Integrand size = 10, antiderivative size = 47 \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx=\frac {b \cos (a) \operatorname {CosIntegral}(b x)}{a}-\frac {b \operatorname {CosIntegral}(a+b x)}{a}-\frac {\operatorname {CosIntegral}(a+b x)}{x}-\frac {b \sin (a) \text {Si}(b x)}{a} \] Output:
b*cos(a)*Ci(b*x)/a-b*Ci(b*x+a)/a-Ci(b*x+a)/x-b*sin(a)*Si(b*x)/a
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85 \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx=\frac {b x \cos (a) \operatorname {CosIntegral}(b x)-(a+b x) \operatorname {CosIntegral}(a+b x)-b x \sin (a) \text {Si}(b x)}{a x} \] Input:
Integrate[CosIntegral[a + b*x]/x^2,x]
Output:
(b*x*Cos[a]*CosIntegral[b*x] - (a + b*x)*CosIntegral[a + b*x] - b*x*Sin[a] *SinIntegral[b*x])/(a*x)
Time = 0.45 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 7058 |
\(\displaystyle b \int \frac {\cos (a+b x)}{x (a+b x)}dx-\frac {\operatorname {CosIntegral}(a+b x)}{x}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle b \int \left (\frac {\cos (a+b x)}{a x}-\frac {b \cos (a+b x)}{a (a+b x)}\right )dx-\frac {\operatorname {CosIntegral}(a+b x)}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle b \left (-\frac {\operatorname {CosIntegral}(a+b x)}{a}+\frac {\cos (a) \operatorname {CosIntegral}(b x)}{a}-\frac {\sin (a) \text {Si}(b x)}{a}\right )-\frac {\operatorname {CosIntegral}(a+b x)}{x}\) |
Input:
Int[CosIntegral[a + b*x]/x^2,x]
Output:
-(CosIntegral[a + b*x]/x) + b*((Cos[a]*CosIntegral[b*x])/a - CosIntegral[a + b*x]/a - (Sin[a]*SinIntegral[b*x])/a)
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 = 0.90 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00
method | result | size |
parts | \(-\frac {\operatorname {Ci}\left (b x +a \right )}{x}+b \left (-\frac {\operatorname {Ci}\left (b x +a \right )}{a}+\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{a}\right )\) | \(47\) |
derivativedivides | \(b \left (-\frac {\operatorname {Ci}\left (b x +a \right )}{b x}-\frac {\operatorname {Ci}\left (b x +a \right )}{a}+\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{a}\right )\) | \(49\) |
default | \(b \left (-\frac {\operatorname {Ci}\left (b x +a \right )}{b x}-\frac {\operatorname {Ci}\left (b x +a \right )}{a}+\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{a}\right )\) | \(49\) |
Input:
int(Ci(b*x+a)/x^2,x,method=_RETURNVERBOSE)
Output:
-Ci(b*x+a)/x+b*(-1/a*Ci(b*x+a)+1/a*(-Si(b*x)*sin(a)+Ci(b*x)*cos(a)))
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{2}} \,d x } \] Input:
integrate(fresnel_cos(b*x+a)/x^2,x, algorithm="fricas")
Output:
integral(fresnel_cos(b*x + a)/x^2, x)
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {Ci}{\left (a + b x \right )}}{x^{2}}\, dx \] Input:
integrate(Ci(b*x+a)/x**2,x)
Output:
Integral(Ci(a + b*x)/x**2, x)
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{2}} \,d x } \] Input:
integrate(fresnel_cos(b*x+a)/x^2,x, algorithm="maxima")
Output:
integrate(fresnel_cos(b*x + a)/x^2, x)
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{2}} \,d x } \] Input:
integrate(fresnel_cos(b*x+a)/x^2,x, algorithm="giac")
Output:
integrate(fresnel_cos(b*x + a)/x^2, x)
Timed out. \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx=\int \frac {\mathrm {cosint}\left (a+b\,x\right )}{x^2} \,d x \] Input:
int(cosint(a + b*x)/x^2,x)
Output:
int(cosint(a + b*x)/x^2, x)
\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^2} \, dx=\int \frac {\mathit {ci} \left (b x +a \right )}{x^{2}}d x \] Input:
int(Ci(b*x+a)/x^2,x)
Output:
int(ci(a + b*x)/x**2,x)