\(\int x^2 \operatorname {CosIntegral}(a+b x) \, dx\) [87]

Optimal result

Integrand size = 10, antiderivative size = 118 \[ \int x^2 \operatorname {CosIntegral}(a+b x) \, dx=\frac {a \cos (a+b x)}{3 b^3}-\frac {2 x \cos (a+b x)}{3 b^2}+\frac {a^3 \operatorname {CosIntegral}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \operatorname {CosIntegral}(a+b x)+\frac {2 \sin (a+b x)}{3 b^3}-\frac {a^2 \sin (a+b x)}{3 b^3}+\frac {a x \sin (a+b x)}{3 b^2}-\frac {x^2 \sin (a+b x)}{3 b} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.54 \[ \int x^2 \operatorname {CosIntegral}(a+b x) \, dx=\frac {(a-2 b x) \cos (a+b x)+\left (a^3+b^3 x^3\right ) \operatorname {CosIntegral}(a+b x)-\left (-2+a^2-a b x+b^2 x^2\right ) \sin (a+b x)}{3 b^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7058, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7058

\(\displaystyle \frac {1}{3} x^3 \operatorname {CosIntegral}(a+b x)-\frac {1}{3} b \int \frac {x^3 \cos (a+b x)}{a+b x}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{3} x^3 \operatorname {CosIntegral}(a+b x)-\frac {1}{3} b \int \left (-\frac {\cos (a+b x) a^3}{b^3 (a+b x)}+\frac {\cos (a+b x) a^2}{b^3}-\frac {x \cos (a+b x) a}{b^2}+\frac {x^2 \cos (a+b x)}{b}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \operatorname {CosIntegral}(a+b x)-\frac {1}{3} b \left (-\frac {a^3 \operatorname {CosIntegral}(a+b x)}{b^4}+\frac {a^2 \sin (a+b x)}{b^4}-\frac {2 \sin (a+b x)}{b^4}-\frac {a \cos (a+b x)}{b^4}-\frac {a x \sin (a+b x)}{b^3}+\frac {2 x \cos (a+b x)}{b^3}+\frac {x^2 \sin (a+b x)}{b^2}\right )\)

Input:

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

Output:

(x^3*CosIntegral[a + b*x])/3 - (b*(-((a*Cos[a + b*x])/b^4) + (2*x*Cos[a + 
b*x])/b^3 - (a^3*CosIntegral[a + b*x])/b^4 - (2*Sin[a + b*x])/b^4 + (a^2*S 
in[a + b*x])/b^4 - (a*x*Sin[a + b*x])/b^3 + (x^2*Sin[a + b*x])/b^2))/3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7058

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

rule 7293

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

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83

method	result	size
parts	\(\frac {x^{3} \operatorname {Ci}\left (b x +a \right )}{3}-\frac {-a^{3} \operatorname {Ci}\left (b x +a \right )+3 a^{2} \sin \left (b x +a \right )-3 a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )}{3 b^{3}}\)	\(98\)
derivativedivides	\(\frac {\frac {\operatorname {Ci}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {Ci}\left (b x +a \right )}{3}-a^{2} \sin \left (b x +a \right )+a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{2} \sin \left (b x +a \right )}{3}+\frac {2 \sin \left (b x +a \right )}{3}-\frac {2 \left (b x +a \right ) \cos \left (b x +a \right )}{3}}{b^{3}}\)	\(99\)
default	\(\frac {\frac {\operatorname {Ci}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {Ci}\left (b x +a \right )}{3}-a^{2} \sin \left (b x +a \right )+a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{2} \sin \left (b x +a \right )}{3}+\frac {2 \sin \left (b x +a \right )}{3}-\frac {2 \left (b x +a \right ) \cos \left (b x +a \right )}{3}}{b^{3}}\)	\(99\)
orering	\(\frac {\left (b^{5} x^{5}+a^{3} b^{2} x^{2}+8 b^{3} x^{3}+4 a \,b^{2} x^{2}-4 a^{2} b x +6 a^{3}-8 b x -12 a \right ) \operatorname {Ci}\left (b x +a \right )}{3 b^{5} x^{2}}-\frac {\left (5 b^{3} x^{3}+2 a \,b^{2} x^{2}-2 a^{2} b x +4 a^{3}-6 b x -8 a \right ) \left (2 x \,\operatorname {Ci}\left (b x +a \right )+\frac {x^{2} \cos \left (b x +a \right ) b}{b x +a}\right )}{3 b^{5} x^{3}}+\frac {\left (b^{2} x^{2}-b x a +a^{2}-2\right ) \left (b x +a \right ) \left (2 \,\operatorname {Ci}\left (b x +a \right )+\frac {4 x \cos \left (b x +a \right ) b}{b x +a}-\frac {x^{2} b^{2} \sin \left (b x +a \right )}{b x +a}-\frac {x^{2} \cos \left (b x +a \right ) b^{2}}{\left (b x +a \right )^{2}}\right )}{3 b^{5} x^{2}}\)	\(241\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.25 \[ \int x^2 \operatorname {CosIntegral}(a+b x) \, dx=\frac {\pi ^{2} b^{4} x^{3} \operatorname {C}\left (b x + a\right ) + \pi ^{2} a^{3} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 3 \, \pi a \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) - {\left (\pi b^{3} x^{2} - \pi a b^{2} x + \pi a^{2} b\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{3 \, \pi ^{2} b^{4}} \] Input:

