\(\int x^2 \operatorname {CosIntegral}(b x) \, dx\) [71]

Optimal result

Integrand size = 8, antiderivative size = 49 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)+\frac {2 \sin (b x)}{3 b^3}-\frac {x^2 \sin (b x)}{3 b} \]

[Out]

1/3*x^3*Ci(b*x)-2/3*x*cos(b*x)/b^2+2/3*sin(b*x)/b^3-1/3*x^2*sin(b*x)/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6639, 12, 3377, 2717} \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\frac {2 \sin (b x)}{3 b^3}-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {x^2 \sin (b x)}{3 b} \]

[In]

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

[Out]

(-2*x*Cos[b*x])/(3*b^2) + (x^3*CosIntegral[b*x])/3 + (2*Sin[b*x])/(3*b^3) - (x^2*Sin[b*x])/(3*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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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

Rule 6639

Int[CosIntegral[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(CosIntegr
al[a + b*x]/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(Cos[a + b*x]/(a + b*x)), x], x] /; F
reeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {1}{3} b \int \frac {x^2 \cos (b x)}{b} \, dx \\ & = \frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {1}{3} \int x^2 \cos (b x) \, dx \\ & = \frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {x^2 \sin (b x)}{3 b}+\frac {2 \int x \sin (b x) \, dx}{3 b} \\ & = -\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {x^2 \sin (b x)}{3 b}+\frac {2 \int \cos (b x) \, dx}{3 b^2} \\ & = -\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)+\frac {2 \sin (b x)}{3 b^3}-\frac {x^2 \sin (b x)}{3 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {\left (-2+b^2 x^2\right ) \sin (b x)}{3 b^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86

method	result	size
parts	\(\frac {x^{3} \operatorname {Ci}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sin \left (b x \right )-2 \sin \left (b x \right )+2 b x \cos \left (b x \right )}{3 b^{3}}\)	\(42\)
derivativedivides	\(\frac {\frac {b^{3} x^{3} \operatorname {Ci}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sin \left (b x \right )}{3}+\frac {2 \sin \left (b x \right )}{3}-\frac {2 b x \cos \left (b x \right )}{3}}{b^{3}}\)	\(44\)
default	\(\frac {\frac {b^{3} x^{3} \operatorname {Ci}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sin \left (b x \right )}{3}+\frac {2 \sin \left (b x \right )}{3}-\frac {2 b x \cos \left (b x \right )}{3}}{b^{3}}\)	\(44\)
meijerg	\(\frac {2 \sqrt {\pi }\, \left (-\frac {b^{5} x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{2}\right ], \left [\frac {3}{2}, 2, 2, \frac {7}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{40 \sqrt {\pi }}+\frac {\left (-\frac {2}{3}+2 \gamma +2 \ln \left (x \right )+2 \ln \left (b \right )\right ) x^{3} b^{3}}{12 \sqrt {\pi }}\right )}{b^{3}}\)	\(63\)

[In]

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

[Out]

1/3*x^3*Ci(b*x)-1/3/b^3*(b^2*x^2*sin(b*x)-2*sin(b*x)+2*b*x*cos(b*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\frac {\pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) - \pi b^{2} x^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{3 \, \pi ^{2} b^{3}} \]

[In]

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

[Out]

1/3*(pi^2*b^3*x^3*fresnel_cos(b*x) - pi*b^2*x^2*sin(1/2*pi*b^2*x^2) - 2*cos(1/2*pi*b^2*x^2))/(pi^2*b^3)

Sympy [A] (verification not implemented)

Time = 1.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=- \frac {x^{3} \log {\left (b x \right )}}{3} + \frac {x^{3} \log {\left (b^{2} x^{2} \right )}}{6} + \frac {x^{3} \operatorname {Ci}{\left (b x \right )}}{3} - \frac {x^{2} \sin {\left (b x \right )}}{3 b} - \frac {2 x \cos {\left (b x \right )}}{3 b^{2}} + \frac {2 \sin {\left (b x \right )}}{3 b^{3}} \]

[In]

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

[Out]

-x**3*log(b*x)/3 + x**3*log(b**2*x**2)/6 + x**3*Ci(b*x)/3 - x**2*sin(b*x)/(3*b) - 2*x*cos(b*x)/(3*b**2) + 2*si
n(b*x)/(3*b**3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {C}\left (b x\right ) - \frac {\pi b^{2} x^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{3 \, \pi ^{2} b^{3}} \]

[In]

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

[Out]

1/3*x^3*fresnel_cos(b*x) - 1/3*(pi*b^2*x^2*sin(1/2*pi*b^2*x^2) + 2*cos(1/2*pi*b^2*x^2))/(pi^2*b^3)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*fresnel_cos(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\frac {x^3\,\mathrm {cosint}\left (b\,x\right )}{3}-\frac {b^2\,x^2\,\sin \left (b\,x\right )-2\,\sin \left (b\,x\right )+2\,b\,x\,\cos \left (b\,x\right )}{3\,b^3} \]

[In]

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

[Out]

(x^3*cosint(b*x))/3 - (b^2*x^2*sin(b*x) - 2*sin(b*x) + 2*b*x*cos(b*x))/(3*b^3)