\(\int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx\) [114]

Optimal result

Integrand size = 12, antiderivative size = 97 \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-b^2 \operatorname {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \]

[Out]

-1/4*b^2*Ci(b*x)^2-b^2*Ci(2*b*x)-1/2*Ci(b*x)*cos(b*x)/x^2-1/4*cos(b*x)^2/x^2+1/2*b*Ci(b*x)*sin(b*x)/x+1/2*b*co
s(b*x)*sin(b*x)/x+1/4*b*sin(2*b*x)/x

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6651, 6657, 6818, 12, 4491, 3378, 3383, 3395, 29, 3393} \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-b^2 \operatorname {CosIntegral}(2 b x)-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}-\frac {\cos ^2(b x)}{4 x^2}+\frac {b \sin (2 b x)}{4 x}+\frac {b \sin (b x) \cos (b x)}{2 x} \]

[In]

Int[(Cos[b*x]*CosIntegral[b*x])/x^3,x]

[Out]

-1/4*Cos[b*x]^2/x^2 - (Cos[b*x]*CosIntegral[b*x])/(2*x^2) - (b^2*CosIntegral[b*x]^2)/4 - b^2*CosIntegral[2*b*x
] + (b*Cos[b*x]*Sin[b*x])/(2*x) + (b*CosIntegral[b*x]*Sin[b*x])/(2*x) + (b*Sin[2*b*x])/(4*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 3378

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

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 3395

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((b*Si
n[e + f*x])^n/(d*(m + 1))), x] + (Dist[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[f^2*(n^2/(d^2*(m + 1)*(m + 2))), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1)*(m + 2))), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 6651

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

Rule 6657

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

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cos ^2(b x)}{b x^3} \, dx-\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx \\ & = -\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {1}{2} \int \frac {\cos ^2(b x)}{x^3} \, dx-\frac {1}{2} b^2 \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x} \, dx-\frac {1}{2} b^2 \int \frac {\cos (b x) \sin (b x)}{b x^2} \, dx \\ & = -\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}-\frac {1}{2} b \int \frac {\cos (b x) \sin (b x)}{x^2} \, dx+\frac {1}{2} b^2 \int \frac {1}{x} \, dx-b^2 \int \frac {\cos ^2(b x)}{x} \, dx \\ & = -\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2+\frac {1}{2} b^2 \log (x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}-\frac {1}{2} b \int \frac {\sin (2 b x)}{2 x^2} \, dx-b^2 \int \left (\frac {1}{2 x}+\frac {\cos (2 b x)}{2 x}\right ) \, dx \\ & = -\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}-\frac {1}{4} b \int \frac {\sin (2 b x)}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\cos (2 b x)}{x} \, dx \\ & = -\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-\frac {1}{2} b^2 \operatorname {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x}-\frac {1}{2} b^2 \int \frac {\cos (2 b x)}{x} \, dx \\ & = -\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-b^2 \operatorname {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-b^2 \operatorname {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \]

[In]

Integrate[(Cos[b*x]*CosIntegral[b*x])/x^3,x]

[Out]

-1/4*Cos[b*x]^2/x^2 - (Cos[b*x]*CosIntegral[b*x])/(2*x^2) - (b^2*CosIntegral[b*x]^2)/4 - b^2*CosIntegral[2*b*x
] + (b*Cos[b*x]*Sin[b*x])/(2*x) + (b*CosIntegral[b*x]*Sin[b*x])/(2*x) + (b*Sin[2*b*x])/(4*x)

Maple [F]

\[\int \frac {\operatorname {Ci}\left (b x \right ) \cos \left (b x \right )}{x^{3}}d x\]

[In]

int(Ci(b*x)*cos(b*x)/x^3,x)

[Out]

int(Ci(b*x)*cos(b*x)/x^3,x)

Fricas [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*cos(b*x)/x^3,x, algorithm="fricas")

[Out]

integral(cos(b*x)*fresnel_cos(b*x)/x^3, x)

Sympy [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int \frac {\cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \]

[In]

integrate(Ci(b*x)*cos(b*x)/x**3,x)

[Out]

Integral(cos(b*x)*Ci(b*x)/x**3, x)

Maxima [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*cos(b*x)/x^3,x, algorithm="maxima")

[Out]

integrate(cos(b*x)*fresnel_cos(b*x)/x^3, x)

Giac [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*cos(b*x)/x^3,x, algorithm="giac")

[Out]

integrate(cos(b*x)*fresnel_cos(b*x)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right )}{x^3} \,d x \]

[In]

int((cosint(b*x)*cos(b*x))/x^3,x)

[Out]

int((cosint(b*x)*cos(b*x))/x^3, x)