Integrand size = 10, antiderivative size = 155 \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=-\frac {\cos (2 a+2 b x)}{4 b^2}-\frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {a \text {Si}(2 a+2 b x)}{b^2} \]
-1/2*a*(b*x+a)*Ci(b*x+a)^2/b^2+1/2*x*(b*x+a)*Ci(b*x+a)^2/b+1/2*Ci(2*b*x+2* a)/b^2-Ci(b*x+a)*cos(b*x+a)/b^2-1/4*cos(2*b*x+2*a)/b^2+1/2*ln(b*x+a)/b^2-a *Si(2*b*x+2*a)/b^2+a*Ci(b*x+a)*sin(b*x+a)/b^2-x*Ci(b*x+a)*sin(b*x+a)/b
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.62 \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=-\frac {\cos (2 (a+b x))+2 \left (a^2-b^2 x^2\right ) \operatorname {CosIntegral}(a+b x)^2-2 \operatorname {CosIntegral}(2 (a+b x))-2 \log (a+b x)+4 \operatorname {CosIntegral}(a+b x) (\cos (a+b x)+(-a+b x) \sin (a+b x))+4 a \text {Si}(2 (a+b x))}{4 b^2} \]
-1/4*(Cos[2*(a + b*x)] + 2*(a^2 - b^2*x^2)*CosIntegral[a + b*x]^2 - 2*CosI ntegral[2*(a + b*x)] - 2*Log[a + b*x] + 4*CosIntegral[a + b*x]*(Cos[a + b* x] + (-a + b*x)*Sin[a + b*x]) + 4*a*SinIntegral[2*(a + b*x)])/b^2
Time = 1.70 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.21, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.600, Rules used = {7064, 7060, 7066, 4906, 27, 3042, 3780, 7068, 5084, 7072, 3042, 3793, 2009, 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.
\(\displaystyle \int x \operatorname {CosIntegral}(a+b x)^2 \, dx\) |
\(\Big \downarrow \) 7064 |
\(\displaystyle -\frac {a \int \operatorname {CosIntegral}(a+b x)^2dx}{2 b}-\int x \cos (a+b x) \operatorname {CosIntegral}(a+b x)dx+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}\) |
\(\Big \downarrow \) 7060 |
\(\displaystyle -\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \int \cos (a+b x) \operatorname {CosIntegral}(a+b x)dx\right )}{2 b}-\int x \cos (a+b x) \operatorname {CosIntegral}(a+b x)dx+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}\) |
\(\Big \downarrow \) 7066 |
\(\displaystyle -\int x \cos (a+b x) \operatorname {CosIntegral}(a+b x)dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\int \frac {\cos (a+b x) \sin (a+b x)}{a+b x}dx\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\int \frac {\sin (2 a+2 b x)}{2 (a+b x)}dx\right )\right )}{2 b}-\int x \cos (a+b x) \operatorname {CosIntegral}(a+b x)dx+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{a+b x}dx\right )\right )}{2 b}-\int x \cos (a+b x) \operatorname {CosIntegral}(a+b x)dx+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{a+b x}dx\right )\right )}{2 b}-\int x \cos (a+b x) \operatorname {CosIntegral}(a+b x)dx+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle -\int x \cos (a+b x) \operatorname {CosIntegral}(a+b x)dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}\) |
\(\Big \downarrow \) 7068 |
\(\displaystyle \frac {\int \operatorname {CosIntegral}(a+b x) \sin (a+b x)dx}{b}+\int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x}dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}\) |
\(\Big \downarrow \) 5084 |
\(\displaystyle \frac {\int \operatorname {CosIntegral}(a+b x) \sin (a+b x)dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}\) |
\(\Big \downarrow \) 7072 |
\(\displaystyle \frac {\int \frac {\cos ^2(a+b x)}{a+b x}dx-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^2}{a+b x}dx-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\int \left (\frac {\cos (2 a+2 b x)}{2 (a+b x)}+\frac {1}{2 (a+b x)}\right )dx-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {1}{2} \int \frac {x \sin (2 a+2 b x)}{a+b x}dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\sin (2 a+2 b x)}{b}+\frac {a \sin (2 a+2 b x)}{b (-a-b x)}\right )dx-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a \text {Si}(2 a+2 b x)}{b^2}-\frac {\cos (2 a+2 b x)}{2 b^2}\right )-\frac {a \left (\frac {(a+b x) \operatorname {CosIntegral}(a+b x)^2}{b}-2 \left (\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}\right )\right )}{2 b}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}\) |
(x*(a + b*x)*CosIntegral[a + b*x]^2)/(2*b) + (-((Cos[a + b*x]*CosIntegral[ a + b*x])/b) + CosIntegral[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*Si nIntegral[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)
3.1.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]))
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]
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]
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]
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]
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]
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]
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]
Time = 0.90 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x +a \right )^{2} \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-2 \,\operatorname {Ci}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\frac {\cos \left (b x +a \right )}{2}+\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}\right )-a \,\operatorname {Si}\left (2 b x +2 a \right )-\frac {\cos \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}}{b^{2}}\) | \(113\) |
default | \(\frac {\operatorname {Ci}\left (b x +a \right )^{2} \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-2 \,\operatorname {Ci}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\frac {\cos \left (b x +a \right )}{2}+\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}\right )-a \,\operatorname {Si}\left (2 b x +2 a \right )-\frac {\cos \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}}{b^{2}}\) | \(113\) |
1/b^2*(Ci(b*x+a)^2*(-(b*x+a)*a+1/2*(b*x+a)^2)-2*Ci(b*x+a)*(-a*sin(b*x+a)+1 /2*cos(b*x+a)+1/2*(b*x+a)*sin(b*x+a))-a*Si(2*b*x+2*a)-1/2*cos(b*x+a)^2+1/2 *ln(b*x+a)+1/2*Ci(2*b*x+2*a))
\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]
\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int x \operatorname {Ci}^{2}{\left (a + b x \right )}\, dx \]
\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]
\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int x\,{\mathrm {cosint}\left (a+b\,x\right )}^2 \,d x \]