Integrand size = 10, antiderivative size = 111 \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=-\frac {b \cos (a+b x)}{2 a x}-\frac {b^2 \cos (a) \operatorname {CosIntegral}(b x)}{2 a^2}+\frac {b^2 \operatorname {CosIntegral}(a+b x)}{2 a^2}-\frac {\operatorname {CosIntegral}(a+b x)}{2 x^2}-\frac {b^2 \operatorname {CosIntegral}(b x) \sin (a)}{2 a}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a}+\frac {b^2 \sin (a) \text {Si}(b x)}{2 a^2} \]
1/2*b^2*Ci(b*x+a)/a^2-1/2*Ci(b*x+a)/x^2-1/2*b^2*Ci(b*x)*cos(a)/a^2-1/2*b*c os(b*x+a)/a/x-1/2*b^2*cos(a)*Si(b*x)/a-1/2*b^2*Ci(b*x)*sin(a)/a+1/2*b^2*Si (b*x)*sin(a)/a^2
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.68 \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=-\frac {\left (a^2-b^2 x^2\right ) \operatorname {CosIntegral}(a+b x)+b^2 x^2 \operatorname {CosIntegral}(b x) (\cos (a)+a \sin (a))+b x (a \cos (a+b x)+b x (a \cos (a)-\sin (a)) \text {Si}(b x))}{2 a^2 x^2} \]
-1/2*((a^2 - b^2*x^2)*CosIntegral[a + b*x] + b^2*x^2*CosIntegral[b*x]*(Cos [a] + a*Sin[a]) + b*x*(a*Cos[a + b*x] + b*x*(a*Cos[a] - Sin[a])*SinIntegra l[b*x]))/(a^2*x^2)
Time = 0.56 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.83, 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.
\(\displaystyle \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 7058 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (a+b x)}{x^2 (a+b x)}dx-\frac {\operatorname {CosIntegral}(a+b x)}{2 x^2}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} b \int \left (\frac {\cos (a+b x) b^2}{a^2 (a+b x)}-\frac {\cos (a+b x) b}{a^2 x}+\frac {\cos (a+b x)}{a x^2}\right )dx-\frac {\operatorname {CosIntegral}(a+b x)}{2 x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} b \left (\frac {b \operatorname {CosIntegral}(a+b x)}{a^2}-\frac {b \cos (a) \operatorname {CosIntegral}(b x)}{a^2}+\frac {b \sin (a) \text {Si}(b x)}{a^2}-\frac {b \sin (a) \operatorname {CosIntegral}(b x)}{a}-\frac {b \cos (a) \text {Si}(b x)}{a}-\frac {\cos (a+b x)}{a x}\right )-\frac {\operatorname {CosIntegral}(a+b x)}{2 x^2}\) |
-1/2*CosIntegral[a + b*x]/x^2 + (b*(-(Cos[a + b*x]/(a*x)) - (b*Cos[a]*CosI ntegral[b*x])/a^2 + (b*CosIntegral[a + b*x])/a^2 - (b*CosIntegral[b*x]*Sin [a])/a - (b*Cos[a]*SinIntegral[b*x])/a + (b*Sin[a]*SinIntegral[b*x])/a^2)) /2
3.1.92.3.1 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 = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.77
method | result | size |
parts | \(-\frac {\operatorname {Ci}\left (b x +a \right )}{2 x^{2}}+\frac {b^{2} \left (\frac {\operatorname {Ci}\left (b x +a \right )}{a^{2}}-\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{a^{2}}+\frac {-\frac {\cos \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \cos \left (a \right )-\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{a}\right )}{2}\) | \(85\) |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {Ci}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\operatorname {Ci}\left (b x +a \right )}{2 a^{2}}-\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{2 a^{2}}+\frac {-\frac {\cos \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \cos \left (a \right )-\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{2 a}\right )\) | \(88\) |
default | \(b^{2} \left (-\frac {\operatorname {Ci}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\operatorname {Ci}\left (b x +a \right )}{2 a^{2}}-\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{2 a^{2}}+\frac {-\frac {\cos \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \cos \left (a \right )-\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{2 a}\right )\) | \(88\) |
-1/2*Ci(b*x+a)/x^2+1/2*b^2*(1/a^2*Ci(b*x+a)-1/a^2*(-Si(b*x)*sin(a)+Ci(b*x) *cos(a))+1/a*(-cos(b*x+a)/b/x-Si(b*x)*cos(a)-Ci(b*x)*sin(a)))
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{3}} \,d x } \]
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {Ci}{\left (a + b x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{3}} \,d x } \]
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (a+b\,x\right )}{x^3} \,d x \]