Optimal. Leaf size=185 \[ \frac {1}{5} e^{x/2} x^2 \sin (x)-\frac {3}{37} e^{x/2} x^2 \sin (3 x)+\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {8}{125} e^{x/2} \sin (x)+\frac {792 e^{x/2} \sin (3 x)}{50653}+\frac {6}{25} e^{x/2} x \cos (x)-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {44}{125} e^{x/2} \cos (x)+\frac {428 e^{x/2} \cos (3 x)}{50653} \]
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Rubi [A] time = 0.36, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {4470, 4433, 4466, 14, 4432, 4465} \[ \frac {1}{5} e^{x/2} x^2 \sin (x)-\frac {3}{37} e^{x/2} x^2 \sin (3 x)+\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {8}{125} e^{x/2} \sin (x)+\frac {792 e^{x/2} \sin (3 x)}{50653}+\frac {6}{25} e^{x/2} x \cos (x)-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {44}{125} e^{x/2} \cos (x)+\frac {428 e^{x/2} \cos (3 x)}{50653} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4432
Rule 4433
Rule 4465
Rule 4466
Rule 4470
Rubi steps
\begin {align*} \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx &=\int \left (\frac {1}{4} e^{x/2} x^2 \cos (x)-\frac {1}{4} e^{x/2} x^2 \cos (3 x)\right ) \, dx\\ &=\frac {1}{4} \int e^{x/2} x^2 \cos (x) \, dx-\frac {1}{4} \int e^{x/2} x^2 \cos (3 x) \, dx\\ &=\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {1}{74} e^{x/2} x^2 \cos (3 x)+\frac {1}{5} e^{x/2} x^2 \sin (x)-\frac {3}{37} e^{x/2} x^2 \sin (3 x)-\frac {1}{2} \int x \left (\frac {2}{5} e^{x/2} \cos (x)+\frac {4}{5} e^{x/2} \sin (x)\right ) \, dx+\frac {1}{2} \int x \left (\frac {2}{37} e^{x/2} \cos (3 x)+\frac {12}{37} e^{x/2} \sin (3 x)\right ) \, dx\\ &=\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {1}{74} e^{x/2} x^2 \cos (3 x)+\frac {1}{5} e^{x/2} x^2 \sin (x)-\frac {3}{37} e^{x/2} x^2 \sin (3 x)-\frac {1}{2} \int \left (\frac {2}{5} e^{x/2} x \cos (x)+\frac {4}{5} e^{x/2} x \sin (x)\right ) \, dx+\frac {1}{2} \int \left (\frac {2}{37} e^{x/2} x \cos (3 x)+\frac {12}{37} e^{x/2} x \sin (3 x)\right ) \, dx\\ &=\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {1}{74} e^{x/2} x^2 \cos (3 x)+\frac {1}{5} e^{x/2} x^2 \sin (x)-\frac {3}{37} e^{x/2} x^2 \sin (3 x)+\frac {1}{37} \int e^{x/2} x \cos (3 x) \, dx+\frac {6}{37} \int e^{x/2} x \sin (3 x) \, dx-\frac {1}{5} \int e^{x/2} x \cos (x) \, dx-\frac {2}{5} \int e^{x/2} x \sin (x) \, dx\\ &=\frac {6}{25} e^{x/2} x \cos (x)+\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {1}{5} e^{x/2} x^2 \sin (x)+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {3}{37} e^{x/2} x^2 \sin (3 x)-\frac {1}{37} \int \left (\frac {2}{37} e^{x/2} \cos (3 x)+\frac {12}{37} e^{x/2} \sin (3 x)\right ) \, dx-\frac {6}{37} \int \left (-\frac {12}{37} e^{x/2} \cos (3 x)+\frac {2}{37} e^{x/2} \sin (3 x)\right ) \, dx+\frac {1}{5} \int \left (\frac {2}{5} e^{x/2} \cos (x)+\frac {4}{5} e^{x/2} \sin (x)\right ) \, dx+\frac {2}{5} \int \left (-\frac {4}{5} e^{x/2} \cos (x)+\frac {2}{5} e^{x/2} \sin (x)\right ) \, dx\\ &=\frac {6}{25} e^{x/2} x \cos (x)+\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {1}{5} e^{x/2} x^2 \sin (x)+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {3}{37} e^{x/2} x^2 \sin (3 x)-\frac {2 \int e^{x/2} \cos (3 x) \, dx}{1369}-2 \frac {12 \int e^{x/2} \sin (3 x) \, dx}{1369}+\frac {72 \int e^{x/2} \cos (3 x) \, dx}{1369}+\frac {2}{25} \int e^{x/2} \cos (x) \, dx+2 \left (\frac {4}{25} \int e^{x/2} \sin (x) \, dx\right )-\frac {8}{25} \int e^{x/2} \cos (x) \, dx\\ &=-\frac {12}{125} e^{x/2} \cos (x)+\frac {6}{25} e^{x/2} x \cos (x)+\frac {1}{10} e^{x/2} x^2 \cos (x)+\frac {140 e^{x/2} \cos (3 x)}{50653}-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {24}{125} e^{x/2} \sin (x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {1}{5} e^{x/2} x^2 \sin (x)+2 \left (-\frac {16}{125} e^{x/2} \cos (x)+\frac {8}{125} e^{x/2} \sin (x)\right )+\frac {840 e^{x/2} \sin (3 x)}{50653}+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {3}{37} e^{x/2} x^2 \sin (3 x)-2 \left (-\frac {144 e^{x/2} \cos (3 x)}{50653}+\frac {24 e^{x/2} \sin (3 x)}{50653}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 76, normalized size = 0.41 \[ \frac {e^{x/2} \left (50653 \left (2 \left (25 x^2-40 x-8\right ) \sin (x)+\left (25 x^2+60 x-88\right ) \cos (x)\right )-125 \left (6 \left (1369 x^2-296 x-264\right ) \sin (3 x)+\left (1369 x^2+5180 x-856\right ) \cos (3 x)\right )\right )}{12663250} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.15, size = 72, normalized size = 0.39 \[ -\frac {4}{6331625} \, {\left (375 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \relax (x)^{2} - 444925 \, x^{2} + 534280 \, x + 126056\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \relax (x) - \frac {2}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \relax (x)^{3} - {\left (444925 \, x^{2} + 1245420 \, x - 1194616\right )} \cos \relax (x)\right )} e^{\left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 73, normalized size = 0.39 \[ -\frac {1}{101306} \, {\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} + \frac {1}{250} \, {\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \relax (x) + 2 \, {\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \relax (x)\right )} e^{\left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 106, normalized size = 0.57
method | result | size |
risch | \(\left (-\frac {1}{202612}+\frac {3 i}{101306}\right ) \left (1369 x^{2}+888 i x -148 x -96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}+3 i\right ) x}+\left (\frac {1}{500}-\frac {i}{250}\right ) \left (25 x^{2}+40 i x -20 x -32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}+i\right ) x}+\left (\frac {1}{500}+\frac {i}{250}\right ) \left (25 x^{2}-40 i x -20 x +32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}-i\right ) x}+\left (-\frac {1}{202612}-\frac {3 i}{101306}\right ) \left (1369 x^{2}-888 i x -148 x +96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}-3 i\right ) x}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 77, normalized size = 0.42 \[ -\frac {1}{101306} \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {1}{250} \, {\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \relax (x) e^{\left (\frac {1}{2} \, x\right )} - \frac {3}{50653} \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac {1}{125} \, {\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 83, normalized size = 0.45 \[ \frac {{\mathrm {e}}^{x/2}\,\left (107000\,\cos \left (3\,x\right )+198000\,\sin \left (3\,x\right )-4457464\,\cos \relax (x)-810448\,\sin \relax (x)-647500\,x\,\cos \left (3\,x\right )+1266325\,x^2\,\cos \relax (x)+222000\,x\,\sin \left (3\,x\right )+2532650\,x^2\,\sin \relax (x)-171125\,x^2\,\cos \left (3\,x\right )-1026750\,x^2\,\sin \left (3\,x\right )+3039180\,x\,\cos \relax (x)-4052240\,x\,\sin \relax (x)\right )}{12663250} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.81, size = 202, normalized size = 1.09 \[ \frac {52 x^{2} e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{185} + \frac {26 x^{2} e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{185} - \frac {8 x^{2} e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{185} + \frac {16 x^{2} e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{185} - \frac {11552 x e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{34225} + \frac {13464 x e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{34225} - \frac {9152 x e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{34225} + \frac {6464 x e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{34225} - \frac {504224 e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{6331625} - \frac {2389232 e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{6331625} - \frac {108224 e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{6331625} - \frac {2175232 e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{6331625} \]
Verification of antiderivative is not currently implemented for this CAS.
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