\(\int x \operatorname {CosIntegral}(a+b x)^2 \, dx\) [95]

Optimal result

Integrand size = 10, antiderivative size = 155 \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=-\frac {\cos (2 a+2 b x)}{4 b^2}-\frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {a \text {Si}(2 a+2 b x)}{b^2} \]

[Out]

-1/2*a*(b*x+a)*Ci(b*x+a)^2/b^2+1/2*x*(b*x+a)*Ci(b*x+a)^2/b+1/2*Ci(2*b*x+2*a)/b^2-Ci(b*x+a)*cos(b*x+a)/b^2-1/4*
cos(2*b*x+2*a)/b^2+1/2*ln(b*x+a)/b^2-a*Si(2*b*x+2*a)/b^2+a*Ci(b*x+a)*sin(b*x+a)/b^2-x*Ci(b*x+a)*sin(b*x+a)/b

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {6645, 6649, 4669, 6873, 6874, 2718, 3380, 6653, 3393, 3383, 6641, 6647, 4491, 12} \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=-\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b^2}-\frac {a \text {Si}(2 a+2 b x)}{b^2}+\frac {\log (a+b x)}{2 b^2}-\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b} \]

[In]

Int[x*CosIntegral[a + b*x]^2,x]

[Out]

-1/4*Cos[2*a + 2*b*x]/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 12

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

Rule 2718

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

Rule 3380

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 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 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 4669

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 6641

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 6645

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 6647

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 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]

Rule 6873

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

Rule 6874

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

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.62 \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=-\frac {\cos (2 (a+b x))+2 \left (a^2-b^2 x^2\right ) \operatorname {CosIntegral}(a+b x)^2-2 \operatorname {CosIntegral}(2 (a+b x))-2 \log (a+b x)+4 \operatorname {CosIntegral}(a+b x) (\cos (a+b x)+(-a+b x) \sin (a+b x))+4 a \text {Si}(2 (a+b x))}{4 b^2} \]

[In]

Integrate[x*CosIntegral[a + b*x]^2,x]

[Out]

-1/4*(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)])/b^2

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x +a \right )^{2} \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-2 \,\operatorname {Ci}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\frac {\cos \left (b x +a \right )}{2}+\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}\right )-a \,\operatorname {Si}\left (2 b x +2 a \right )-\frac {\cos \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(113\)
default	\(\frac {\operatorname {Ci}\left (b x +a \right )^{2} \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-2 \,\operatorname {Ci}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\frac {\cos \left (b x +a \right )}{2}+\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}\right )-a \,\operatorname {Si}\left (2 b x +2 a \right )-\frac {\cos \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(113\)

[In]

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

[Out]

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

Fricas [F]

\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

integral(x*fresnel_cos(b*x + a)^2, x)

Sympy [F]

\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int x \operatorname {Ci}^{2}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral(x*Ci(a + b*x)**2, x)

Maxima [F]

\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(x*fresnel_cos(b*x + a)^2, x)

Giac [F]

\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(x*fresnel_cos(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int x\,{\mathrm {cosint}\left (a+b\,x\right )}^2 \,d x \]

[In]

int(x*cosint(a + b*x)^2,x)

[Out]

int(x*cosint(a + b*x)^2, x)