3.95 \(\int x \text{CosIntegral}(a+b x)^2 \, dx\)

Optimal. Leaf size=155 \[ -\frac{a (a+b x) \text{CosIntegral}(a+b x)^2}{2 b^2}+\frac{\text{CosIntegral}(2 a+2 b x)}{2 b^2}+\frac{a \text{CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac{\text{CosIntegral}(a+b x) \cos (a+b x)}{b^2}-\frac{a \text{Si}(2 a+2 b x)}{b^2}+\frac{\log (a+b x)}{2 b^2}-\frac{\cos (2 a+2 b x)}{4 b^2}+\frac{x (a+b x) \text{CosIntegral}(a+b x)^2}{2 b}-\frac{x \text{CosIntegral}(a+b x) \sin (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.337556, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.4, Rules used = {6510, 6514, 4573, 6741, 6742, 2638, 3299, 6518, 3312, 3302, 6506, 6512, 4406, 12} \[ -\frac{a (a+b x) \text{CosIntegral}(a+b x)^2}{2 b^2}+\frac{\text{CosIntegral}(2 a+2 b x)}{2 b^2}+\frac{a \text{CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac{\text{CosIntegral}(a+b x) \cos (a+b x)}{b^2}-\frac{a \text{Si}(2 a+2 b x)}{b^2}+\frac{\log (a+b x)}{2 b^2}-\frac{\cos (2 a+2 b x)}{4 b^2}+\frac{x (a+b x) \text{CosIntegral}(a+b x)^2}{2 b}-\frac{x \text{CosIntegral}(a+b x) \sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[x*CosIntegral[a + b*x]^2,x]

[Out]

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

Rule 6510

Int[CosIntegral[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)*(c + d*x)^m*CosI
ntegral[a + b*x]^2)/(b*(m + 1)), x] + (-Dist[2/(m + 1), Int[(c + d*x)^m*Cos[a + b*x]*CosIntegral[a + b*x], x],
x] + Dist[((b*c - a*d)*m)/(b*(m + 1)), Int[(c + d*x)^(m - 1)*CosIntegral[a + b*x]^2, x], x]) /; FreeQ[{a, b,
c, d}, 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 4573

Int[Cos[w_]^(p_.)*(u_.)*Sin[v_]^(p_.), x_Symbol] :> Dist[1/2^p, Int[u*Sin[2*v]^p, x], x] /; EqQ[w, v] && Integ
erQ[p]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

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]

Rule 6506

Int[CosIntegral[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*CosIntegral[a + b*x]^2)/b, x] - Dist[2, In
t[Cos[a + b*x]*CosIntegral[a + b*x], x], x] /; FreeQ[{a, b}, x]

Rule 6512

Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(Sin[a + b*x]*CosIntegral[c + d
*x])/b, x] - Dist[d/b, Int[(Sin[a + b*x]*Cos[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, 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 12

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

Rubi steps

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

Mathematica [A]  time = 0.327434, size = 96, normalized size = 0.62 \[ -\frac{2 \left (a^2-b^2 x^2\right ) \text{CosIntegral}(a+b x)^2-2 \text{CosIntegral}(2 (a+b x))+4 \text{CosIntegral}(a+b x) ((b x-a) \sin (a+b x)+\cos (a+b x))+4 a \text{Si}(2 (a+b x))-2 \log (a+b x)+\cos (2 (a+b x))}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*CosIntegral[a + b*x]^2,x]

[Out]

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

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Maple [A]  time = 0.061, size = 136, normalized size = 0.9 \begin{align*}{\frac{{x}^{2} \left ({\it Ci} \left ( bx+a \right ) \right ) ^{2}}{2}}-{\frac{ \left ({\it Ci} \left ( bx+a \right ) \right ) ^{2}{a}^{2}}{2\,{b}^{2}}}-{\frac{x{\it Ci} \left ( bx+a \right ) \sin \left ( bx+a \right ) }{b}}+{\frac{{\it Ci} \left ( bx+a \right ) a\sin \left ( bx+a \right ) }{{b}^{2}}}-{\frac{{\it Ci} \left ( bx+a \right ) \cos \left ( bx+a \right ) }{{b}^{2}}}-{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{2\,{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) }{2\,{b}^{2}}}+{\frac{{\it Ci} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{2}}}-{\frac{a{\it Si} \left ( 2\,bx+2\,a \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*Ci(b*x+a)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Ci(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*cos_integral(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(b*x+a)**2,x)

[Out]

Integral(x*Ci(a + b*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Ci(b*x + a)^2, x)