Integrand size = 116, antiderivative size = 19 \[ \int \frac {20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))}{\log (2)} \, dx=5 \left (e^x+e^{2+x+\frac {x}{\log (2)}}\right )^4 \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(19)=38\).
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.95, number of steps used = 11, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {12, 2225, 2259} \[ \int \frac {20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))}{\log (2)} \, dx=5 e^{4 x}+5 e^{4 \left (\frac {x (1+\log (2))}{\log (2)}+2\right )}+30 e^{\frac {2 x (1+\log (4))}{\log (2)}+4}+20 e^{\frac {x (1+\log (16))}{\log (2)}+2}+20 e^{\frac {x (3+\log (16))}{\log (2)}+6} \]
[In]
[Out]
Rule 12
Rule 2225
Rule 2259
Rubi steps \begin{align*} \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{align*}
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(19)=38\).
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.63 \[ \int \frac {20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))}{\log (2)} \, dx=5 \left (e^{4 x}+6 e^{4+x \left (4+\frac {2}{\log (2)}\right )}+e^{4 \left (2+x+\frac {x}{\log (2)}\right )}+4 e^{2+\frac {x (1+\log (16))}{\log (2)}}+4 e^{6+\frac {x (3+\log (16))}{\log (2)}}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(17)=34\).
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.58
method | result | size |
risch | \(5 \,{\mathrm e}^{4 x}+20 \,{\mathrm e}^{\frac {4 x \ln \left (2\right )+2 \ln \left (2\right )+x}{\ln \left (2\right )}}+30 \,{\mathrm e}^{\frac {4 x \ln \left (2\right )+4 \ln \left (2\right )+2 x}{\ln \left (2\right )}}+20 \,{\mathrm e}^{\frac {4 x \ln \left (2\right )+6 \ln \left (2\right )+3 x}{\ln \left (2\right )}}+5 \,{\mathrm e}^{\frac {4 x \ln \left (2\right )+8 \ln \left (2\right )+4 x}{\ln \left (2\right )}}\) | \(87\) |
parallelrisch | \(\frac {5 \ln \left (2\right ) {\mathrm e}^{4 x}+20 \,{\mathrm e}^{3 x} {\mathrm e}^{\frac {\left (2+x \right ) \ln \left (2\right )+x}{\ln \left (2\right )}} \ln \left (2\right )+30 \,{\mathrm e}^{2 x} {\mathrm e}^{\frac {2 \left (2+x \right ) \ln \left (2\right )+2 x}{\ln \left (2\right )}} \ln \left (2\right )+20 \,{\mathrm e}^{x} {\mathrm e}^{\frac {3 \left (2+x \right ) \ln \left (2\right )+3 x}{\ln \left (2\right )}} \ln \left (2\right )+5 \ln \left (2\right ) {\mathrm e}^{\frac {4 \left (2+x \right ) \ln \left (2\right )+4 x}{\ln \left (2\right )}}}{\ln \left (2\right )}\) | \(103\) |
parts | \(5 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{\frac {4 \left (2+x \right ) \ln \left (2\right )+4 x}{\ln \left (2\right )}}+\frac {60 \left (1+2 \ln \left (2\right )\right ) {\mathrm e}^{4+\left (2+\frac {2+2 \ln \left (2\right )}{\ln \left (2\right )}\right ) x}}{\ln \left (2\right ) \left (2+\frac {2+2 \ln \left (2\right )}{\ln \left (2\right )}\right )}+\frac {20 \left (1+4 \ln \left (2\right )\right ) {\mathrm e}^{2+\left (3+\frac {1+\ln \left (2\right )}{\ln \left (2\right )}\right ) x}}{\ln \left (2\right ) \left (3+\frac {1+\ln \left (2\right )}{\ln \left (2\right )}\right )}+\frac {20 \left (4 \ln \left (2\right )+3\right ) {\mathrm e}^{6+\left (1+\frac {3 \ln \left (2\right )+3}{\ln \left (2\right )}\right ) x}}{\ln \left (2\right ) \left (1+\frac {3 \ln \left (2\right )+3}{\ln \left (2\right )}\right )}\) | \(153\) |
default | \(\frac {\frac {\left (20 \ln \left (2\right )+20\right ) {\mathrm e}^{\frac {4 \left (2+x \right ) \ln \left (2\right )+4 x}{\ln \left (2\right )}} \ln \left (2\right )}{4 \ln \left (2\right )+4}+\frac {\left (80 \ln \left (2\right )+20\right ) {\mathrm e}^{2+\left (3+\frac {1+\ln \left (2\right )}{\ln \left (2\right )}\right ) x}}{3+\frac {1+\ln \left (2\right )}{\ln \left (2\right )}}+\frac {\left (80 \ln \left (2\right )+60\right ) {\mathrm e}^{6+\left (1+\frac {3 \ln \left (2\right )+3}{\ln \left (2\right )}\right ) x}}{1+\frac {3 \ln \left (2\right )+3}{\ln \left (2\right )}}+\frac {\left (120 \ln \left (2\right )+60\right ) {\mathrm e}^{4+\left (2+\frac {2+2 \ln \left (2\right )}{\ln \left (2\right )}\right ) x}}{2+\frac {2+2 \ln \left (2\right )}{\ln \left (2\right )}}+5 \ln \left (2\right ) {\mathrm e}^{4 x}}{\ln \left (2\right )}\) | \(159\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 8.58 \[ \int \frac {20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))}{\log (2)} \, dx=5 \, {\left (e^{\left (\frac {4 \, {\left (2 \, {\left (2 \, x + 3\right )} \log \left (2\right ) + 3 \, x\right )}}{\log \left (2\right )}\right )} + 4 \, e^{\left (\frac {3 \, {\left (2 \, {\left (2 \, x + 3\right )} \log \left (2\right ) + 3 \, x\right )}}{\log \left (2\right )} + \frac {4 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )} + 6 \, e^{\left (\frac {2 \, {\left (2 \, {\left (2 \, x + 3\right )} \log \left (2\right ) + 3 \, x\right )}}{\log \left (2\right )} + \frac {8 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )} + 4 \, e^{\left (\frac {2 \, {\left (2 \, x + 3\right )} \log \left (2\right ) + 3 \, x}{\log \left (2\right )} + \frac {12 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )} + e^{\left (\frac {16 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )}\right )} e^{\left (-\frac {12 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (15) = 30\).
