Optimal. Leaf size=29 \[ \frac{1}{10} \text{CosIntegral}(3 x)+\frac{1}{10} \text{CosIntegral}(7 x)-\frac{1}{5} \text{CosIntegral}(2 x) \cos (5 x) \]
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Rubi [A] time = 0.0557009, 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 = {6518, 12, 4429, 3302} \[ \frac{1}{10} \text{CosIntegral}(3 x)+\frac{1}{10} \text{CosIntegral}(7 x)-\frac{1}{5} \text{CosIntegral}(2 x) \cos (5 x) \]
Antiderivative was successfully verified.
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Rule 6518
Rule 12
Rule 4429
Rule 3302
Rubi steps
\begin{align*} \int \text{Ci}(2 x) \sin (5 x) \, dx &=-\frac{1}{5} \cos (5 x) \text{Ci}(2 x)+\frac{2}{5} \int \frac{\cos (2 x) \cos (5 x)}{2 x} \, dx\\ &=-\frac{1}{5} \cos (5 x) \text{Ci}(2 x)+\frac{1}{5} \int \frac{\cos (2 x) \cos (5 x)}{x} \, dx\\ &=-\frac{1}{5} \cos (5 x) \text{Ci}(2 x)+\frac{1}{5} \int \left (\frac{\cos (3 x)}{2 x}+\frac{\cos (7 x)}{2 x}\right ) \, dx\\ &=-\frac{1}{5} \cos (5 x) \text{Ci}(2 x)+\frac{1}{10} \int \frac{\cos (3 x)}{x} \, dx+\frac{1}{10} \int \frac{\cos (7 x)}{x} \, dx\\ &=-\frac{1}{5} \cos (5 x) \text{Ci}(2 x)+\frac{\text{Ci}(3 x)}{10}+\frac{\text{Ci}(7 x)}{10}\\ \end{align*}
Mathematica [A] time = 0.0262833, size = 23, normalized size = 0.79 \[ \frac{1}{10} (\text{CosIntegral}(3 x)+\text{CosIntegral}(7 x)-2 \text{CosIntegral}(2 x) \cos (5 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.104, size = 24, normalized size = 0.8 \begin{align*}{\frac{{\it Ci} \left ( 3\,x \right ) }{10}}+{\frac{{\it Ci} \left ( 7\,x \right ) }{10}}-{\frac{{\it Ci} \left ( 2\,x \right ) \cos \left ( 5\,x \right ) }{5}} \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 Ci}\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{Ci}{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21775, size = 31, normalized size = 1.07 \begin{align*} -\frac{1}{5} \, \cos \left (5 \, x\right ) \operatorname{Ci}\left (2 \, x\right ) + \frac{1}{10} \, \operatorname{Ci}\left (7 \, x\right ) + \frac{1}{10} \, \operatorname{Ci}\left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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