3.5 \(\int \text{Si}(b x) \, dx\)

Optimal. Leaf size=15 \[ x \text{Si}(b x)+\frac{\cos (b x)}{b} \]

[Out]

Cos[b*x]/b + x*SinIntegral[b*x]

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Rubi [A]  time = 0.0038514, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6499} \[ x \text{Si}(b x)+\frac{\cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[SinIntegral[b*x],x]

[Out]

Cos[b*x]/b + x*SinIntegral[b*x]

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

Mathematica [A]  time = 0.0014557, size = 15, normalized size = 1. \[ x \text{Si}(b x)+\frac{\cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[SinIntegral[b*x],x]

[Out]

Cos[b*x]/b + x*SinIntegral[b*x]

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Maple [A]  time = 0.05, size = 17, normalized size = 1.1 \begin{align*}{\frac{bx{\it Si} \left ( bx \right ) +\cos \left ( bx \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Si(b*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Si(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin_integral(b*x), x)

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Sympy [A]  time = 0.802061, size = 12, normalized size = 0.8 \begin{align*} x \operatorname{Si}{\left (b x \right )} + \frac{\cos{\left (b x \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x),x)

[Out]

x*Si(b*x) + cos(b*x)/b

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Giac [A]  time = 1.25101, size = 20, normalized size = 1.33 \begin{align*} x \operatorname{Si}\left (b x\right ) + \frac{\cos \left (b x\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin_integral(b*x) + cos(b*x)/b