Integrand size = 8, antiderivative size = 71 \[ \int x \operatorname {CosIntegral}(a+b x) \, dx=-\frac {\cos (a+b x)}{2 b^2}-\frac {a^2 \operatorname {CosIntegral}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \operatorname {CosIntegral}(a+b x)+\frac {a \sin (a+b x)}{2 b^2}-\frac {x \sin (a+b x)}{2 b} \]
-1/2*a^2*Ci(b*x+a)/b^2+1/2*x^2*Ci(b*x+a)-1/2*cos(b*x+a)/b^2+1/2*a*sin(b*x+ a)/b^2-1/2*x*sin(b*x+a)/b
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.69 \[ \int x \operatorname {CosIntegral}(a+b x) \, dx=\frac {-\cos (a+b x)+\left (-a^2+b^2 x^2\right ) \operatorname {CosIntegral}(a+b x)+(a-b x) \sin (a+b x)}{2 b^2} \]
Time = 0.45 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, 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.
\(\displaystyle \int x \operatorname {CosIntegral}(a+b x) \, dx\) |
\(\Big \downarrow \) 7058 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {CosIntegral}(a+b x)-\frac {1}{2} b \int \frac {x^2 \cos (a+b x)}{a+b x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {CosIntegral}(a+b x)-\frac {1}{2} b \int \left (\frac {\cos (a+b x) a^2}{b^2 (a+b x)}-\frac {\cos (a+b x) a}{b^2}+\frac {x \cos (a+b x)}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {CosIntegral}(a+b x)-\frac {1}{2} b \left (\frac {a^2 \operatorname {CosIntegral}(a+b x)}{b^3}-\frac {a \sin (a+b x)}{b^3}+\frac {\cos (a+b x)}{b^3}+\frac {x \sin (a+b x)}{b^2}\right )\) |
(x^2*CosIntegral[a + b*x])/2 - (b*(Cos[a + b*x]/b^3 + (a^2*CosIntegral[a + b*x])/b^3 - (a*Sin[a + b*x])/b^3 + (x*Sin[a + b*x])/b^2))/2
3.1.88.3.1 Defintions of rubi rules used
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]
Time = 0.47 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79
method | result | size |
parts | \(\frac {x^{2} \operatorname {Ci}\left (b x +a \right )}{2}-\frac {a^{2} \operatorname {Ci}\left (b x +a \right )-2 a \sin \left (b x +a \right )+\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )}{2 b^{2}}\) | \(56\) |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x +a \right ) \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )+a \sin \left (b x +a \right )-\frac {\cos \left (b x +a \right )}{2}-\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}}{b^{2}}\) | \(60\) |
default | \(\frac {\operatorname {Ci}\left (b x +a \right ) \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )+a \sin \left (b x +a \right )-\frac {\cos \left (b x +a \right )}{2}-\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}}{b^{2}}\) | \(60\) |
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.46 \[ \int x \operatorname {CosIntegral}(a+b x) \, dx=\frac {\pi b^{3} x^{2} \operatorname {C}\left (b x + a\right ) - \pi a^{2} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (b^{2} x - a b\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{2 \, \pi b^{3}} \]
1/2*(pi*b^3*x^2*fresnel_cos(b*x + a) - pi*a^2*sqrt(b^2)*fresnel_cos(sqrt(b ^2)*(b*x + a)/b) - (b^2*x - a*b)*sin(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^ 2) + sqrt(b^2)*fresnel_sin(sqrt(b^2)*(b*x + a)/b))/(pi*b^3)
\[ \int x \operatorname {CosIntegral}(a+b x) \, dx=\int x \operatorname {Ci}{\left (a + b x \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.38 \[ \int x \operatorname {CosIntegral}(a+b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {C}\left (b x + a\right ) + \frac {{\left (8 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a b x + 8 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{2} - \sqrt {2 \, \pi b^{2} x^{2} + 4 \, \pi a b x + 2 \, \pi a^{2}} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )} + \left (i + 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )}\right )} a^{2} + \left (2 i + 2\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) - \left (2 i - 2\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )}\right )} b}{16 \, {\left (\pi ^{2} b^{4} x + \pi ^{2} a b^{3}\right )}} \]
1/2*x^2*fresnel_cos(b*x + a) + 1/16*(8*(-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*b*x + 8*(-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 - sqrt(2*pi*b^2 *x^2 + 4*pi*a*b*x + 2*pi*a^2)*((-(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) + (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^2 + (2*I + 2)*sqrt(2)*gamma(3/2, 1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2) - (2 *I - 2)*sqrt(2)*gamma(3/2, -1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi*a^2)) )*b/(pi^2*b^4*x + pi^2*a*b^3)
\[ \int x \operatorname {CosIntegral}(a+b x) \, dx=\int { x \operatorname {C}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int x \operatorname {CosIntegral}(a+b x) \, dx=\frac {x^2\,\mathrm {cosint}\left (a+b\,x\right )}{2}-\frac {\cos \left (a+b\,x\right )-a\,\sin \left (a+b\,x\right )+a^2\,\mathrm {cosint}\left (a+b\,x\right )+b\,x\,\sin \left (a+b\,x\right )}{2\,b^2} \]