Optimal. Leaf size=34 \[ \frac{\text{Si}(2 a+2 b x)}{2 b}-\frac{\text{Si}(a+b x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.0541703, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6511, 4406, 12, 3299} \[ \frac{\text{Si}(2 a+2 b x)}{2 b}-\frac{\text{Si}(a+b x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6511
Rule 4406
Rule 12
Rule 3299
Rubi steps
\begin{align*} \int \sin (a+b x) \text{Si}(a+b x) \, dx &=-\frac{\cos (a+b x) \text{Si}(a+b x)}{b}+\int \frac{\cos (a+b x) \sin (a+b x)}{a+b x} \, dx\\ &=-\frac{\cos (a+b x) \text{Si}(a+b x)}{b}+\int \frac{\sin (2 a+2 b x)}{2 (a+b x)} \, dx\\ &=-\frac{\cos (a+b x) \text{Si}(a+b x)}{b}+\frac{1}{2} \int \frac{\sin (2 a+2 b x)}{a+b x} \, dx\\ &=-\frac{\cos (a+b x) \text{Si}(a+b x)}{b}+\frac{\text{Si}(2 a+2 b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.01693, size = 33, normalized size = 0.97 \[ \frac{\text{Si}(2 (a+b x))}{2 b}-\frac{\text{Si}(a+b x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 31, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( -\cos \left ( bx+a \right ){\it Si} \left ( bx+a \right ) +{\frac{{\it Si} \left ( 2\,bx+2\,a \right ) }{2}} \right ) } \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 (b x + a\right ) \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \left (b x + a\right ) \operatorname{Si}\left (b x + a\right ), x\right ) \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 (a + b x \right )} \operatorname{Si}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.19919, size = 77, normalized size = 2.26 \begin{align*} -\frac{\cos \left (b x + a\right ) \operatorname{Si}\left (b x + a\right )}{b} + \frac{\Im \left ( \operatorname{Ci}\left (2 \, b x + 2 \, a\right ) \right ) - \Im \left ( \operatorname{Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 2 \, \operatorname{Si}\left (2 \, b x + 2 \, a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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