\(\int \operatorname {CosIntegral}(b x)^2 \, dx\) [81]

Optimal result

Integrand size = 6, antiderivative size = 31 \[ \int \operatorname {CosIntegral}(b x)^2 \, dx=x \operatorname {CosIntegral}(b x)^2-\frac {2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b} \]

[Out]

x*Ci(b*x)^2+Si(2*b*x)/b-2*Ci(b*x)*sin(b*x)/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6641, 6647, 12, 4491, 3380} \[ \int \operatorname {CosIntegral}(b x)^2 \, dx=x \operatorname {CosIntegral}(b x)^2-\frac {2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b} \]

[In]

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

[Out]

x*CosIntegral[b*x]^2 - (2*CosIntegral[b*x]*Sin[b*x])/b + SinIntegral[2*b*x]/b

Rule 12

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \operatorname {CosIntegral}(b x)^2 \, dx=x \operatorname {CosIntegral}(b x)^2-\frac {2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b} \]

[In]

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

[Out]

x*CosIntegral[b*x]^2 - (2*CosIntegral[b*x]*Sin[b*x])/b + SinIntegral[2*b*x]/b

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x \right )^{2} b x -2 \,\operatorname {Ci}\left (b x \right ) \sin \left (b x \right )+\operatorname {Si}\left (2 b x \right )}{b}\)	\(30\)
default	\(\frac {\operatorname {Ci}\left (b x \right )^{2} b x -2 \,\operatorname {Ci}\left (b x \right ) \sin \left (b x \right )+\operatorname {Si}\left (2 b x \right )}{b}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \operatorname {CosIntegral}(b x)^2 \, dx=\frac {2 \, \pi b^{2} x \operatorname {C}\left (b x\right )^{2} - 4 \, b \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{2 \, \pi b^{2}} \]

[In]

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

[Out]

1/2*(2*pi*b^2*x*fresnel_cos(b*x)^2 - 4*b*fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2) + sqrt(2)*sqrt(b^2)*fresnel_sin(
sqrt(2)*sqrt(b^2)*x))/(pi*b^2)

Sympy [F]

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

[In]

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

[Out]

Integral(Ci(b*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

integrate(fresnel_cos(b*x)^2, x)

Giac [F]

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

[In]

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

[Out]

integrate(fresnel_cos(b*x)^2, x)

Mupad [F(-1)]

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

[In]

int(cosint(b*x)^2,x)

[Out]

int(cosint(b*x)^2, x)