3.110 \(\int \text{CosIntegral}(b x) \sin (b x) \, dx\)

Optimal. Leaf size=35 \[ \frac{\text{CosIntegral}(2 b x)}{2 b}-\frac{\text{CosIntegral}(b x) \cos (b x)}{b}+\frac{\log (x)}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0612325, antiderivative size = 35, 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 = {6518, 12, 3312, 3302} \[ \frac{\text{CosIntegral}(2 b x)}{2 b}-\frac{\text{CosIntegral}(b x) \cos (b x)}{b}+\frac{\log (x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[CosIntegral[b*x]*Sin[b*x],x]

[Out]

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

Rule 6518

Int[CosIntegral[(c_.) + (d_.)*(x_)]*Sin[(a_.) + (b_.)*(x_)], x_Symbol] :> -Simp[(Cos[a + b*x]*CosIntegral[c +
d*x])/b, x] + Dist[d/b, Int[(Cos[a + b*x]*Cos[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 \text{Ci}(b x) \sin (b x) \, dx &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\int \frac{\cos ^2(b x)}{b x} \, dx\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\frac{\int \frac{\cos ^2(b x)}{x} \, dx}{b}\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\frac{\int \left (\frac{1}{2 x}+\frac{\cos (2 b x)}{2 x}\right ) \, dx}{b}\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\frac{\log (x)}{2 b}+\frac{\int \frac{\cos (2 b x)}{x} \, dx}{2 b}\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\frac{\text{Ci}(2 b x)}{2 b}+\frac{\log (x)}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0132629, size = 37, normalized size = 1.06 \[ \frac{\text{CosIntegral}(2 b x)}{2 b}-\frac{\text{CosIntegral}(b x) \cos (b x)}{b}+\frac{\log (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[CosIntegral[b*x]*Sin[b*x],x]

[Out]

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

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Maple [A]  time = 0.052, size = 34, normalized size = 1. \begin{align*} -{\frac{{\it Ci} \left ( bx \right ) \cos \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(Ci(b*x)*sin(b*x),x)

[Out]

-Ci(b*x)*cos(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 Ci}\left (b x\right ) \sin \left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*sin(b*x),x, algorithm="maxima")

[Out]

integrate(Ci(b*x)*sin(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*sin(b*x),x, algorithm="fricas")

[Out]

integral(cos_integral(b*x)*sin(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*sin(b*x),x)

[Out]

Integral(sin(b*x)*Ci(b*x), x)

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Giac [A]  time = 1.16776, size = 46, normalized size = 1.31 \begin{align*} -\frac{\cos \left (b x\right ) \operatorname{Ci}\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(Ci(b*x)*sin(b*x),x, algorithm="giac")

[Out]

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