Optimal. Leaf size=27 \[ \frac{(a+b x) \text{CosIntegral}(a+b x)}{b}-\frac{\sin (a+b x)}{b} \]
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Rubi [A] time = 0.0056449, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6500} \[ \frac{(a+b x) \text{CosIntegral}(a+b x)}{b}-\frac{\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6500
Rubi steps
\begin{align*} \int \text{Ci}(a+b x) \, dx &=\frac{(a+b x) \text{Ci}(a+b x)}{b}-\frac{\sin (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0363525, size = 42, normalized size = 1.56 \[ x \text{CosIntegral}(a+b x)+\frac{a \text{CosIntegral}(a+b x)}{b}-\frac{\sin (a) \cos (b x)}{b}-\frac{\cos (a) \sin (b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 26, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ){\it Ci} \left ( bx+a \right ) -\sin \left ( bx+a \right ) }{b}} \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 (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 (\operatorname{Ci}\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 \operatorname{Ci}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13353, size = 93, normalized size = 3.44 \begin{align*} x \operatorname{Ci}\left (b x + a\right ) + \frac{a \cos \left (a\right )^{2} \operatorname{Ci}\left (b x + a\right ) + a \cos \left (a\right )^{2} \operatorname{Ci}\left (-b x - a\right ) + a \operatorname{Ci}\left (b x + a\right ) \sin \left (a\right )^{2} + a \operatorname{Ci}\left (-b x - a\right ) \sin \left (a\right )^{2}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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