3.78 \(\int x^3 \text{CosIntegral}(b x)^2 \, dx\)

Optimal. Leaf size=163 \[ -\frac{3 x^2 \text{CosIntegral}(b x) \cos (b x)}{2 b^2}-\frac{3 \text{CosIntegral}(2 b x)}{2 b^4}+\frac{3 x \text{CosIntegral}(b x) \sin (b x)}{b^3}+\frac{3 \text{CosIntegral}(b x) \cos (b x)}{b^4}+\frac{x^2}{4 b^2}+\frac{x^2 \sin ^2(b x)}{4 b^2}-\frac{3 \log (x)}{2 b^4}-\frac{13 \sin ^2(b x)}{8 b^4}+\frac{3 \cos ^2(b x)}{8 b^4}+\frac{x \sin (b x) \cos (b x)}{b^3}+\frac{1}{4} x^4 \text{CosIntegral}(b x)^2-\frac{x^3 \text{CosIntegral}(b x) \sin (b x)}{2 b} \]

[Out]

x^2/(4*b^2) + (3*Cos[b*x]^2)/(8*b^4) + (3*Cos[b*x]*CosIntegral[b*x])/b^4 - (3*x^2*Cos[b*x]*CosIntegral[b*x])/(
2*b^2) + (x^4*CosIntegral[b*x]^2)/4 - (3*CosIntegral[2*b*x])/(2*b^4) - (3*Log[x])/(2*b^4) + (x*Cos[b*x]*Sin[b*
x])/b^3 + (3*x*CosIntegral[b*x]*Sin[b*x])/b^3 - (x^3*CosIntegral[b*x]*Sin[b*x])/(2*b) - (13*Sin[b*x]^2)/(8*b^4
) + (x^2*Sin[b*x]^2)/(4*b^2)

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Rubi [A]  time = 0.228525, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {6508, 6514, 12, 3443, 3310, 30, 6520, 2564, 6518, 3312, 3302} \[ -\frac{3 x^2 \text{CosIntegral}(b x) \cos (b x)}{2 b^2}-\frac{3 \text{CosIntegral}(2 b x)}{2 b^4}+\frac{3 x \text{CosIntegral}(b x) \sin (b x)}{b^3}+\frac{3 \text{CosIntegral}(b x) \cos (b x)}{b^4}+\frac{x^2}{4 b^2}+\frac{x^2 \sin ^2(b x)}{4 b^2}-\frac{3 \log (x)}{2 b^4}-\frac{13 \sin ^2(b x)}{8 b^4}+\frac{3 \cos ^2(b x)}{8 b^4}+\frac{x \sin (b x) \cos (b x)}{b^3}+\frac{1}{4} x^4 \text{CosIntegral}(b x)^2-\frac{x^3 \text{CosIntegral}(b x) \sin (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*CosIntegral[b*x]^2,x]

[Out]

x^2/(4*b^2) + (3*Cos[b*x]^2)/(8*b^4) + (3*Cos[b*x]*CosIntegral[b*x])/b^4 - (3*x^2*Cos[b*x]*CosIntegral[b*x])/(
2*b^2) + (x^4*CosIntegral[b*x]^2)/4 - (3*CosIntegral[2*b*x])/(2*b^4) - (3*Log[x])/(2*b^4) + (x*Cos[b*x]*Sin[b*
x])/b^3 + (3*x*CosIntegral[b*x]*Sin[b*x])/b^3 - (x^3*CosIntegral[b*x]*Sin[b*x])/(2*b) - (13*Sin[b*x]^2)/(8*b^4
) + (x^2*Sin[b*x]^2)/(4*b^2)

Rule 6508

Int[CosIntegral[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*CosIntegral[b*x]^2)/(m + 1), x] - Dist[
2/(m + 1), Int[x^m*Cos[b*x]*CosIntegral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]

Rule 6514

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 12

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

Rule 3443

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m - n
+ 1)*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 3310

