+2\right )}+30 e^{\frac {2 x (1+\log (4))}{\log (2)}+4}+20 e^{\frac {x (1+\log (16))}{\log (2)}+2}+20 e^{\frac {x (3+\log (16))}{\log (2)}+6} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2194
Rule 2227
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))\right ) \, dx}{\log (2)}\\ &=20 \int e^{4 x} \, dx+\frac {(20 (1+\log (2))) \int e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} \, dx}{\log (2)}+\frac {(60 (1+\log (4))) \int e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} \, dx}{\log (2)}+\frac {(20 (1+\log (16))) \int e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} \, dx}{\log (2)}+\frac {(20 (3+\log (16))) \int e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} \, dx}{\log (2)}\\ &=5 e^{4 x}+\frac {(20 (1+\log (2))) \int e^{4 \left (2+\frac {x (1+\log (2))}{\log (2)}\right )} \, dx}{\log (2)}+\frac {(60 (1+\log (4))) \int e^{4+\frac {2 x (1+\log (4))}{\log (2)}} \, dx}{\log (2)}+\frac {(20 (1+\log (16))) \int e^{2+\frac {x (1+\log (16))}{\log (2)}} \, dx}{\log (2)}+\frac {(20 (3+\log (16))) \int e^{6+\frac {x (3+\log (16))}{\log (2)}} \, dx}{\log (2)}\\ &=5 e^{4 x}+5 e^{4 \left (2+\frac {x (1+\log (2))}{\log (2)}\right )}+30 e^{4+\frac {2 x (1+\log (4))}{\log (2)}}+20 e^{2+\frac {x (1+\log (16))}{\log (2)}}+20 e^{6+\frac {x (3+\log (16))}{\log (2)}}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.28, size = 69, normalized size = 3.63 \begin {gather*} 5 \left (e^{4 x}+6 e^{4+x \left (4+\frac {2}{\log (2)}\right )}+e^{4 \left (2+x+\frac {x}{\log (2)}\right )}+4 e^{2+\frac {x (1+\log (16))}{\log (2)}}+4 e^{6+\frac {x (3+\log (16))}{\log (2)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 163, normalized size = 8.58 \begin {gather*} 5 \, {\left (e^{\left (\frac {4 \, {\left (2 \, {\left (2 \, x + 3\right )} \log \relax (2) + 3 \, x\right )}}{\log \relax (2)}\right )} + 4 \, e^{\left (\frac {3 \, {\left (2 \, {\left (2 \, x + 3\right )} \log \relax (2) + 3 \, x\right )}}{\log \relax (2)} + \frac {4 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )} + 6 \, e^{\left (\frac {2 \, {\left (2 \, {\left (2 \, x + 3\right )} \log \relax (2) + 3 \, x\right )}}{\log \relax (2)} + \frac {8 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )} + 4 \, e^{\left (\frac {2 \, {\left (2 \, x + 3\right )} \log \relax (2) + 3 \, x}{\log \relax (2)} + \frac {12 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )} + e^{\left (\frac {16 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )}\right )} e^{\left (-\frac {12 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 150, normalized size = 7.89 \begin {gather*} \frac {5 \, {\left (e^{\left (4 \, x\right )} \log \relax (2) + e^{\left (\frac {4 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )} \log \relax (2) + \frac {4 \, {\left (4 \, \log \relax (2) + 1\right )} e^{\left (3 \, x + \frac {{\left (x + 2\right )} \log \relax (2) + x}{\log \relax (2)}\right )}}{\frac {\log \relax (2) + 1}{\log \relax (2)} + 3} + \frac {6 \, {\left (2 \, \log \relax (2) + 1\right )} e^{\left (2 \, x + \frac {2 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )}}{\frac {\log \relax (2) + 1}{\log \relax (2)} + 1} + \frac {4 \, {\left (4 \, \log \relax (2) + 3\right )} e^{\left (x + \frac {3 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )}}{\frac {3 \, {\left (\log \relax (2) + 1\right )}}{\log \relax (2)} + 1}\right )}}{\log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 87, normalized size = 4.58
