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

Optimal result

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

[Out]

-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)+EulerGamma*ln(x)+1/2*l
n(b*x)^2

Rubi [A] (verified)

Time = 0.02 (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 = {6666} \[ \int \frac {\text {Chi}(b x)}{x} \, dx=-\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;-b x)+\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;b x)+\frac {1}{2} \log ^2(b x)+\gamma \log (x) \]

[In]

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

[Out]

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

Rule 6666

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

Rubi steps \begin{align*} \text {integral}& = -\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) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {\text {Chi}(b x)}{x} \, dx=-\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;-b x)+\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;b x)+\gamma \log (x)+\frac {1}{2} \log ^2(b x) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(cosh_integral(b*x)/x, x)

Sympy [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.81 \[ \int \frac {\text {Chi}(b x)}{x} \, dx=\frac {b^{2} x^{2} {{}_{3}F_{4}\left (\begin {matrix} 1, 1, 1 \\ \frac {3}{2}, 2, 2, 2 \end {matrix}\middle | {\frac {b^{2} x^{2}}{4}} \right )}}{8} + \frac {\log {\left (b^{2} x^{2} \right )}^{2}}{8} + \frac {\gamma \log {\left (b^{2} x^{2} \right )}}{2} \]

[In]

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

[Out]

b**2*x**2*hyper((1, 1, 1), (3/2, 2, 2, 2), b**2*x**2/4)/8 + log(b**2*x**2)**2/8 + EulerGamma*log(b**2*x**2)/2

Maxima [F]

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

[In]

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

[Out]

integrate(Chi(b*x)/x, x)

Giac [F]

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

[In]

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

[Out]

integrate(Chi(b*x)/x, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(coshint(b*x)/x, x)