Optimal. Leaf size=187 \[ -\frac {132}{125} e^{x/2} \cos (x)+\frac {18}{25} e^{x/2} x \cos (x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {428 e^{x/2} \cos (3 x)}{50653}+\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {24}{125} e^{x/2} \sin (x)-\frac {24}{25} e^{x/2} x \sin (x)+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-\frac {792 e^{x/2} \sin (3 x)}{50653}-\frac {24 e^{x/2} x \sin (3 x)}{1369} \]
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Rubi [A]
time = 0.34, antiderivative size = 253, normalized size of antiderivative = 1.35, number of steps
used = 31, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {4520, 4518,
4554, 14, 4517, 4557, 4553, 4558} \begin {gather*} \frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)-\frac {1218672 e^{x/2} \sin (x)}{6331625}-\frac {32556 e^{x/2} x \sin (x)}{34225}-\frac {816 e^{x/2} \sin (3 x)}{50653}-\frac {12 e^{x/2} x \sin (3 x)}{1369}+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}-\frac {6687696 e^{x/2} \cos (x)}{6331625}+\frac {24792 e^{x/2} x \cos (x)}{34225}-\frac {432 e^{x/2} \cos (3 x)}{50653}+\frac {72 e^{x/2} x \cos (3 x)}{1369}+\frac {96 e^{x/2} \sin (x) \cos ^2(x)}{50653}-\frac {48 e^{x/2} x \sin (x) \cos ^2(x)}{1369} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 4517
Rule 4518
Rule 4520
Rule 4553
Rule 4554
Rule 4557
Rule 4558
Rubi steps
\begin {align*} \int e^{x/2} x^2 \cos ^3(x) \, dx &=\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-2 \int x \left (\frac {48}{185} e^{x/2} \cos (x)+\frac {2}{37} e^{x/2} \cos ^3(x)+\frac {96}{185} e^{x/2} \sin (x)+\frac {12}{37} e^{x/2} \cos ^2(x) \sin (x)\right ) \, dx\\ &=\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-2 \int \left (\frac {48}{185} e^{x/2} x \cos (x)+\frac {2}{37} e^{x/2} x \cos ^3(x)+\frac {96}{185} e^{x/2} x \sin (x)+\frac {12}{37} e^{x/2} x \cos ^2(x) \sin (x)\right ) \, dx\\ &=\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-\frac {4}{37} \int e^{x/2} x \cos ^3(x) \, dx-\frac {96}{185} \int e^{x/2} x \cos (x) \, dx-\frac {24}{37} \int e^{x/2} x \cos ^2(x) \sin (x) \, dx-\frac {192}{185} \int e^{x/2} x \sin (x) \, dx\\ &=\frac {20352 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {30336 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+\frac {4}{37} \int \left (\frac {48}{185} e^{x/2} \cos (x)+\frac {2}{37} e^{x/2} \cos ^3(x)+\frac {96}{185} e^{x/2} \sin (x)+\frac {12}{37} e^{x/2} \cos ^2(x) \sin (x)\right ) \, dx+\frac {96}{185} \int \left (\frac {2}{5} e^{x/2} \cos (x)+\frac {4}{5} e^{x/2} \sin (x)\right ) \, dx-\frac {24}{37} \int \left (\frac {1}{4} e^{x/2} x \sin (x)+\frac {1}{4} e^{x/2} x \sin (3 x)\right ) \, dx+\frac {192}{185} \int \left (-\frac {4}{5} e^{x/2} \cos (x)+\frac {2}{5} e^{x/2} \sin (x)\right ) \, dx\\ &=\frac {20352 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {30336 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+\frac {8 \int e^{x/2} \cos ^3(x) \, dx}{1369}+\frac {192 \int e^{x/2} \cos (x) \, dx}{6845}+\frac {48 \int e^{x/2} \cos ^2(x) \sin (x) \, dx}{1369}+\frac {384 \int e^{x/2} \sin (x) \, dx}{6845}-\frac {6}{37} \int e^{x/2} x \sin (x) \, dx-\frac {6}{37} \int e^{x/2} x \sin (3 x) \, dx+\frac {192}{925} \int e^{x/2} \cos (x) \, dx+2 \left (\frac {384}{925} \int e^{x/2} \sin (x) \, dx\right )-\frac {768}{925} \int e^{x/2} \cos (x) \, dx\\ &=-\frac {48384 e^{x/2} \cos (x)}{171125}+\frac {24792 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {72 e^{x/2} x \cos (3 x)}{1369}-\frac {77568 e^{x/2} \sin (x)}{171125}-\frac {32556 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac {1536 e^{x/2} \cos (x)}{4625}+\frac {768 e^{x/2} \sin (x)}{4625}\right )-\frac {12 e^{x/2} x \sin (3 x)}{1369}+\frac {192 \int e^{x/2} \cos (x) \, dx}{50653}+\frac {48 \int \left (\frac {1}{4} e^{x/2} \sin (x)+\frac {1}{4} e^{x/2} \sin (3 x)\right ) \, dx}{1369}+\frac {6}{37} \int \left (-\frac {4}{5} e^{x/2} \cos (x)+\frac {2}{5} e^{x/2} \sin (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 {1780608 e^{x/2} \cos (x)}{6331625}+\frac {24792 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {72 e^{x/2} x \cos (3 x)}{1369}-\frac {2850816 e^{x/2} \sin (x)}{6331625}-\frac {32556 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac {1536 e^{x/2} \cos (x)}{4625}+\frac {768 e^{x/2} \sin (x)}{4625}\right )-\frac {12 e^{x/2} x \sin (3 x)}{1369}+\frac {12 \int e^{x/2} \sin (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 {12}{185} \int e^{x/2} \sin (x) \, dx-\frac {24}{185} \int e^{x/2} \cos (x) \, dx\\ &=-\frac {2482128 e^{x/2} \cos (x)}{6331625}+\frac {24792 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {144 e^{x/2} \cos (3 x)}{50653}+\frac {72 e^{x/2} x \cos (3 x)}{1369}-\frac {3321456 e^{x/2} \sin (x)}{6331625}-\frac {32556 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac {1536 e^{x/2} \cos (x)}{4625}+\frac {768 e^{x/2} \sin (x)}{4625}\right )-\frac {864 e^{x/2} \sin (3 x)}{50653}-\frac {12 e^{x/2} x \sin (3 x)}{1369}+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 = 0.11, size = 72, normalized size = 0.39 \begin {gather*} \frac {e^{x/2} \left (151959 \left (-88+60 x+25 x^2\right ) \cos (x)+125 \left (-856+5180 x+1369 x^2\right ) \cos (3 x)+303918 \left (-8-40 x+25 x^2\right ) \sin (x)+750 \left (-264-296 x+1369 x^2\right ) \sin (3 x)\right )}{12663250} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 78, normalized size = 0.42
method | result | size |
default | \(\frac {\left (\frac {2}{37} x^{2}+\frac {280}{1369} x -\frac {1712}{50653}\right ) {\mathrm e}^{\frac {x}{2}} \cos \left (3 x \right )}{4}-\frac {\left (-\frac {12}{37} x^{2}+\frac {96}{1369} x +\frac {3168}{50653}\right ) {\mathrm e}^{\frac {x}{2}} \sin \left (3 x \right )}{4}+\frac {3 \left (\frac {2}{5} x^{2}+\frac {24}{25} x -\frac {176}{125}\right ) {\mathrm e}^{\frac {x}{2}} \cos \left (x \right )}{4}-\frac {3 \left (-\frac {4}{5} x^{2}+\frac {32}{25} x +\frac {32}{125}\right ) {\mathrm e}^{\frac {x}{2}} \sin \left (x \right )}{4}\) | \(78\) |
risch | \(\left (\frac {1}{202612}-\frac {3 i}{101306}\right ) \left (1369 x^{2}+888 i x -148 x -96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}+3 i\right ) x}+\left (\frac {3}{500}-\frac {3 i}{250}\right ) \left (25 x^{2}+40 i x -20 x -32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}+i\right ) x}+\left (\frac {3}{500}+\frac {3 i}{250}\right ) \left (25 x^{2}-40 i x -20 x +32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}-i\right ) x}+\left (\frac {1}{202612}+\frac {3 i}{101306}\right ) \left (1369 x^{2}-888 i x -148 x +96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}-3 i\right ) x}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.13, size = 77, normalized size = 0.41 \begin {gather*} \frac {1}{101306} \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{250} \, {\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) 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 {3}{125} \, {\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 72, normalized size = 0.39 \begin {gather*} \frac {12}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \left (x\right )^{2} + 273800 \, x^{2} - 497280 \, x - 93056\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) + \frac {2}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (x\right )^{3} + 24 \, {\left (34225 \, x^{2} + 74740 \, x - 135952\right )} \cos \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.33, size = 202, normalized size = 1.08 \begin {gather*} \frac {96 x^{2} e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{185} + \frac {48 x^{2} e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{185} + \frac {156 x^{2} e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{185} + \frac {58 x^{2} e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{185} - \frac {32256 x e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{34225} + \frac {19392 x e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{34225} - \frac {34656 x e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{34225} + \frac {26392 x e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{34225} - \frac {1116672 e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{6331625} - \frac {6525696 e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{6331625} - \frac {1512672 e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{6331625} - \frac {6739696 e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{6331625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.62, size = 73, normalized size = 0.39 \begin {gather*} \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 {3}{250} \, {\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) + 2 \, {\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 83, normalized size = 0.44 \begin {gather*} -\frac {{\mathrm {e}}^{x/2}\,\left (107000\,\cos \left (3\,x\right )+198000\,\sin \left (3\,x\right )+13372392\,\cos \left (x\right )+2431344\,\sin \left (x\right )-647500\,x\,\cos \left (3\,x\right )-3798975\,x^2\,\cos \left (x\right )+222000\,x\,\sin \left (3\,x\right )-7597950\,x^2\,\sin \left (x\right )-171125\,x^2\,\cos \left (3\,x\right )-1026750\,x^2\,\sin \left (3\,x\right )-9117540\,x\,\cos \left (x\right )+12156720\,x\,\sin \left (x\right )\right )}{12663250} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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