method | result | size |
risch | \(5 \,{\mathrm e}^{4 x}+20 \,{\mathrm e}^{\frac {4 x \ln \relax (2)+2 \ln \relax (2)+x}{\ln \relax (2)}}+30 \,{\mathrm e}^{\frac {4 x \ln \relax (2)+4 \ln \relax (2)+2 x}{\ln \relax (2)}}+20 \,{\mathrm e}^{\frac {4 x \ln \relax (2)+6 \ln \relax (2)+3 x}{\ln \relax (2)}}+5 \,{\mathrm e}^{\frac {4 x \ln \relax (2)+8 \ln \relax (2)+4 x}{\ln \relax (2)}}\) | \(87\) |
default | \(\frac {\frac {5 \,{\mathrm e}^{\frac {4 \left (2+x \right ) \ln \relax (2)+4 x}{\ln \relax (2)}} \ln \relax (2)^{2}}{1+\ln \relax (2)}+\frac {5 \,{\mathrm e}^{\frac {4 \left (2+x \right ) \ln \relax (2)+4 x}{\ln \relax (2)}} \ln \relax (2)}{1+\ln \relax (2)}+\frac {80 \,{\mathrm e}^{3 x +\frac {\left (2+x \right ) \ln \relax (2)+x}{\ln \relax (2)}} \ln \relax (2)}{4+\frac {1}{\ln \relax (2)}}+\frac {20 \,{\mathrm e}^{3 x +\frac {\left (2+x \right ) \ln \relax (2)+x}{\ln \relax (2)}}}{4+\frac {1}{\ln \relax (2)}}+\frac {80 \,{\mathrm e}^{x +\frac {3 \left (2+x \right ) \ln \relax (2)+3 x}{\ln \relax (2)}} \ln \relax (2)}{4+\frac {3}{\ln \relax (2)}}+\frac {60 \,{\mathrm e}^{x +\frac {3 \left (2+x \right ) \ln \relax (2)+3 x}{\ln \relax (2)}}}{4+\frac {3}{\ln \relax (2)}}+\frac {120 \,{\mathrm e}^{2 x +\frac {2 \left (2+x \right ) \ln \relax (2)+2 x}{\ln \relax (2)}} \ln \relax (2)}{4+\frac {2}{\ln \relax (2)}}+\frac {60 \,{\mathrm e}^{2 x +\frac {2 \left (2+x \right ) \ln \relax (2)+2 x}{\ln \relax (2)}}}{4+\frac {2}{\ln \relax (2)}}+5 \ln \relax (2) {\mathrm e}^{4 x}}{\ln \relax (2)}\) | \(251\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 150, normalized size = 7.89 \begin {gather*} \frac {5 \, {\left (e^{\left (4 \, x\right )} \log \relax (2) + e^{\left (\frac {4 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )} \log \relax (2) + \frac {4 \, {\left (4 \, \log \relax (2) + 1\right )} e^{\left (3 \, x + \frac {{\left (x + 2\right )} \log \relax (2) + x}{\log \relax (2)}\right )}}{\frac {\log \relax (2) + 1}{\log \relax (2)} + 3} + \frac {6 \, {\left (2 \, \log \relax (2) + 1\right )} e^{\left (2 \, x + \frac {2 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )}}{\frac {\log \relax (2) + 1}{\log \relax (2)} + 1} + \frac {4 \, {\left (4 \, \log \relax (2) + 3\right )} e^{\left (x + \frac {3 \, {\left ({\left (x + 2\right )} \log \relax (2) + x\right )}}{\log \relax (2)}\right )}}{\frac {3 \, {\left (\log \relax (2) + 1\right )}}{\log \relax (2)} + 1}\right )}}{\log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 66, normalized size = 3.47 \begin {gather*} 5\,{\mathrm {e}}^{4\,x}+20\,{\mathrm {e}}^{4\,x+\frac {x}{\ln \relax (2)}+2}+30\,{\mathrm {e}}^{4\,x+\frac {2\,x}{\ln \relax (2)}+4}+20\,{\mathrm {e}}^{4\,x+\frac {3\,x}{\ln \relax (2)}+6}+5\,{\mathrm {e}}^{4\,x+\frac {4\,x}{\ln \relax (2)}+8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 20.98, size = 99, normalized size = 5.21 \begin {gather*} \frac {5 e^{8} e^{4 x} e^{\frac {4 x}{\log {\relax (2 )}}} \log {\relax (2 )} + 20 e^{6} e^{4 x} e^{\frac {3 x}{\log {\relax (2 )}}} \log {\relax (2 )} + 30 e^{4} e^{4 x} e^{\frac {2 x}{\log {\relax (2 )}}} \log {\relax (2 )} + 20 e^{2} e^{4 x} e^{\frac {x}{\log {\relax (2 )}}} \log {\relax (2 )} + 5 e^{4 x} \log {\relax (2 )}}{\log {\relax (2 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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