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

Output:

1/3*(pi^2*b^4*x^3*fresnel_cos(b*x + a) + pi^2*a^3*sqrt(b^2)*fresnel_cos(sq 
rt(b^2)*(b*x + a)/b) - 3*pi*a*sqrt(b^2)*fresnel_sin(sqrt(b^2)*(b*x + a)/b) 
 - 2*b*cos(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2) - (pi*b^3*x^2 - pi*a*b^ 
2*x + pi*a^2*b)*sin(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2))/(pi^2*b^4)
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.97 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.58 \[ \int x^2 \operatorname {CosIntegral}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {C}\left (b x + a\right ) - \frac {{\left (12 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{3} + 4 \, {\left (3 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{2} + 2 \, \Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + 2 \, \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} b x + 8 \, a {\left (\Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} + \sqrt {2 \, \pi b^{2} x^{2} + 4 \, \pi a b x + 2 \, \pi a^{2}} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )}\right )} a^{3} + 6 \, {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a\right )}\right )} b}{24 \, {\left (\pi ^{2} b^{5} x + \pi ^{2} a b^{4}\right )}} \] Input:

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

Output:

1/3*x^3*fresnel_cos(b*x + a) - 1/24*(12*(-I*pi*e^(1/2*I*pi*b^2*x^2 + I*pi* 
a*b*x + 1/2*I*pi*a^2) + I*pi*e^(-1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi* 
a^2))*a^3 + 4*(3*(-I*pi*e^(1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2) + 
 I*pi*e^(-1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi*a^2))*a^2 + 2*gamma(2, 
1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2) + 2*gamma(2, -1/2*I*pi*b^2*x 
^2 - I*pi*a*b*x - 1/2*I*pi*a^2))*b*x + 8*a*(gamma(2, 1/2*I*pi*b^2*x^2 + I* 
pi*a*b*x + 1/2*I*pi*a^2) + gamma(2, -1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I 
*pi*a^2)) + sqrt(2*pi*b^2*x^2 + 4*pi*a*b*x + 2*pi*a^2)*(((I - 1)*sqrt(2)*p 
i^(3/2)*(erf(sqrt(1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2)) - 1) - (I 
 + 1)*sqrt(2)*pi^(3/2)*(erf(sqrt(-1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi 
*a^2)) - 1))*a^3 + 6*(-(I + 1)*sqrt(2)*gamma(3/2, 1/2*I*pi*b^2*x^2 + I*pi* 
a*b*x + 1/2*I*pi*a^2) + (I - 1)*sqrt(2)*gamma(3/2, -1/2*I*pi*b^2*x^2 - I*p 
i*a*b*x - 1/2*I*pi*a^2))*a))*b/(pi^2*b^5*x + pi^2*a*b^4)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.76 \[ \int x^2 \operatorname {CosIntegral}(a+b x) \, dx=\frac {\mathit {ci} \left (b x +a \right ) a^{3}+\mathit {ci} \left (b x +a \right ) b^{3} x^{3}+\cos \left (b x +a \right ) a -2 \cos \left (b x +a \right ) b x -\sin \left (b x +a \right ) a^{2}+\sin \left (b x +a \right ) a b x -\sin \left (b x +a \right ) b^{2} x^{2}+2 \sin \left (b x +a \right )}{3 b^{3}} \] Input:

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

Output:

(ci(a + b*x)*a**3 + ci(a + b*x)*b**3*x**3 + cos(a + b*x)*a - 2*cos(a + b*x 
)*b*x - sin(a + b*x)*a**2 + sin(a + b*x)*a*b*x - sin(a + b*x)*b**2*x**2 + 
2*sin(a + b*x))/(3*b**3)