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 = 0.0127517, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 3443
Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.0028449, 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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Maple [A] time = 0.001, size = 18, normalized size = 0.8 \begin{align*} -{\frac{t \left ( \cos \left ( t \right ) \right ) ^{2}}{2}}+{\frac{\cos \left ( t \right ) \sin \left ( t \right ) }{4}}+{\frac{t}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938485, size = 19, normalized size = 0.83 \begin{align*} -\frac{1}{4} \, t \cos \left (2 \, t\right ) + \frac{1}{8} \, \sin \left (2 \, t\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95604, size = 61, normalized size = 2.65 \begin{align*} -\frac{1}{2} \, t \cos \left (t\right )^{2} + \frac{1}{4} \, \cos \left (t\right ) \sin \left (t\right ) + \frac{1}{4} \, t \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.313081, size = 24, normalized size = 1.04 \begin{align*} \frac{t \sin ^{2}{\left (t \right )}}{4} - \frac{t \cos ^{2}{\left (t \right )}}{4} + \frac{\sin{\left (t \right )} \cos{\left (t \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05105, size = 19, normalized size = 0.83 \begin{align*} -\frac{1}{4} \, t \cos \left (2 \, t\right ) + \frac{1}{8} \, \sin \left (2 \, t\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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