Optimal. Leaf size=29 \[ -\frac{\text{Si}(3 x)}{10}+\frac{\text{Si}(7 x)}{10}-\frac{1}{5} \text{Si}(2 x) \cos (5 x) \]
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Rubi [A] time = 0.0550338, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6511, 12, 4430, 3299} \[ -\frac{\text{Si}(3 x)}{10}+\frac{\text{Si}(7 x)}{10}-\frac{1}{5} \text{Si}(2 x) \cos (5 x) \]
Antiderivative was successfully verified.
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Rule 6511
Rule 12
Rule 4430
Rule 3299
Rubi steps
\begin{align*} \int \sin (5 x) \text{Si}(2 x) \, dx &=-\frac{1}{5} \cos (5 x) \text{Si}(2 x)+\frac{2}{5} \int \frac{\cos (5 x) \sin (2 x)}{2 x} \, dx\\ &=-\frac{1}{5} \cos (5 x) \text{Si}(2 x)+\frac{1}{5} \int \frac{\cos (5 x) \sin (2 x)}{x} \, dx\\ &=-\frac{1}{5} \cos (5 x) \text{Si}(2 x)+\frac{1}{5} \int \left (-\frac{\sin (3 x)}{2 x}+\frac{\sin (7 x)}{2 x}\right ) \, dx\\ &=-\frac{1}{5} \cos (5 x) \text{Si}(2 x)-\frac{1}{10} \int \frac{\sin (3 x)}{x} \, dx+\frac{1}{10} \int \frac{\sin (7 x)}{x} \, dx\\ &=-\frac{1}{5} \cos (5 x) \text{Si}(2 x)-\frac{\text{Si}(3 x)}{10}+\frac{\text{Si}(7 x)}{10}\\ \end{align*}
Mathematica [A] time = 0.0260674, size = 25, normalized size = 0.86 \[ \frac{1}{10} (-\text{Si}(3 x)+\text{Si}(7 x)-2 \text{Si}(2 x) \cos (5 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.124, size = 24, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( 5\,x \right ){\it Si} \left ( 2\,x \right ) }{5}}-{\frac{{\it Si} \left ( 3\,x \right ) }{10}}+{\frac{{\it Si} \left ( 7\,x \right ) }{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Si}\left (2 \, x\right ) \sin \left (5 \, x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (5 x \right )} \operatorname{Si}{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20063, size = 31, normalized size = 1.07 \begin{align*} -\frac{1}{5} \, \cos \left (5 \, x\right ) \operatorname{Si}\left (2 \, x\right ) + \frac{1}{10} \, \operatorname{Si}\left (7 \, x\right ) - \frac{1}{10} \, \operatorname{Si}\left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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