\(\int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx\) [92]

Optimal result

Integrand size = 10, antiderivative size = 111 \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=-\frac {b \cos (a+b x)}{2 a x}-\frac {b^2 \cos (a) \operatorname {CosIntegral}(b x)}{2 a^2}+\frac {b^2 \operatorname {CosIntegral}(a+b x)}{2 a^2}-\frac {\operatorname {CosIntegral}(a+b x)}{2 x^2}-\frac {b^2 \operatorname {CosIntegral}(b x) \sin (a)}{2 a}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a}+\frac {b^2 \sin (a) \text {Si}(b x)}{2 a^2} \] Output:

-1/2*b*cos(b*x+a)/a/x-1/2*b^2*cos(a)*Ci(b*x)/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)*sin(a)/a-1/2*b^2*cos(a)*Si(b*x)/a+1/2*b^2*s 
in(a)*Si(b*x)/a^2
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.68 \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=-\frac {\left (a^2-b^2 x^2\right ) \operatorname {CosIntegral}(a+b x)+b^2 x^2 \operatorname {CosIntegral}(b x) (\cos (a)+a \sin (a))+b x (a \cos (a+b x)+b x (a \cos (a)-\sin (a)) \text {Si}(b x))}{2 a^2 x^2} \] Input:

Integrate[CosIntegral[a + b*x]/x^3,x]
 

Output:

-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)
 

Rubi [A] (verified)

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

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

Input:

Int[CosIntegral[a + b*x]/x^3,x]
 

Output:

-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
 

Defintions of rubi rules used

rule 2009

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

rule 7058

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]
 

rule 7293

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

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.77

method	result	size
parts	\(-\frac {\operatorname {Ci}\left (b x +a \right )}{2 x^{2}}+\frac {b^{2} \left (-\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{a^{2}}+\frac {\operatorname {Ci}\left (b x +a \right )}{a^{2}}+\frac {-\frac {\cos \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \cos \left (a \right )-\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{a}\right )}{2}\)	\(85\)
derivativedivides	\(b^{2} \left (-\frac {\operatorname {Ci}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{2 a^{2}}+\frac {\operatorname {Ci}\left (b x +a \right )}{2 a^{2}}+\frac {-\frac {\cos \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \cos \left (a \right )-\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{2 a}\right )\)	\(88\)
default	\(b^{2} \left (-\frac {\operatorname {Ci}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {-\operatorname {Si}\left (b x \right ) \sin \left (a \right )+\operatorname {Ci}\left (b x \right ) \cos \left (a \right )}{2 a^{2}}+\frac {\operatorname {Ci}\left (b x +a \right )}{2 a^{2}}+\frac {-\frac {\cos \left (b x +a \right )}{b x}-\operatorname {Si}\left (b x \right ) \cos \left (a \right )-\operatorname {Ci}\left (b x \right ) \sin \left (a \right )}{2 a}\right )\)	\(88\)

Input:

int(Ci(b*x+a)/x^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*Ci(b*x+a)/x^2+1/2*b^2*(-1/a^2*(-Si(b*x)*sin(a)+Ci(b*x)*cos(a))+1/a^2* 
Ci(b*x+a)+1/a*(-cos(b*x+a)/b/x-Si(b*x)*cos(a)-Ci(b*x)*sin(a)))
 

Fricas [F]

\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{3}} \,d x } \] Input:

integrate(fresnel_cos(b*x+a)/x^3,x, algorithm="fricas")
 

Output:

integral(fresnel_cos(b*x + a)/x^3, x)
 

Sympy [F]

\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {Ci}{\left (a + b x \right )}}{x^{3}}\, dx \] Input:

integrate(Ci(b*x+a)/x**3,x)
 

Output:

Integral(Ci(a + b*x)/x**3, x)
 

Maxima [F]

\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{3}} \,d x } \] Input:

integrate(fresnel_cos(b*x+a)/x^3,x, algorithm="maxima")
 

Output:

integrate(fresnel_cos(b*x + a)/x^3, x)
 

Giac [F]

\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x + a\right )}{x^{3}} \,d x } \] Input:

integrate(fresnel_cos(b*x+a)/x^3,x, algorithm="giac")
 

Output:

integrate(fresnel_cos(b*x + a)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (a+b\,x\right )}{x^3} \,d x \] Input:

int(cosint(a + b*x)/x^3,x)
 

Output:

int(cosint(a + b*x)/x^3, x)
 

Reduce [F]

\[ \int \frac {\operatorname {CosIntegral}(a+b x)}{x^3} \, dx=\int \frac {\mathit {ci} \left (b x +a \right )}{x^{3}}d x \] Input:

int(Ci(b*x+a)/x^3,x)
                                                                                    
                                                                                    
 

Output:

int(ci(a + b*x)/x**3,x)