\(\int \frac {\text {Chi}(a+b x)}{x^3} \, dx\) [92]

Optimal result

Integrand size = 10, antiderivative size = 111 \[ \int \frac {\text {Chi}(a+b x)}{x^3} \, dx=-\frac {b \cosh (a+b x)}{2 a x}-\frac {b^2 \cosh (a) \text {Chi}(b x)}{2 a^2}+\frac {b^2 \text {Chi}(a+b x)}{2 a^2}-\frac {\text {Chi}(a+b x)}{2 x^2}+\frac {b^2 \text {Chi}(b x) \sinh (a)}{2 a}+\frac {b^2 \cosh (a) \text {Shi}(b x)}{2 a}-\frac {b^2 \sinh (a) \text {Shi}(b x)}{2 a^2} \]

[Out]

1/2*b^2*Chi(b*x+a)/a^2-1/2*Chi(b*x+a)/x^2-1/2*b^2*Chi(b*x)*cosh(a)/a^2-1/2*b*cosh(b*x+a)/a/x+1/2*b^2*cosh(a)*S
hi(b*x)/a+1/2*b^2*Chi(b*x)*sinh(a)/a-1/2*b^2*Shi(b*x)*sinh(a)/a^2

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6668, 6874, 3378, 3384, 3379, 3382} \[ \int \frac {\text {Chi}(a+b x)}{x^3} \, dx=\frac {b^2 \text {Chi}(a+b x)}{2 a^2}-\frac {b^2 \cosh (a) \text {Chi}(b x)}{2 a^2}-\frac {b^2 \sinh (a) \text {Shi}(b x)}{2 a^2}+\frac {b^2 \sinh (a) \text {Chi}(b x)}{2 a}+\frac {b^2 \cosh (a) \text {Shi}(b x)}{2 a}-\frac {\text {Chi}(a+b x)}{2 x^2}-\frac {b \cosh (a+b x)}{2 a x} \]

[In]

Int[CoshIntegral[a + b*x]/x^3,x]

[Out]

-1/2*(b*Cosh[a + b*x])/(a*x) - (b^2*Cosh[a]*CoshIntegral[b*x])/(2*a^2) + (b^2*CoshIntegral[a + b*x])/(2*a^2) -
CoshIntegral[a + b*x]/(2*x^2) + (b^2*CoshIntegral[b*x]*Sinh[a])/(2*a) + (b^2*Cosh[a]*SinhIntegral[b*x])/(2*a)
- (b^2*Sinh[a]*SinhIntegral[b*x])/(2*a^2)

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 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

