3.1.5 \(\int \text {Si}(b x) \, dx\) [5]

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

[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 = {6634} \begin {gather*} x \text {Si}(b x)+\frac {\cos (b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[SinIntegral[b*x],x]

[Out]

Cos[b*x]/b + x*SinIntegral[b*x]

Rule 6634

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 \begin {gather*} \frac {\cos (b x)}{b}+x \text {Si}(b x) \end {gather*}

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 = 0.20, size = 17, normalized size = 1.13

method	result	size
derivativedivides	\(\frac {\sinIntegral \left (b x \right ) b x +\cos \left (b x \right )}{b}\)	\(17\)
default	\(\frac {\sinIntegral \left (b x \right ) b x +\cos \left (b x \right )}{b}\)	\(17\)
meijerg	\(\frac {\sqrt {\pi }\, \left (-\frac {2}{\sqrt {\pi }}+\frac {2 \cos \left (b x \right )}{\sqrt {\pi }}+\frac {2 b x \sinIntegral \left (b x \right )}{\sqrt {\pi }}\right )}{2 b}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Si(b*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 16, normalized size = 1.07 \begin {gather*} \frac {b x \operatorname {Si}\left (b x\right ) + \cos \left (b x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*sin_integral(b*x) + cos(b*x))/b

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Fricas [A]
time = 0.37, size = 16, normalized size = 1.07 \begin {gather*} \frac {b x \operatorname {Si}\left (b x\right ) + \cos \left (b x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*sin_integral(b*x) + cos(b*x))/b

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Sympy [A]
time = 0.45, size = 12, normalized size = 0.80 \begin {gather*} x \operatorname {Si}{\left (b x \right )} + \frac {\cos {\left (b x \right )}}{b} \end {gather*}

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 = 0.40, size = 15, normalized size = 1.00 \begin {gather*} x \operatorname {Si}\left (b x\right ) + \frac {\cos \left (b x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin_integral(b*x) + cos(b*x)/b

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

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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