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*Si(b*x)

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ x \text {Si}(b x)+\frac {\cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[SinIntegral[b*x],x]

[Out]

Cos[b*x]/b + x*SinIntegral[b*x]

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Si}\left (b x\right ), x\right ) \]

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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giac [A]  time = 0.62, size = 15, normalized size = 1.00 \[ x \operatorname {Si}\left (b x\right ) + \frac {\cos \left (b x\right )}{b} \]

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

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maple [A]  time = 0.02, size = 17, normalized size = 1.13 \[ \frac {b x \Si \left (b x \right )+\cos \left (b x \right )}{b} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Si}\left (b x\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ x\,\mathrm {sinint}\left (b\,x\right )+\frac {\cos \left (b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinint(b*x),x)

[Out]

x*sinint(b*x) + cos(b*x)/b

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sympy [A]  time = 0.71, size = 12, normalized size = 0.80 \[ x \operatorname {Si}{\left (b x \right )} + \frac {\cos {\left (b x \right )}}{b} \]

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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