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

Rule 30

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

Rule 6520

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

Rule 2564

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 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 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 x^3 \text{Ci}(b x)^2 \, dx &=\frac{1}{4} x^4 \text{Ci}(b x)^2-\frac{1}{2} \int x^3 \cos (b x) \text{Ci}(b x) \, dx\\ &=\frac{1}{4} x^4 \text{Ci}(b x)^2-\frac{x^3 \text{Ci}(b x) \sin (b x)}{2 b}+\frac{1}{2} \int \frac{x^2 \cos (b x) \sin (b x)}{b} \, dx+\frac{3 \int x^2 \text{Ci}(b x) \sin (b x) \, dx}{2 b}\\ &=-\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)^2-\frac{x^3 \text{Ci}(b x) \sin (b x)}{2 b}+\frac{3 \int x \cos (b x) \text{Ci}(b x) \, dx}{b^2}+\frac{\int x^2 \cos (b x) \sin (b x) \, dx}{2 b}+\frac{3 \int \frac{x \cos ^2(b x)}{b} \, dx}{2 b}\\ &=-\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)^2+\frac{3 x \text{Ci}(b x) \sin (b x)}{b^3}-\frac{x^3 \text{Ci}(b x) \sin (b x)}{2 b}+\frac{x^2 \sin ^2(b x)}{4 b^2}-\frac{3 \int \text{Ci}(b x) \sin (b x) \, dx}{b^3}-\frac{\int x \sin ^2(b x) \, dx}{2 b^2}+\frac{3 \int x \cos ^2(b x) \, dx}{2 b^2}-\frac{3 \int \frac{\cos (b x) \sin (b x)}{b} \, dx}{b^2}\\ &=\frac{3 \cos ^2(b x)}{8 b^4}+\frac{3 \cos (b x) \text{Ci}(b x)}{b^4}-\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)^2+\frac{x \cos (b x) \sin (b x)}{b^3}+\frac{3 x \text{Ci}(b x) \sin (b x)}{b^3}-\frac{x^3 \text{Ci}(b x) \sin (b x)}{2 b}-\frac{\sin ^2(b x)}{8 b^4}+\frac{x^2 \sin ^2(b x)}{4 b^2}-\frac{3 \int \frac{\cos ^2(b x)}{b x} \, dx}{b^3}-\frac{3 \int \cos (b x) \sin (b x) \, dx}{b^3}-\frac{\int x \, dx}{4 b^2}+\frac{3 \int x \, dx}{4 b^2}\\ &=\frac{x^2}{4 b^2}+\frac{3 \cos ^2(b x)}{8 b^4}+\frac{3 \cos (b x) \text{Ci}(b x)}{b^4}-\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)^2+\frac{x \cos (b x) \sin (b x)}{b^3}+\frac{3 x \text{Ci}(b x) \sin (b x)}{b^3}-\frac{x^3 \text{Ci}(b x) \sin (b x)}{2 b}-\frac{\sin ^2(b x)}{8 b^4}+\frac{x^2 \sin ^2(b x)}{4 b^2}-\frac{3 \int \frac{\cos ^2(b x)}{x} \, dx}{b^4}-\frac{3 \operatorname{Subst}(\int x \, dx,x,\sin (b x))}{b^4}\\ &=\frac{x^2}{4 b^2}+\frac{3 \cos ^2(b x)}{8 b^4}+\frac{3 \cos (b x) \text{Ci}(b x)}{b^4}-\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)^2+\frac{x \cos (b x) \sin (b x)}{b^3}+\frac{3 x \text{Ci}(b x) \sin (b x)}{b^3}-\frac{x^3 \text{Ci}(b x) \sin (b x)}{2 b}-\frac{13 \sin ^2(b x)}{8 b^4}+\frac{x^2 \sin ^2(b x)}{4 b^2}-\frac{3 \int \left (\frac{1}{2 x}+\frac{\cos (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=\frac{x^2}{4 b^2}+\frac{3 \cos ^2(b x)}{8 b^4}+\frac{3 \cos (b x) \text{Ci}(b x)}{b^4}-\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)^2-\frac{3 \log (x)}{2 b^4}+\frac{x \cos (b x) \sin (b x)}{b^3}+\frac{3 x \text{Ci}(b x) \sin (b x)}{b^3}-\frac{x^3 \text{Ci}(b x) \sin (b x)}{2 b}-\frac{13 \sin ^2(b x)}{8 b^4}+\frac{x^2 \sin ^2(b x)}{4 b^2}-\frac{3 \int \frac{\cos (2 b x)}{x} \, dx}{2 b^4}\\ &=\frac{x^2}{4 b^2}+\frac{3 \cos ^2(b x)}{8 b^4}+\frac{3 \cos (b x) \text{Ci}(b x)}{b^4}-\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)^2-\frac{3 \text{Ci}(2 b x)}{2 b^4}-\frac{3 \log (x)}{2 b^4}+\frac{x \cos (b x) \sin (b x)}{b^3}+\frac{3 x \text{Ci}(b x) \sin (b x)}{b^3}-\frac{x^3 \text{Ci}(b x) \sin (b x)}{2 b}-\frac{13 \sin ^2(b x)}{8 b^4}+\frac{x^2 \sin ^2(b x)}{4 b^2}\\ \end{align*}

