3.21 \(\int \text{Si}(a+b x) \, dx\)

Optimal. Leaf size=26 \[ \frac{(a+b x) \text{Si}(a+b x)}{b}+\frac{\cos (a+b x)}{b} \]

[Out]

Cos[a + b*x]/b + ((a + b*x)*SinIntegral[a + b*x])/b

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Rubi [A]  time = 0.0054842, antiderivative size = 26, 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 = {6499} \[ \frac{(a+b x) \text{Si}(a+b x)}{b}+\frac{\cos (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[SinIntegral[a + b*x],x]

[Out]

Cos[a + b*x]/b + ((a + b*x)*SinIntegral[a + b*x])/b

Rule 6499

Int[SinIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*SinIntegral[a + b*x])/b, x] + Simp[Cos[a + b
*x]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \text{Si}(a+b x) \, dx &=\frac{\cos (a+b x)}{b}+\frac{(a+b x) \text{Si}(a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0359392, size = 41, normalized size = 1.58 \[ x \text{Si}(a+b x)+\frac{a \text{Si}(a+b x)}{b}-\frac{\sin (a) \sin (b x)}{b}+\frac{\cos (a) \cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[SinIntegral[a + b*x],x]

[Out]

(Cos[a]*Cos[b*x])/b - (Sin[a]*Sin[b*x])/b + (a*SinIntegral[a + b*x])/b + x*SinIntegral[a + b*x]

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Maple [A]  time = 0.049, size = 24, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ){\it Si} \left ( bx+a \right ) +\cos \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Si(b*x+a),x)

[Out]

1/b*((b*x+a)*Si(b*x+a)+cos(b*x+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Si}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x+a),x, algorithm="maxima")

[Out]

integrate(Si(b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{Si}\left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x+a),x, algorithm="fricas")

[Out]

integral(sin_integral(b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Si}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x+a),x)

[Out]

Integral(Si(a + b*x), x)

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Giac [C]  time = 1.34509, size = 62, normalized size = 2.38 \begin{align*} x \operatorname{Si}\left (b x + a\right ) + \frac{a \Im \left ( \operatorname{Ci}\left (b x + a\right ) \right ) - a \Im \left ( \operatorname{Ci}\left (-b x - a\right ) \right ) + 2 \, a \operatorname{Si}\left (b x + a\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x+a),x, algorithm="giac")

[Out]

x*sin_integral(b*x + a) + 1/2*(a*imag_part(cos_integral(b*x + a)) - a*imag_part(cos_integral(-b*x - a)) + 2*a*
sin_integral(b*x + a))/b