\(\int x^2 \operatorname {CosIntegral}(a+b x)^2 \, dx\) [94]

Optimal result

Integrand size = 12, antiderivative size = 334 \[ \int x^2 \operatorname {CosIntegral}(a+b x)^2 \, dx=\frac {2 x}{3 b^2}+\frac {a \cos (2 a+2 b x)}{6 b^3}+\frac {(a-b x) \cos (2 a+2 b x)}{6 b^3}+\frac {2 a \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{3 b^3}-\frac {4 x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \operatorname {CosIntegral}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \operatorname {CosIntegral}(a+b x)^2}{3 b}-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{b^3}-\frac {a \log (a+b x)}{b^3}+\frac {2 \cos (a+b x) \sin (a+b x)}{3 b^3}+\frac {4 \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{3 b^3}-\frac {2 a^2 \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{3 b^3}+\frac {2 a x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{3 b^2}-\frac {2 x^2 \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{3 b}+\frac {\sin (2 a+2 b x)}{12 b^3}-\frac {2 \text {Si}(2 a+2 b x)}{3 b^3}+\frac {a^2 \text {Si}(2 a+2 b x)}{b^3} \] Output:

2/3*x/b^2+1/6*a*cos(2*b*x+2*a)/b^3+1/6*(-b*x+a)*cos(2*b*x+2*a)/b^3+2/3*a*c 
os(b*x+a)*Ci(b*x+a)/b^3-4/3*x*cos(b*x+a)*Ci(b*x+a)/b^2+1/3*a^2*(b*x+a)*Ci( 
b*x+a)^2/b^3-1/3*a*x*(b*x+a)*Ci(b*x+a)^2/b^2+1/3*x^2*(b*x+a)*Ci(b*x+a)^2/b 
-a*Ci(2*b*x+2*a)/b^3-a*ln(b*x+a)/b^3+2/3*cos(b*x+a)*sin(b*x+a)/b^3+4/3*Ci( 
b*x+a)*sin(b*x+a)/b^3-2/3*a^2*Ci(b*x+a)*sin(b*x+a)/b^3+2/3*a*x*Ci(b*x+a)*s 
in(b*x+a)/b^2-2/3*x^2*Ci(b*x+a)*sin(b*x+a)/b+1/12*sin(2*b*x+2*a)/b^3-2/3*S 
i(2*b*x+2*a)/b^3+a^2*Si(2*b*x+2*a)/b^3
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.48 \[ \int x^2 \operatorname {CosIntegral}(a+b x)^2 \, dx=\frac {8 a+8 b x+4 a \cos (2 (a+b x))-2 b x \cos (2 (a+b x))+4 \left (a^3+b^3 x^3\right ) \operatorname {CosIntegral}(a+b x)^2-12 a \operatorname {CosIntegral}(2 (a+b x))-12 a \log (a+b x)-8 \operatorname {CosIntegral}(a+b x) \left (-((a-2 b x) \cos (a+b x))+\left (-2+a^2-a b x+b^2 x^2\right ) \sin (a+b x)\right )+5 \sin (2 (a+b x))-8 \text {Si}(2 (a+b x))+12 a^2 \text {Si}(2 (a+b x))}{12 b^3} \] Input:

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

Output:

(8*a + 8*b*x + 4*a*Cos[2*(a + b*x)] - 2*b*x*Cos[2*(a + b*x)] + 4*(a^3 + b^ 
3*x^3)*CosIntegral[a + b*x]^2 - 12*a*CosIntegral[2*(a + b*x)] - 12*a*Log[a 
 + b*x] - 8*CosIntegral[a + b*x]*(-((a - 2*b*x)*Cos[a + b*x]) + (-2 + a^2 
- a*b*x + b^2*x^2)*Sin[a + b*x]) + 5*Sin[2*(a + b*x)] - 8*SinIntegral[2*(a 
 + b*x)] + 12*a^2*SinIntegral[2*(a + b*x)])/(12*b^3)
 

Rubi [A] (verified)

Time = 4.54 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.30, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.917, Rules used = {7064, 7064, 7060, 7066, 4906, 27, 3042, 3780, 7068, 5084, 7072, 3042, 3793, 2009, 7074, 7066, 4906, 27, 3042, 3780, 7292, 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.

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

\(\Big \downarrow \) 7064

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

\(\Big \downarrow \) 7064

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

\(\Big \downarrow \) 7060

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

\(\Big \downarrow \) 7066

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7068

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

\(\Big \downarrow \) 5084

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

\(\Big \downarrow \) 7072

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7074

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

\(\Big \downarrow \) 7066

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(x^2*(a + b*x)*CosIntegral[a + b*x]^2)/(3*b) - (2*a*((x*(a + b*x)*CosInteg 
ral[a + b*x]^2)/(2*b) + (-((Cos[a + b*x]*CosIntegral[a + b*x])/b) + CosInt 
egral[2*a + 2*b*x]/(2*b) + Log[a + b*x]/(2*b))/b - (x*CosIntegral[a + b*x] 
*Sin[a + b*x])/b + (-1/2*Cos[2*a + 2*b*x]/b^2 - (a*SinIntegral[2*a + 2*b*x 
])/b^2)/2 - (a*(((a + b*x)*CosIntegral[a + b*x]^2)/b - 2*((CosIntegral[a + 
 b*x]*Sin[a + b*x])/b - SinIntegral[2*a + 2*b*x]/(2*b))))/(2*b)))/(3*b) - 
(2*((x^2*CosIntegral[a + b*x]*Sin[a + b*x])/b + (-1/2*(a*Cos[2*a + 2*b*x]) 
/b^3 + (x*Cos[2*a + 2*b*x])/(2*b^2) - Sin[2*a + 2*b*x]/(4*b^3) - (a^2*SinI 
ntegral[2*a + 2*b*x])/b^3)/2 - (2*(x/(2*b) - (x*Cos[a + b*x]*CosIntegral[a 
 + b*x])/b - (a*CosIntegral[2*a + 2*b*x])/(2*b^2) - (a*Log[a + b*x])/(2*b^ 
2) + (Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + ((CosIntegral[a + b*x]*Sin[a + 
b*x])/b - SinIntegral[2*a + 2*b*x]/(2*b))/b))/b))/3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3780

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[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 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG 
tQ[p, 0]
 

rule 5084

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

rule 7060

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

rule 7064

Int[CosIntegral[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(a + b*x)*(c + d*x)^m*(CosIntegral[a + b*x]^2/(b*(m + 1))), x] + (- 
Simp[2/(m + 1)   Int[(c + d*x)^m*Cos[a + b*x]*CosIntegral[a + b*x], x], x] 
+ Simp[(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 7066

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

rule 7068

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] + (-Simp[d/b   Int[(e + f*x)^m*Sin[a + b*x]*(Cos[c + d*x]/(c + d* 
x)), x], x] - Simp[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 7072

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

rule 7074

Int[CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + 
(b_.)*(x_)], x_Symbol] :> Simp[(-(e + f*x)^m)*Cos[a + b*x]*(CosIntegral[c + 
 d*x]/b), x] + (Simp[d/b   Int[(e + f*x)^m*Cos[a + b*x]*(Cos[c + d*x]/(c + 
d*x)), x], x] + Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Cos[a + b*x]*CosIntegr 
al[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

rule 7292

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

rule 7293

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

Maple [F]

\[\int x^{2} \operatorname {Ci}\left (b x +a \right )^{2}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(ci(a + b*x)**2*x**2,x)