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

Optimal. Leaf size=34 \[ \frac{\text{CosIntegral}(2 b x)}{2 b}+\frac{\text{Si}(b x) \sin (b x)}{b}-\frac{\log (x)}{2 b} \]

[Out]

CosIntegral[2*b*x]/(2*b) - Log[x]/(2*b) + (Sin[b*x]*SinIntegral[b*x])/b

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Rubi [A]  time = 0.0577979, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6517, 12, 3312, 3302} \[ \frac{\text{CosIntegral}(2 b x)}{2 b}+\frac{\text{Si}(b x) \sin (b x)}{b}-\frac{\log (x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[b*x]*SinIntegral[b*x],x]

[Out]

CosIntegral[2*b*x]/(2*b) - Log[x]/(2*b) + (Sin[b*x]*SinIntegral[b*x])/b

Rule 6517

Int[Cos[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(Sin[a + b*x]*SinIntegral[c + d
*x])/b, x] - Dist[d/b, Int[(Sin[a + b*x]*Sin[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \cos (b x) \text{Si}(b x) \, dx &=\frac{\sin (b x) \text{Si}(b x)}{b}-\int \frac{\sin ^2(b x)}{b x} \, dx\\ &=\frac{\sin (b x) \text{Si}(b x)}{b}-\frac{\int \frac{\sin ^2(b x)}{x} \, dx}{b}\\ &=\frac{\sin (b x) \text{Si}(b x)}{b}-\frac{\int \left (\frac{1}{2 x}-\frac{\cos (2 b x)}{2 x}\right ) \, dx}{b}\\ &=-\frac{\log (x)}{2 b}+\frac{\sin (b x) \text{Si}(b x)}{b}+\frac{\int \frac{\cos (2 b x)}{x} \, dx}{2 b}\\ &=\frac{\text{Ci}(2 b x)}{2 b}-\frac{\log (x)}{2 b}+\frac{\sin (b x) \text{Si}(b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0129354, size = 36, normalized size = 1.06 \[ \frac{\text{CosIntegral}(2 b x)}{2 b}+\frac{\text{Si}(b x) \sin (b x)}{b}-\frac{\log (b x)}{2 b} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[b*x]*SinIntegral[b*x],x]

[Out]

CosIntegral[2*b*x]/(2*b) - Log[b*x]/(2*b) + (Sin[b*x]*SinIntegral[b*x])/b

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Maple [A]  time = 0.053, size = 33, normalized size = 1. \begin{align*}{\frac{{\it Si} \left ( bx \right ) \sin \left ( bx \right ) }{b}}-{\frac{\ln \left ( bx \right ) }{2\,b}}+{\frac{{\it Ci} \left ( 2\,bx \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x)*Si(b*x),x)

[Out]

Si(b*x)*sin(b*x)/b-1/2/b*ln(b*x)+1/2*Ci(2*b*x)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Si(b*x)*cos(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(b*x)*sin_integral(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x)*Si(b*x),x)

[Out]

Integral(cos(b*x)*Si(b*x), x)

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Giac [A]  time = 1.18222, size = 45, normalized size = 1.32 \begin{align*} \frac{\sin \left (b x\right ) \operatorname{Si}\left (b x\right )}{b} + \frac{\operatorname{Ci}\left (2 \, b x\right ) + \operatorname{Ci}\left (-2 \, b x\right ) - 2 \, \log \left (x\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(b*x)*sin_integral(b*x)/b + 1/4*(cos_integral(2*b*x) + cos_integral(-2*b*x) - 2*log(x))/b