\(\int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx\) [74]

Optimal result

Integrand size = 8, antiderivative size = 61 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=-\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;i b x)+\gamma \log (x)+\frac {1}{2} \log ^2(b x) \] Output:

-1/2*I*b*x*hypergeom([1, 1, 1],[2, 2, 2],-I*b*x)+1/2*I*b*x*hypergeom([1, 1 
, 1],[2, 2, 2],I*b*x)+gamma*ln(x)+1/2*ln(b*x)^2
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.54 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=\frac {1}{2} \left (-i b x \, _3F_3(1,1,1;2,2,2;-i b x)+i b x \, _3F_3(1,1,1;2,2,2;i b x)+\log (x) (2 \gamma +2 \operatorname {CosIntegral}(b x)+\Gamma (0,-i b x)+\Gamma (0,i b x)-\log (x)+\log (-i b x)+\log (i b x))\right ) \] Input:

Integrate[CosIntegral[b*x]/x,x]
 

Output:

((-I)*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-I)*b*x] + I*b*x*Hyperg 
eometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x] + Log[x]*(2*EulerGamma + 2*CosInt 
egral[b*x] + Gamma[0, (-I)*b*x] + Gamma[0, I*b*x] - Log[x] + Log[(-I)*b*x] 
 + Log[I*b*x]))/2
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {7056}

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.

\(\displaystyle \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx\)

\(\Big \downarrow \) 7056

\(\displaystyle -\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;i b x)+\frac {1}{2} \log ^2(b x)+\gamma \log (x)\)

Input:

Int[CosIntegral[b*x]/x,x]
 

Output:

(-1/2*I)*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-I)*b*x] + (I/2)*b*x 
*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x] + EulerGamma*Log[x] + Log[ 
b*x]^2/2
 

Defintions of rubi rules used

rule 7056

Int[CosIntegral[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[(-2^(-1))*I*b*x*Hyperge 
ometricPFQ[{1, 1, 1}, {2, 2, 2}, (-I)*b*x], x] + (Simp[(1/2)*I*b*x*Hypergeo 
metricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x], x] + Simp[EulerGamma*Log[x], x] + S 
imp[(1/2)*Log[b*x]^2, x]) /; FreeQ[b, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(51)=102\).

Time = 0.80 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.59

method	result	size
meijerg	\(\frac {\sqrt {\pi }\, \left (\frac {-\frac {\pi ^{2}}{3}+4 \ln \left (x \right )^{2}+\left (-\gamma -2 \ln \left (2\right )\right )^{2}+4 \ln \left (b \right )^{2}+8 \ln \left (x \right ) \ln \left (b \right )-4 \ln \left (x \right ) \left (-\gamma -2 \ln \left (2\right )\right )+4 \ln \left (2\right ) \left (-\gamma -2 \ln \left (2\right )\right )+4 \gamma \ln \left (x \right )-4 \ln \left (b \right ) \left (-\gamma -2 \ln \left (2\right )\right )-2 \gamma \left (-\gamma -2 \ln \left (2\right )\right )-8 \ln \left (2\right ) \ln \left (b \right )-4 \ln \left (2\right ) \gamma +4 \ln \left (b \right ) \gamma +\gamma ^{2}-8 \ln \left (x \right ) \ln \left (2\right )+4 \ln \left (2\right )^{2}}{2 \sqrt {\pi }}-\frac {b^{2} x^{2} \operatorname {hypergeom}\left (\left [1, 1, 1\right ], \left [\frac {3}{2}, 2, 2, 2\right ], -\frac {b^{2} x^{2}}{4}\right )}{2 \sqrt {\pi }}\right )}{4}\)	\(158\)

Input:

int(Ci(b*x)/x,x,method=_RETURNVERBOSE)
 

Output:

1/4*Pi^(1/2)*(1/2*(-1/3*Pi^2+4*ln(x)^2+(-gamma-2*ln(2))^2+4*ln(b)^2+8*ln(x 
)*ln(b)-4*ln(x)*(-gamma-2*ln(2))+4*ln(2)*(-gamma-2*ln(2))+4*gamma*ln(x)-4* 
ln(b)*(-gamma-2*ln(2))-2*gamma*(-gamma-2*ln(2))-8*ln(2)*ln(b)-4*ln(2)*gamm 
a+4*ln(b)*gamma+gamma^2-8*ln(x)*ln(2)+4*ln(2)^2)/Pi^(1/2)-1/2/Pi^(1/2)*b^2 
*x^2*hypergeom([1,1,1],[3/2,2,2,2],-1/4*b^2*x^2))
 

Fricas [F]

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

integrate(fresnel_cos(b*x)/x,x, algorithm="fricas")
 

Output:

integral(fresnel_cos(b*x)/x, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=\text {Timed out} \] Input:

integrate(Ci(b*x)/x,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate(fresnel_cos(b*x)/x,x, algorithm="maxima")
 

Output:

integrate(fresnel_cos(b*x)/x, x)
 

Giac [F]

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

integrate(fresnel_cos(b*x)/x,x, algorithm="giac")
 

Output:

integrate(fresnel_cos(b*x)/x, x)
 

Mupad [F(-1)]

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

int(cosint(b*x)/x,x)
 

Output:

int(cosint(b*x)/x, x)
 

Reduce [F]

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

int(Ci(b*x)/x,x)
 

Output:

int(ci(b*x)/x,x)