\(\int \frac {\text {Chi}(b x)}{x^2} \, dx\) [75]

Optimal result

Integrand size = 8, antiderivative size = 25 \[ \int \frac {\text {Chi}(b x)}{x^2} \, dx=-\frac {\cosh (b x)}{x}-\frac {\text {Chi}(b x)}{x}+b \text {Shi}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6668, 12, 3378, 3379} \[ \int \frac {\text {Chi}(b x)}{x^2} \, dx=-\frac {\text {Chi}(b x)}{x}+b \text {Shi}(b x)-\frac {\cosh (b x)}{x} \]

[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 3378

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 3379

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 6668

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*} \text {integral}& = -\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] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\text {Chi}(b x)}{x^2} \, dx=-\frac {\cosh (b x)}{x}-\frac {\text {Chi}(b x)}{x}+b \text {Shi}(b x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20

method	result	size
parts	\(-\frac {\operatorname {Chi}\left (b x \right )}{x}+b \left (-\frac {\cosh \left (b x \right )}{b x}+\operatorname {Shi}\left (b x \right )\right )\)	\(30\)
derivativedivides	\(b \left (-\frac {\operatorname {Chi}\left (b x \right )}{b x}-\frac {\cosh \left (b x \right )}{b x}+\operatorname {Shi}\left (b x \right )\right )\)	\(32\)
default	\(b \left (-\frac {\operatorname {Chi}\left (b x \right )}{b x}-\frac {\cosh \left (b x \right )}{b x}+\operatorname {Shi}\left (b x \right )\right )\)	\(32\)

[In]

int(Chi(b*x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\text {Chi}(b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(cosh_integral(b*x)/x^2, x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).

Time = 0.60 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {\text {Chi}(b x)}{x^2} \, dx=\frac {b^{2} x {{}_{3}F_{4}\left (\begin {matrix} \frac {1}{2}, 1, 1 \\ \frac {3}{2}, \frac {3}{2}, 2, 2 \end {matrix}\middle | {\frac {b^{2} x^{2}}{4}} \right )}}{4} - \frac {\log {\left (b^{2} x^{2} \right )}}{2 x} - \frac {1}{x} - \frac {\gamma }{x} \]

[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

Maxima [F]

\[ \int \frac {\text {Chi}(b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(Chi(b*x)/x^2, x)

Giac [F]

\[ \int \frac {\text {Chi}(b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(Chi(b*x)/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Chi}(b x)}{x^2} \, dx=\int \frac {\mathrm {coshint}\left (b\,x\right )}{x^2} \,d x \]

[In]

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

[Out]

int(coshint(b*x)/x^2, x)