3.114 \(\int \frac{\cos (b x) \text{CosIntegral}(b x)}{x^3} \, dx\)

Optimal. Leaf size=97 \[ -\frac{1}{4} b^2 \text{CosIntegral}(b x)^2-b^2 \text{CosIntegral}(2 b x)-\frac{\text{CosIntegral}(b x) \cos (b x)}{2 x^2}+\frac{b \text{CosIntegral}(b x) \sin (b x)}{2 x}-\frac{\cos ^2(b x)}{4 x^2}+\frac{b \sin (2 b x)}{4 x}+\frac{b \sin (b x) \cos (b x)}{2 x} \]

[Out]

-Cos[b*x]^2/(4*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)

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Rubi [A]  time = 0.198593, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6516, 6522, 6686, 12, 4406, 3297, 3302, 3314, 29, 3312} \[ -\frac{1}{4} b^2 \text{CosIntegral}(b x)^2-b^2 \text{CosIntegral}(2 b x)-\frac{\text{CosIntegral}(b x) \cos (b x)}{2 x^2}+\frac{b \text{CosIntegral}(b x) \sin (b x)}{2 x}-\frac{\cos ^2(b x)}{4 x^2}+\frac{b \sin (2 b x)}{4 x}+\frac{b \sin (b x) \cos (b x)}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[b*x]^2/(4*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 6516

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 6522

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 6686

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]

Rule 12

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

Rule 4406

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 3297

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 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]

Rule 3314

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 29

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

Rubi steps

\begin{align*} \int \frac{\cos (b x) \text{Ci}(b x)}{x^3} \, dx &=-\frac{\cos (b x) \text{Ci}(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{\text{Ci}(b x) \sin (b x)}{x^2} \, dx\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{2 x^2}+\frac{b \text{Ci}(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) \text{Ci}(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) \text{Ci}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Ci}(b x)^2+\frac{b \cos (b x) \sin (b x)}{2 x}+\frac{b \text{Ci}(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) \text{Ci}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Ci}(b x)^2+\frac{1}{2} b^2 \log (x)+\frac{b \cos (b x) \sin (b x)}{2 x}+\frac{b \text{Ci}(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) \text{Ci}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Ci}(b x)^2+\frac{b \cos (b x) \sin (b x)}{2 x}+\frac{b \text{Ci}(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) \text{Ci}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Ci}(b x)^2-\frac{1}{2} b^2 \text{Ci}(2 b x)+\frac{b \cos (b x) \sin (b x)}{2 x}+\frac{b \text{Ci}(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) \text{Ci}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Ci}(b x)^2-b^2 \text{Ci}(2 b x)+\frac{b \cos (b x) \sin (b x)}{2 x}+\frac{b \text{Ci}(b x) \sin (b x)}{2 x}+\frac{b \sin (2 b x)}{4 x}\\ \end{align*}

Mathematica [A]  time = 0.0138336, size = 97, normalized size = 1. \[ -\frac{1}{4} b^2 \text{CosIntegral}(b x)^2-b^2 \text{CosIntegral}(2 b x)-\frac{\text{CosIntegral}(b x) \cos (b x)}{2 x^2}+\frac{b \text{CosIntegral}(b x) \sin (b x)}{2 x}-\frac{\cos ^2(b x)}{4 x^2}+\frac{b \sin (2 b x)}{4 x}+\frac{b \sin (b x) \cos (b x)}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[b*x]^2/(4*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)

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Ci} \left ( bx \right ) \cos \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right ) \cos \left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)*cos(b*x)/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (b x\right ) \operatorname{Ci}\left (b x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (b x \right )} \operatorname{Ci}{\left (b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(b*x)*Ci(b*x)/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right ) \cos \left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)*cos(b*x)/x^3, x)