Optimal. Leaf size=87 \[ \frac {1}{10} e^{2 x} x^2 \sin (4 x)-\frac {1}{5} e^{2 x} x^2 \cos (4 x)+\frac {3}{50} e^{2 x} x \sin (4 x)-\frac {11}{500} e^{2 x} \sin (4 x)+\frac {2}{25} e^{2 x} x \cos (4 x)+\frac {1}{250} e^{2 x} \cos (4 x) \]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {4432, 4465, 14, 4433, 4466} \[ \frac {1}{10} e^{2 x} x^2 \sin (4 x)-\frac {1}{5} e^{2 x} x^2 \cos (4 x)+\frac {3}{50} e^{2 x} x \sin (4 x)-\frac {11}{500} e^{2 x} \sin (4 x)+\frac {2}{25} e^{2 x} x \cos (4 x)+\frac {1}{250} e^{2 x} \cos (4 x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 4432
Rule 4433
Rule 4465
Rule 4466
Rubi steps
\begin {align*} \int e^{2 x} x^2 \sin (4 x) \, dx &=-\frac {1}{5} e^{2 x} x^2 \cos (4 x)+\frac {1}{10} e^{2 x} x^2 \sin (4 x)-2 \int x \left (-\frac {1}{5} e^{2 x} \cos (4 x)+\frac {1}{10} e^{2 x} \sin (4 x)\right ) \, dx\\ &=-\frac {1}{5} e^{2 x} x^2 \cos (4 x)+\frac {1}{10} e^{2 x} x^2 \sin (4 x)-2 \int \left (-\frac {1}{5} e^{2 x} x \cos (4 x)+\frac {1}{10} e^{2 x} x \sin (4 x)\right ) \, dx\\ &=-\frac {1}{5} e^{2 x} x^2 \cos (4 x)+\frac {1}{10} e^{2 x} x^2 \sin (4 x)-\frac {1}{5} \int e^{2 x} x \sin (4 x) \, dx+\frac {2}{5} \int e^{2 x} x \cos (4 x) \, dx\\ &=\frac {2}{25} e^{2 x} x \cos (4 x)-\frac {1}{5} e^{2 x} x^2 \cos (4 x)+\frac {3}{50} e^{2 x} x \sin (4 x)+\frac {1}{10} e^{2 x} x^2 \sin (4 x)+\frac {1}{5} \int \left (-\frac {1}{5} e^{2 x} \cos (4 x)+\frac {1}{10} e^{2 x} \sin (4 x)\right ) \, dx-\frac {2}{5} \int \left (\frac {1}{10} e^{2 x} \cos (4 x)+\frac {1}{5} e^{2 x} \sin (4 x)\right ) \, dx\\ &=\frac {2}{25} e^{2 x} x \cos (4 x)-\frac {1}{5} e^{2 x} x^2 \cos (4 x)+\frac {3}{50} e^{2 x} x \sin (4 x)+\frac {1}{10} e^{2 x} x^2 \sin (4 x)+\frac {1}{50} \int e^{2 x} \sin (4 x) \, dx-2 \left (\frac {1}{25} \int e^{2 x} \cos (4 x) \, dx\right )-\frac {2}{25} \int e^{2 x} \sin (4 x) \, dx\\ &=\frac {3}{250} e^{2 x} \cos (4 x)+\frac {2}{25} e^{2 x} x \cos (4 x)-\frac {1}{5} e^{2 x} x^2 \cos (4 x)-\frac {3}{500} e^{2 x} \sin (4 x)+\frac {3}{50} e^{2 x} x \sin (4 x)+\frac {1}{10} e^{2 x} x^2 \sin (4 x)-2 \left (\frac {1}{250} e^{2 x} \cos (4 x)+\frac {1}{125} e^{2 x} \sin (4 x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 40, normalized size = 0.46 \[ \frac {1}{500} e^{2 x} \left (\left (50 x^2+30 x-11\right ) \sin (4 x)+\left (-100 x^2+40 x+2\right ) \cos (4 x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{2 x} x^2 \sin (4 x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.27, size = 41, normalized size = 0.47 \[ -\frac {1}{250} \, {\left (50 \, x^{2} - 20 \, x - 1\right )} \cos \left (4 \, x\right ) e^{\left (2 \, x\right )} + \frac {1}{500} \, {\left (50 \, x^{2} + 30 \, x - 11\right )} e^{\left (2 \, x\right )} \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.57, size = 39, normalized size = 0.45 \[ -\frac {1}{500} \, {\left (2 \, {\left (50 \, x^{2} - 20 \, x - 1\right )} \cos \left (4 \, x\right ) - {\left (50 \, x^{2} + 30 \, x - 11\right )} \sin \left (4 \, x\right )\right )} e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 40, normalized size = 0.46
method | result | size |
default | \(\left (-\frac {1}{5} x^{2}+\frac {2}{25} x +\frac {1}{250}\right ) {\mathrm e}^{2 x} \cos \left (4 x \right )+\left (\frac {1}{10} x^{2}+\frac {3}{50} x -\frac {11}{500}\right ) {\mathrm e}^{2 x} \sin \left (4 x \right )\) | \(40\) |
risch | \(\left (-\frac {1}{500}-\frac {i}{1000}\right ) \left (50 x^{2}+20 i x -10 x -4 i-3\right ) {\mathrm e}^{\left (2+4 i\right ) x}+\left (-\frac {1}{500}+\frac {i}{1000}\right ) \left (50 x^{2}-20 i x -10 x +4 i-3\right ) {\mathrm e}^{\left (2-4 i\right ) x}\) | \(54\) |
norman | \(\frac {\frac {2 x \,{\mathrm e}^{2 x}}{25}-\frac {{\mathrm e}^{2 x} x^{2}}{5}-\frac {11 \,{\mathrm e}^{2 x} \tan \left (2 x \right )}{250}-\frac {{\mathrm e}^{2 x} \left (\tan ^{2}\left (2 x \right )\right )}{250}+\frac {3 x \,{\mathrm e}^{2 x} \tan \left (2 x \right )}{25}-\frac {2 x \,{\mathrm e}^{2 x} \left (\tan ^{2}\left (2 x \right )\right )}{25}+\frac {{\mathrm e}^{2 x} x^{2} \tan \left (2 x \right )}{5}+\frac {{\mathrm e}^{2 x} x^{2} \left (\tan ^{2}\left (2 x \right )\right )}{5}+\frac {{\mathrm e}^{2 x}}{250}}{1+\tan ^{2}\left (2 x \right )}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.45, size = 41, normalized size = 0.47 \[ -\frac {1}{250} \, {\left (50 \, x^{2} - 20 \, x - 1\right )} \cos \left (4 \, x\right ) e^{\left (2 \, x\right )} + \frac {1}{500} \, {\left (50 \, x^{2} + 30 \, x - 11\right )} e^{\left (2 \, x\right )} \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.35, size = 51, normalized size = 0.59 \[ \frac {{\mathrm {e}}^{2\,x}\,\left (2\,\cos \left (4\,x\right )-11\,\sin \left (4\,x\right )+40\,x\,\cos \left (4\,x\right )+30\,x\,\sin \left (4\,x\right )-100\,x^2\,\cos \left (4\,x\right )+50\,x^2\,\sin \left (4\,x\right )\right )}{500} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.11, size = 85, normalized size = 0.98 \[ \frac {x^{2} e^{2 x} \sin {\left (4 x \right )}}{10} - \frac {x^{2} e^{2 x} \cos {\left (4 x \right )}}{5} + \frac {3 x e^{2 x} \sin {\left (4 x \right )}}{50} + \frac {2 x e^{2 x} \cos {\left (4 x \right )}}{25} - \frac {11 e^{2 x} \sin {\left (4 x \right )}}{500} + \frac {e^{2 x} \cos {\left (4 x \right )}}{250} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________