Optimal. Leaf size=50 \[ \frac {1}{2} e^x x^2 \sin (x)-\frac {1}{2} e^x x^2 \cos (x)-\frac {1}{2} e^x \sin (x)+e^x x \cos (x)-\frac {1}{2} e^x \cos (x) \]
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Rubi [A] time = 0.11, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4432, 4465, 14, 4433, 4466} \[ \frac {1}{2} e^x x^2 \sin (x)-\frac {1}{2} e^x x^2 \cos (x)-\frac {1}{2} e^x \sin (x)+e^x x \cos (x)-\frac {1}{2} e^x \cos (x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 4432
Rule 4433
Rule 4465
Rule 4466
Rubi steps
\begin {align*} \int e^x x^2 \sin (x) \, dx &=-\frac {1}{2} e^x x^2 \cos (x)+\frac {1}{2} e^x x^2 \sin (x)-2 \int x \left (-\frac {1}{2} e^x \cos (x)+\frac {1}{2} e^x \sin (x)\right ) \, dx\\ &=-\frac {1}{2} e^x x^2 \cos (x)+\frac {1}{2} e^x x^2 \sin (x)-2 \int \left (-\frac {1}{2} e^x x \cos (x)+\frac {1}{2} e^x x \sin (x)\right ) \, dx\\ &=-\frac {1}{2} e^x x^2 \cos (x)+\frac {1}{2} e^x x^2 \sin (x)+\int e^x x \cos (x) \, dx-\int e^x x \sin (x) \, dx\\ &=e^x x \cos (x)-\frac {1}{2} e^x x^2 \cos (x)+\frac {1}{2} e^x x^2 \sin (x)+\int \left (-\frac {1}{2} e^x \cos (x)+\frac {1}{2} e^x \sin (x)\right ) \, dx-\int \left (\frac {1}{2} e^x \cos (x)+\frac {1}{2} e^x \sin (x)\right ) \, dx\\ &=e^x x \cos (x)-\frac {1}{2} e^x x^2 \cos (x)+\frac {1}{2} e^x x^2 \sin (x)-2 \left (\frac {1}{2} \int e^x \cos (x) \, dx\right )\\ &=e^x x \cos (x)-\frac {1}{2} e^x x^2 \cos (x)+\frac {1}{2} e^x x^2 \sin (x)-2 \left (\frac {1}{4} e^x \cos (x)+\frac {1}{4} e^x \sin (x)\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 25, normalized size = 0.50 \[ \frac {1}{2} e^x \left (\left (x^2-1\right ) \sin (x)-(x-1)^2 \cos (x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^x x^2 \sin (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.40, size = 26, normalized size = 0.52 \[ -\frac {1}{2} \, {\left (x^{2} - 2 \, x + 1\right )} \cos \relax (x) e^{x} + \frac {1}{2} \, {\left (x^{2} - 1\right )} e^{x} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 25, normalized size = 0.50 \[ -\frac {1}{2} \, {\left ({\left (x^{2} - 2 \, x + 1\right )} \cos \relax (x) - {\left (x^{2} - 1\right )} \sin \relax (x)\right )} e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 27, normalized size = 0.54
method | result | size |
default | \(\left (-\frac {1}{2} x^{2}+x -\frac {1}{2}\right ) {\mathrm e}^{x} \cos \relax (x )+\left (\frac {x^{2}}{2}-\frac {1}{2}\right ) {\mathrm e}^{x} \sin \relax (x )\) | \(27\) |
risch | \(\left (-\frac {1}{4}-\frac {i}{4}\right ) \left (x^{2}+i x -x -i\right ) {\mathrm e}^{\left (1+i\right ) x}+\left (-\frac {1}{4}+\frac {i}{4}\right ) \left (x^{2}-i x -x +i\right ) {\mathrm e}^{\left (1-i\right ) x}\) | \(48\) |
norman | \(\frac {{\mathrm e}^{x} x +{\mathrm e}^{x} x^{2} \tan \left (\frac {x}{2}\right )-\frac {{\mathrm e}^{x} x^{2}}{2}-{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )+\frac {{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-{\mathrm e}^{x} x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {{\mathrm e}^{x} x^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {{\mathrm e}^{x}}{2}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 26, normalized size = 0.52 \[ -\frac {1}{2} \, {\left (x^{2} - 2 \, x + 1\right )} \cos \relax (x) e^{x} + \frac {1}{2} \, {\left (x^{2} - 1\right )} e^{x} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 21, normalized size = 0.42 \[ \frac {{\mathrm {e}}^x\,\left (x-1\right )\,\left (\cos \relax (x)+\sin \relax (x)-x\,\cos \relax (x)+x\,\sin \relax (x)\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.06, size = 48, normalized size = 0.96 \[ \frac {x^{2} e^{x} \sin {\relax (x )}}{2} - \frac {x^{2} e^{x} \cos {\relax (x )}}{2} + x e^{x} \cos {\relax (x )} - \frac {e^{x} \sin {\relax (x )}}{2} - \frac {e^{x} \cos {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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