Time = 1.79 (sec) , antiderivative size = 99, normalized size of antiderivative = 5.21 \[ \int \frac {20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))}{\log (2)} \, dx=\frac {5 e^{8} e^{4 x} e^{\frac {4 x}{\log {\left (2 \right )}}} \log {\left (2 \right )} + 20 e^{6} e^{4 x} e^{\frac {3 x}{\log {\left (2 \right )}}} \log {\left (2 \right )} + 30 e^{4} e^{4 x} e^{\frac {2 x}{\log {\left (2 \right )}}} \log {\left (2 \right )} + 20 e^{2} e^{4 x} e^{\frac {x}{\log {\left (2 \right )}}} \log {\left (2 \right )} + 5 e^{4 x} \log {\left (2 \right )}}{\log {\left (2 \right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (17) = 34\).
Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 7.89 \[ \int \frac {20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))}{\log (2)} \, dx=\frac {5 \, {\left (e^{\left (4 \, x\right )} \log \left (2\right ) + e^{\left (\frac {4 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )} \log \left (2\right ) + \frac {4 \, {\left (4 \, \log \left (2\right ) + 1\right )} e^{\left (3 \, x + \frac {{\left (x + 2\right )} \log \left (2\right ) + x}{\log \left (2\right )}\right )}}{\frac {\log \left (2\right ) + 1}{\log \left (2\right )} + 3} + \frac {6 \, {\left (2 \, \log \left (2\right ) + 1\right )} e^{\left (2 \, x + \frac {2 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )}}{\frac {\log \left (2\right ) + 1}{\log \left (2\right )} + 1} + \frac {4 \, {\left (4 \, \log \left (2\right ) + 3\right )} e^{\left (x + \frac {3 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )}}{\frac {3 \, {\left (\log \left (2\right ) + 1\right )}}{\log \left (2\right )} + 1}\right )}}{\log \left (2\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (17) = 34\).
Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 7.89 \[ \int \frac {20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))}{\log (2)} \, dx=\frac {5 \, {\left (e^{\left (4 \, x\right )} \log \left (2\right ) + e^{\left (\frac {4 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )} \log \left (2\right ) + \frac {4 \, {\left (4 \, \log \left (2\right ) + 1\right )} e^{\left (3 \, x + \frac {{\left (x + 2\right )} \log \left (2\right ) + x}{\log \left (2\right )}\right )}}{\frac {\log \left (2\right ) + 1}{\log \left (2\right )} + 3} + \frac {6 \, {\left (2 \, \log \left (2\right ) + 1\right )} e^{\left (2 \, x + \frac {2 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )}}{\frac {\log \left (2\right ) + 1}{\log \left (2\right )} + 1} + \frac {4 \, {\left (4 \, \log \left (2\right ) + 3\right )} e^{\left (x + \frac {3 \, {\left ({\left (x + 2\right )} \log \left (2\right ) + x\right )}}{\log \left (2\right )}\right )}}{\frac {3 \, {\left (\log \left (2\right ) + 1\right )}}{\log \left (2\right )} + 1}\right )}}{\log \left (2\right )} \]
[In]
[Out]
Time = 1.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.47 \[ \int \frac {20 e^{4 x} \log (2)+e^{\frac {4 (x+(2+x) \log (2))}{\log (2)}} (20+20 \log (2))+e^{3 x+\frac {x+(2+x) \log (2)}{\log (2)}} (20+80 \log (2))+e^{x+\frac {3 (x+(2+x) \log (2))}{\log (2)}} (60+80 \log (2))+e^{2 x+\frac {2 (x+(2+x) \log (2))}{\log (2)}} (60+120 \log (2))}{\log (2)} \, dx=5\,{\mathrm {e}}^{4\,x}+20\,{\mathrm {e}}^{4\,x+\frac {x}{\ln \left (2\right )}+2}+30\,{\mathrm {e}}^{4\,x+\frac {2\,x}{\ln \left (2\right )}+4}+20\,{\mathrm {e}}^{4\,x+\frac {3\,x}{\ln \left (2\right )}+6}+5\,{\mathrm {e}}^{4\,x+\frac {4\,x}{\ln \left (2\right )}+8} \]
[In]
[Out]