Integrand size = 37, antiderivative size = 25 \[ \int \frac {1}{16} \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx=-3+3 x+\left (6-e^x+\frac {x^2}{8}\right )^2+x \log (4) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {12, 2225, 2227, 2207} \[ \int \frac {1}{16} \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx=\frac {x^4}{64}-\frac {e^x x^2}{4}+\frac {3 x^2}{2}-12 e^x+e^{2 x}+x (3+\log (4)) \]
[In]
[Out]
Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{16} \int \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx \\ & = \frac {3 x^2}{2}+\frac {x^4}{64}+x (3+\log (4))+\frac {1}{16} \int e^x \left (-192-8 x-4 x^2\right ) \, dx+2 \int e^{2 x} \, dx \\ & = e^{2 x}+\frac {3 x^2}{2}+\frac {x^4}{64}+x (3+\log (4))+\frac {1}{16} \int \left (-192 e^x-8 e^x x-4 e^x x^2\right ) \, dx \\ & = e^{2 x}+\frac {3 x^2}{2}+\frac {x^4}{64}+x (3+\log (4))-\frac {1}{4} \int e^x x^2 \, dx-\frac {1}{2} \int e^x x \, dx-12 \int e^x \, dx \\ & = -12 e^x+e^{2 x}-\frac {e^x x}{2}+\frac {3 x^2}{2}-\frac {e^x x^2}{4}+\frac {x^4}{64}+x (3+\log (4))+\frac {\int e^x \, dx}{2}+\frac {1}{2} \int e^x x \, dx \\ & = -\frac {23 e^x}{2}+e^{2 x}+\frac {3 x^2}{2}-\frac {e^x x^2}{4}+\frac {x^4}{64}+x (3+\log (4))-\frac {\int e^x \, dx}{2} \\ & = -12 e^x+e^{2 x}+\frac {3 x^2}{2}-\frac {e^x x^2}{4}+\frac {x^4}{64}+x (3+\log (4)) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {1}{16} \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx=e^{2 x}+3 x+\frac {3 x^2}{2}+\frac {x^4}{64}-\frac {1}{4} e^x \left (48+x^2\right )+x \log (4) \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40
method | result | size |
default | \(3 x +\frac {x^{4}}{64}-\frac {{\mathrm e}^{x} x^{2}}{4}-12 \,{\mathrm e}^{x}+\frac {3 x^{2}}{2}+{\mathrm e}^{2 x}+2 x \ln \left (2\right )\) | \(35\) |
norman | \({\mathrm e}^{2 x}+x \left (2 \ln \left (2\right )+3\right )+\frac {3 x^{2}}{2}+\frac {x^{4}}{64}-\frac {{\mathrm e}^{x} x^{2}}{4}-12 \,{\mathrm e}^{x}\) | \(35\) |
risch | \(3 x +\frac {x^{4}}{64}-\frac {{\mathrm e}^{x} x^{2}}{4}-12 \,{\mathrm e}^{x}+\frac {3 x^{2}}{2}+{\mathrm e}^{2 x}+2 x \ln \left (2\right )\) | \(35\) |
parallelrisch | \({\mathrm e}^{2 x}+x \left (2 \ln \left (2\right )+3\right )+\frac {3 x^{2}}{2}+\frac {x^{4}}{64}-\frac {{\mathrm e}^{x} x^{2}}{4}-12 \,{\mathrm e}^{x}\) | \(35\) |
parts | \(3 x +\frac {x^{4}}{64}-\frac {{\mathrm e}^{x} x^{2}}{4}-12 \,{\mathrm e}^{x}+\frac {3 x^{2}}{2}+{\mathrm e}^{2 x}+2 x \ln \left (2\right )\) | \(35\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {1}{16} \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx=\frac {1}{64} \, x^{4} + \frac {3}{2} \, x^{2} - \frac {1}{4} \, {\left (x^{2} + 48\right )} e^{x} + 2 \, x \log \left (2\right ) + 3 \, x + e^{\left (2 \, x\right )} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {1}{16} \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx=\frac {x^{4}}{64} + \frac {3 x^{2}}{2} + x \left (2 \log {\left (2 \right )} + 3\right ) + \frac {\left (- x^{2} - 48\right ) e^{x}}{4} + e^{2 x} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {1}{16} \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx=\frac {1}{64} \, x^{4} + \frac {3}{2} \, x^{2} - \frac {1}{4} \, {\left (x^{2} + 48\right )} e^{x} + 2 \, x \log \left (2\right ) + 3 \, x + e^{\left (2 \, x\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {1}{16} \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx=\frac {1}{64} \, x^{4} + \frac {3}{2} \, x^{2} - \frac {1}{4} \, {\left (x^{2} + 48\right )} e^{x} + 2 \, x \log \left (2\right ) + 3 \, x + e^{\left (2 \, x\right )} \]
[In]
[Out]
Time = 7.59 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {1}{16} \left (48+32 e^{2 x}+48 x+x^3+e^x \left (-192-8 x-4 x^2\right )+16 \log (4)\right ) \, dx={\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^x+x\,\left (2\,\ln \left (2\right )+3\right )-\frac {x^2\,{\mathrm {e}}^x}{4}+\frac {3\,x^2}{2}+\frac {x^4}{64} \]
[In]
[Out]