\(\int \frac {\text {Chi}(b x)}{x} \, dx\) [74]

Optimal result

Integrand size = 8, antiderivative size = 52 \[ \int \frac {\text {Chi}(b x)}{x} \, dx=-\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;-b x)+\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;b x)+\gamma \log (x)+\frac {1}{2} \log ^2(b x) \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {\text {Chi}(b x)}{x} \, dx=-\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;-b x)+\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;b x)+\gamma \log (x)+\frac {1}{2} \log ^2(b x) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 52, 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 = {7085}

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.

\(\displaystyle \int \frac {\text {Chi}(b x)}{x} \, dx\)

\(\Big \downarrow \) 7085

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 7085

Int[CoshIntegral[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[(-2^(-1))*b*x*Hypergeo 
metricPFQ[{1, 1, 1}, {2, 2, 2}, (-b)*x], x] + (Simp[(1/2)*b*x*Hypergeometri 
cPFQ[{1, 1, 1}, {2, 2, 2}, b*x], x] + Simp[EulerGamma*Log[x], x] + Simp[(1/ 
2)*Log[b*x]^2, x]) /; FreeQ[b, x]
 

Maple [F]

\[\int \frac {\operatorname {Chi}\left (b x \right )}{x}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\text {Chi}(b x)}{x} \, dx=\int { \frac {{\rm Chi}\left (b x\right )}{x} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\text {Chi}(b x)}{x} \, dx=\int { \frac {{\rm Chi}\left (b x\right )}{x} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\text {Chi}(b x)}{x} \, dx=\int { \frac {{\rm Chi}\left (b x\right )}{x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Chi}(b x)}{x} \, dx=\int \frac {\mathrm {coshint}\left (b\,x\right )}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\text {Chi}(b x)}{x} \, dx=\int \frac {\chi \left (b x \right )}{x}d x \] Input:

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

Output:

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