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

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Chi}(a+b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cosh (a+b x)}{x^2 (a+b x)} \, dx \\ & = -\frac {\text {Chi}(a+b x)}{2 x^2}+\frac {1}{2} b \int \left (\frac {\cosh (a+b x)}{a x^2}-\frac {b \cosh (a+b x)}{a^2 x}+\frac {b^2 \cosh (a+b x)}{a^2 (a+b x)}\right ) \, dx \\ & = -\frac {\text {Chi}(a+b x)}{2 x^2}+\frac {b \int \frac {\cosh (a+b x)}{x^2} \, dx}{2 a}-\frac {b^2 \int \frac {\cosh (a+b x)}{x} \, dx}{2 a^2}+\frac {b^3 \int \frac {\cosh (a+b x)}{a+b x} \, dx}{2 a^2} \\ & = -\frac {b \cosh (a+b x)}{2 a x}+\frac {b^2 \text {Chi}(a+b x)}{2 a^2}-\frac {\text {Chi}(a+b x)}{2 x^2}+\frac {b^2 \int \frac {\sinh (a+b x)}{x} \, dx}{2 a}-\frac {\left (b^2 \cosh (a)\right ) \int \frac {\cosh (b x)}{x} \, dx}{2 a^2}-\frac {\left (b^2 \sinh (a)\right ) \int \frac {\sinh (b x)}{x} \, dx}{2 a^2} \\ & = -\frac {b \cosh (a+b x)}{2 a x}-\frac {b^2 \cosh (a) \text {Chi}(b x)}{2 a^2}+\frac {b^2 \text {Chi}(a+b x)}{2 a^2}-\frac {\text {Chi}(a+b x)}{2 x^2}-\frac {b^2 \sinh (a) \text {Shi}(b x)}{2 a^2}+\frac {\left (b^2 \cosh (a)\right ) \int \frac {\sinh (b x)}{x} \, dx}{2 a}+\frac {\left (b^2 \sinh (a)\right ) \int \frac {\cosh (b x)}{x} \, dx}{2 a} \\ & = -\frac {b \cosh (a+b x)}{2 a x}-\frac {b^2 \cosh (a) \text {Chi}(b x)}{2 a^2}+\frac {b^2 \text {Chi}(a+b x)}{2 a^2}-\frac {\text {Chi}(a+b x)}{2 x^2}+\frac {b^2 \text {Chi}(b x) \sinh (a)}{2 a}+\frac {b^2 \cosh (a) \text {Shi}(b x)}{2 a}-\frac {b^2 \sinh (a) \text {Shi}(b x)}{2 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.72 \[ \int \frac {\text {Chi}(a+b x)}{x^3} \, dx=\frac {\left (-a^2+b^2 x^2\right ) \text {Chi}(a+b x)+b^2 x^2 \text {Chi}(b x) (-\cosh (a)+a \sinh (a))+b x (-a \cosh (a+b x)+b x (a \cosh (a)-\sinh (a)) \text {Shi}(b x))}{2 a^2 x^2} \]

[In]

Integrate[CoshIntegral[a + b*x]/x^3,x]

[Out]

((-a^2 + b^2*x^2)*CoshIntegral[a + b*x] + b^2*x^2*CoshIntegral[b*x]*(-Cosh[a] + a*Sinh[a]) + b*x*(-(a*Cosh[a +
b*x]) + b*x*(a*Cosh[a] - Sinh[a])*SinhIntegral[b*x]))/(2*a^2*x^2)

Maple [F]

\[\int \frac {\operatorname {Chi}\left (b x +a \right )}{x^{3}}d x\]

[In]

int(Chi(b*x+a)/x^3,x)

[Out]

int(Chi(b*x+a)/x^3,x)

Fricas [F]

\[ \int \frac {\text {Chi}(a+b x)}{x^3} \, dx=\int { \frac {{\rm Chi}\left (b x + a\right )}{x^{3}} \,d x } \]

[In]

integrate(Chi(b*x+a)/x^3,x, algorithm="fricas")

[Out]

integral(cosh_integral(b*x + a)/x^3, x)

Sympy [F]

\[ \int \frac {\text {Chi}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {Chi}\left (a + b x\right )}{x^{3}}\, dx \]

[In]

integrate(Chi(b*x+a)/x**3,x)

[Out]

Integral(Chi(a + b*x)/x**3, x)

Maxima [F]

\[ \int \frac {\text {Chi}(a+b x)}{x^3} \, dx=\int { \frac {{\rm Chi}\left (b x + a\right )}{x^{3}} \,d x } \]

[In]

integrate(Chi(b*x+a)/x^3,x, algorithm="maxima")

[Out]

integrate(Chi(b*x + a)/x^3, x)

Giac [F]

\[ \int \frac {\text {Chi}(a+b x)}{x^3} \, dx=\int { \frac {{\rm Chi}\left (b x + a\right )}{x^{3}} \,d x } \]

[In]

integrate(Chi(b*x+a)/x^3,x, algorithm="giac")

[Out]

integrate(Chi(b*x + a)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Chi}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {coshint}\left (a+b\,x\right )}{x^3} \,d x \]

[In]

int(coshint(a + b*x)/x^3,x)

[Out]

int(coshint(a + b*x)/x^3, x)