\(\int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx\) [118]

Optimal result

Integrand size = 10, antiderivative size = 60 \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}-\frac {\operatorname {CosIntegral}(2 b x)}{2 b^2}-\frac {\log (x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6649, 12, 2644, 30, 6653, 3393, 3383} \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=-\frac {\operatorname {CosIntegral}(2 b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{b^2}-\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b} \]

[In]

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

[Out]

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

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 3383

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]

Rule 3393

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 6649

Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e +
f*x)^m*Sin[a + b*x]*(CosIntegral[c + d*x]/b), x] + (-Dist[d/b, Int[(e + f*x)^m*Sin[a + b*x]*(Cos[c + d*x]/(c
+ d*x)), x], x] - Dist[f*(m/b), Int[(e + f*x)^(m - 1)*Sin[a + b*x]*CosIntegral[c + d*x], x], x]) /; FreeQ[{a,
b, c, d, e, f}, x] && IGtQ[m, 0]

Rule 6653

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]

Rubi steps \begin{align*} \text {integral}& = \frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \operatorname {CosIntegral}(b x) \sin (b x) \, dx}{b}-\int \frac {\cos (b x) \sin (b x)}{b} \, dx \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos ^2(b x)}{b x} \, dx}{b}-\frac {\int \cos (b x) \sin (b x) \, dx}{b} \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos ^2(b x)}{x} \, dx}{b^2}-\frac {\text {Subst}(\int x \, dx,x,\sin (b x))}{b^2} \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2}-\frac {\int \left (\frac {1}{2 x}+\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^2} \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}-\frac {\log (x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2}-\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b^2} \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}-\frac {\operatorname {CosIntegral}(2 b x)}{2 b^2}-\frac {\log (x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70 \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {\cos (2 b x)-2 \operatorname {CosIntegral}(2 b x)-2 \log (x)+4 \operatorname {CosIntegral}(b x) (\cos (b x)+b x \sin (b x))}{4 b^2} \]

[In]

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

[Out]

(Cos[2*b*x] - 2*CosIntegral[2*b*x] - 2*Log[x] + 4*CosIntegral[b*x]*(Cos[b*x] + b*x*Sin[b*x]))/(4*b^2)

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x \right ) \left (\cos \left (b x \right )+b x \sin \left (b x \right )\right )-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Ci}\left (2 b x \right )}{2}+\frac {\cos \left (b x \right )^{2}}{2}}{b^{2}}\)	\(44\)
default	\(\frac {\operatorname {Ci}\left (b x \right ) \left (\cos \left (b x \right )+b x \sin \left (b x \right )\right )-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Ci}\left (2 b x \right )}{2}+\frac {\cos \left (b x \right )^{2}}{2}}{b^{2}}\)	\(44\)

[In]

int(x*Ci(b*x)*cos(b*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (54) = 108\).

Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.68 \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {2 \, \pi b^{2} x \operatorname {C}\left (b x\right ) \sin \left (b x\right ) + 2 \, \pi b \cos \left (b x\right ) \operatorname {C}\left (b x\right ) - 2 \, b \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \sin \left (b x\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \sin \left (\frac {1}{2 \, \pi }\right ) - \cos \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \sin \left (\frac {1}{2 \, \pi }\right ) - \cos \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right )}{2 \, \pi b^{3}} \]

[In]

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

[Out]

1/2*(2*pi*b^2*x*fresnel_cos(b*x)*sin(b*x) + 2*pi*b*cos(b*x)*fresnel_cos(b*x) - 2*b*sin(1/2*pi*b^2*x^2)*sin(b*x
) - sqrt(b^2)*(pi*cos(1/2/pi) + sin(1/2/pi))*fresnel_cos((pi*b*x + 1)*sqrt(b^2)/(pi*b)) - sqrt(b^2)*(pi*cos(1/
2/pi) + sin(1/2/pi))*fresnel_cos((pi*b*x - 1)*sqrt(b^2)/(pi*b)) - sqrt(b^2)*(pi*sin(1/2/pi) - cos(1/2/pi))*fre
snel_sin((pi*b*x + 1)*sqrt(b^2)/(pi*b)) - sqrt(b^2)*(pi*sin(1/2/pi) - cos(1/2/pi))*fresnel_sin((pi*b*x - 1)*sq
rt(b^2)/(pi*b)))/(pi*b^3)

Sympy [F]

\[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int x \cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]

[In]

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

[Out]

Integral(x*cos(b*x)*Ci(b*x), x)

Maxima [F]

\[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int { x \cos \left (b x\right ) \operatorname {C}\left (b x\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int { x \cos \left (b x\right ) \operatorname {C}\left (b x\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int x\,\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \]

[In]

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

[Out]

int(x*cosint(b*x)*cos(b*x), x)