3.75 \(\int \frac {\text {Chi}(b x)}{x^2} \, dx\)

Optimal. Leaf size=25 \[ -\frac {\text {Chi}(b x)}{x}+b \text {Shi}(b x)-\frac {\cosh (b x)}{x} \]

[Out]

-Chi(b*x)/x-cosh(b*x)/x+b*Shi(b*x)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6533, 12, 3297, 3298} \[ -\frac {\text {Chi}(b x)}{x}+b \text {Shi}(b x)-\frac {\cosh (b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[CoshIntegral[b*x]/x^2,x]

[Out]

-(Cosh[b*x]/x) - CoshIntegral[b*x]/x + b*SinhIntegral[b*x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 6533

Int[CoshIntegral[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*CoshInte
gral[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Cosh[a + b*x])/(a + b*x), x], x] /
; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\text {Chi}(b x)}{x^2} \, dx &=-\frac {\text {Chi}(b x)}{x}+b \int \frac {\cosh (b x)}{b x^2} \, dx\\ &=-\frac {\text {Chi}(b x)}{x}+\int \frac {\cosh (b x)}{x^2} \, dx\\ &=-\frac {\cosh (b x)}{x}-\frac {\text {Chi}(b x)}{x}+b \int \frac {\sinh (b x)}{x} \, dx\\ &=-\frac {\cosh (b x)}{x}-\frac {\text {Chi}(b x)}{x}+b \text {Shi}(b x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ -\frac {\text {Chi}(b x)}{x}+b \text {Shi}(b x)-\frac {\cosh (b x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[CoshIntegral[b*x]/x^2,x]

[Out]

-(Cosh[b*x]/x) - CoshIntegral[b*x]/x + b*SinhIntegral[b*x]

________________________________________________________________________________________

fricas [F]  time = 1.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Chi}\left (b x\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(cosh_integral(b*x)/x^2, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Chi}\left (b x\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(Chi(b*x)/x^2, x)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 32, normalized size = 1.28 \[ b \left (-\frac {\Chi \left (b x \right )}{b x}-\frac {\cosh \left (b x \right )}{b x}+\Shi \left (b x \right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Chi(b*x)/x^2,x)

[Out]

b*(-Chi(b*x)/b/x-1/b/x*cosh(b*x)+Shi(b*x))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Chi}\left (b x\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(Chi(b*x)/x^2, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {coshint}\left (b\,x\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coshint(b*x)/x^2,x)

[Out]

int(coshint(b*x)/x^2, x)

________________________________________________________________________________________

sympy [B]  time = 0.97, size = 39, normalized size = 1.56 \[ \frac {b^{2} x {{}_{3}F_{4}\left (\begin {matrix} \frac {1}{2}, 1, 1 \\ \frac {3}{2}, \frac {3}{2}, 2, 2 \end {matrix}\middle | {\frac {b^{2} x^{2}}{4}} \right )}}{4} - \frac {\log {\left (b^{2} x^{2} \right )}}{2 x} - \frac {1}{x} - \frac {\gamma }{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)/x**2,x)

[Out]

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

________________________________________________________________________________________