Optimal. Leaf size=23 \[ -\frac {t}{4}+\frac {1}{2} t \sin ^2(t)+\frac {1}{4} \sin (t) \cos (t) \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3443, 2635, 8} \[ -\frac {t}{4}+\frac {1}{2} t \sin ^2(t)+\frac {1}{4} \sin (t) \cos (t) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3443
Rubi steps
\begin {align*} \int t \cos (t) \sin (t) \, dt &=\frac {1}{2} t \sin ^2(t)-\frac {1}{2} \int \sin ^2(t) \, dt\\ &=\frac {1}{4} \cos (t) \sin (t)+\frac {1}{2} t \sin ^2(t)-\frac {\int 1 \, dt}{4}\\ &=-\frac {t}{4}+\frac {1}{4} \cos (t) \sin (t)+\frac {1}{2} t \sin ^2(t)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 0.78 \[ \frac {1}{8} \sin (2 t)-\frac {1}{4} t \cos (2 t) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 17, normalized size = 0.74 \[ -\frac {1}{2} \, t \cos \relax (t)^{2} + \frac {1}{4} \, \cos \relax (t) \sin \relax (t) + \frac {1}{4} \, t \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 14, normalized size = 0.61 \[ -\frac {1}{4} \, t \cos \left (2 \, t\right ) + \frac {1}{8} \, \sin \left (2 \, t\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.78 \[ -\frac {t \left (\cos ^{2}\relax (t )\right )}{2}+\frac {\cos \relax (t ) \sin \relax (t )}{4}+\frac {t}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 14, normalized size = 0.61 \[ -\frac {1}{4} \, t \cos \left (2 \, t\right ) + \frac {1}{8} \, \sin \left (2 \, t\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 18, normalized size = 0.78 \[ \frac {\sin \left (2\,t\right )}{8}+\frac {t\,\left (2\,{\sin \relax (t)}^2-1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 24, normalized size = 1.04 \[ \frac {t \sin ^{2}{\relax (t )}}{4} - \frac {t \cos ^{2}{\relax (t )}}{4} + \frac {\sin {\relax (t )} \cos {\relax (t )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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