Mathematica [A]  time = 0.115799, size = 108, normalized size = 0.66 \[ \frac{2 b^4 x^4 \text{CosIntegral}(b x)^2-4 \text{CosIntegral}(b x) \left (b x \left (b^2 x^2-6\right ) \sin (b x)+3 \left (b^2 x^2-2\right ) \cos (b x)\right )+3 b^2 x^2-b^2 x^2 \cos (2 b x)-12 \text{CosIntegral}(2 b x)+4 b x \sin (2 b x)+8 \cos (2 b x)-12 \log (x)}{8 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*CosIntegral[b*x]^2,x]

[Out]

(3*b^2*x^2 + 8*Cos[2*b*x] - b^2*x^2*Cos[2*b*x] + 2*b^4*x^4*CosIntegral[b*x]^2 - 12*CosIntegral[2*b*x] - 12*Log
[x] - 4*CosIntegral[b*x]*(3*(-2 + b^2*x^2)*Cos[b*x] + b*x*(-6 + b^2*x^2)*Sin[b*x]) + 4*b*x*Sin[2*b*x])/(8*b^4)

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Maple [A]  time = 0.069, size = 148, normalized size = 0.9 \begin{align*}{\frac{{x}^{4} \left ({\it Ci} \left ( bx \right ) \right ) ^{2}}{4}}-{\frac{{x}^{3}{\it Ci} \left ( bx \right ) \sin \left ( bx \right ) }{2\,b}}-{\frac{3\,{x}^{2}{\it Ci} \left ( bx \right ) \cos \left ( bx \right ) }{2\,{b}^{2}}}+3\,{\frac{{\it Ci} \left ( bx \right ) \cos \left ( bx \right ) }{{b}^{4}}}+3\,{\frac{x{\it Ci} \left ( bx \right ) \sin \left ( bx \right ) }{{b}^{3}}}-{\frac{{x}^{2} \left ( \cos \left ( bx \right ) \right ) ^{2}}{4\,{b}^{2}}}+{\frac{x\cos \left ( bx \right ) \sin \left ( bx \right ) }{{b}^{3}}}+{\frac{{x}^{2}}{2\,{b}^{2}}}-{\frac{ \left ( \sin \left ( bx \right ) \right ) ^{2}}{2\,{b}^{4}}}-{\frac{3\,\ln \left ( bx \right ) }{2\,{b}^{4}}}-{\frac{3\,{\it Ci} \left ( 2\,bx \right ) }{2\,{b}^{4}}}+{\frac{3\, \left ( \cos \left ( bx \right ) \right ) ^{2}}{2\,{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*Ci(b*x)^2,x)

[Out]

1/4*x^4*Ci(b*x)^2-1/2*x^3*Ci(b*x)*sin(b*x)/b-3/2*x^2*Ci(b*x)*cos(b*x)/b^2+3*Ci(b*x)*cos(b*x)/b^4+3*x*Ci(b*x)*s
in(b*x)/b^3-1/4/b^2*x^2*cos(b*x)^2+x*cos(b*x)*sin(b*x)/b^3+1/2*x^2/b^2-1/2*sin(b*x)^2/b^4-3/2/b^4*ln(b*x)-3/2*
Ci(2*b*x)/b^4+3/2*cos(b*x)^2/b^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm Ci}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*Ci(b*x)^2,x, algorithm="maxima")

[Out]

integrate(x^3*Ci(b*x)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \operatorname{Ci}\left (b x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*Ci(b*x)^2,x, algorithm="fricas")

[Out]

integral(x^3*cos_integral(b*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{Ci}^{2}{\left (b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*Ci(b*x)**2,x)

[Out]

Integral(x**3*Ci(b*x)**2, x)

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Giac [A]  time = 1.18327, size = 119, normalized size = 0.73 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{Ci}\left (b x\right )^{2} - \frac{1}{2} \,{\left (\frac{3 \,{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{4}} + \frac{{\left (b^{3} x^{3} - 6 \, b x\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname{Ci}\left (b x\right ) + \frac{3 \,{\left (b^{2} x^{2} - 2 \, \operatorname{Ci}\left (2 \, b x\right ) - 2 \, \operatorname{Ci}\left (-2 \, b x\right ) - 4 \, \log \left (x\right )\right )}}{8 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*Ci(b*x)^2,x, algorithm="giac")

[Out]

1/4*x^4*cos_integral(b*x)^2 - 1/2*(3*(b^2*x^2 - 2)*cos(b*x)/b^4 + (b^3*x^3 - 6*b*x)*sin(b*x)/b^4)*cos_integral
(b*x) + 3/8*(b^2*x^2 - 2*cos_integral(2*b*x) - 2*cos_integral(-2*b*x) - 4*log(x))/b^4