3.1.46 \(\int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx\) [46]

3.1.46.1 Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=-\frac {b \cosh (2 b x)}{4 x}-\frac {b \sinh ^2(b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \text {Int}\left (\frac {\cosh (b x) \text {Shi}(b x)}{x},x\right ) \]

output

1/2*b^2*CannotIntegrate(cosh(b*x)*Shi(b*x)/x,x)-1/4*b*cosh(2*b*x)/x-1/2*co 
sh(b*x)*Shi(b*x)/x^2+b^2*Shi(2*b*x)-1/2*b*Shi(b*x)*sinh(b*x)/x-1/2*b*sinh( 
b*x)^2/x-1/8*sinh(2*b*x)/x^2
 

3.1.46.2 Mathematica [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \]

input

Integrate[(Cosh[b*x]*SinhIntegral[b*x])/x^3,x]
 

output

Integrate[(Cosh[b*x]*SinhIntegral[b*x])/x^3, x]
 

3.1.46.3 Rubi [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {7104, 27, 5971, 27, 3042, 26, 3778, 3042, 3778, 26, 3042, 26, 3779, 7098, 27, 3042, 25, 3794, 27, 3042, 26, 3779, 7299}

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.

\(\displaystyle \int \frac {\text {Shi}(b x) \cosh (b x)}{x^3} \, dx\)

\(\Big \downarrow \) 7104

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{2} b \int \frac {\cosh (b x) \sinh (b x)}{b x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{2} \int \frac {\cosh (b x) \sinh (b x)}{x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{2} \int \frac {\sinh (2 b x)}{2 x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{4} \int \frac {\sinh (2 b x)}{x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{4} \int -\frac {i \sin (2 i b x)}{x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \int \frac {\sin (2 i b x)}{x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \int \frac {\cosh (2 b x)}{x^2}dx-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \int \frac {\sin \left (2 i b x+\frac {\pi }{2}\right )}{x^2}dx-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \left (-\frac {\cosh (2 b x)}{x}+2 i b \int -\frac {i \sinh (2 b x)}{x}dx\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \left (2 b \int \frac {\sinh (2 b x)}{x}dx-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \left (-\frac {\cosh (2 b x)}{x}+2 b \int -\frac {i \sin (2 i b x)}{x}dx\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \left (-\frac {\cosh (2 b x)}{x}-2 i b \int \frac {\sin (2 i b x)}{x}dx\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 7098

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \int \frac {\sinh ^2(b x)}{b x^2}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+\int \frac {\sinh ^2(b x)}{x^2}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+\int -\frac {\sin (i b x)^2}{x^2}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx-\int \frac {\sin (i b x)^2}{x^2}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3794

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx-2 i b \int \frac {i \sinh (2 b x)}{2 x}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \int \frac {\sinh (2 b x)}{x}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \int -\frac {i \sin (2 i b x)}{x}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx-i b \int \frac {\sin (2 i b x)}{x}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \text {Shi}(2 b x)-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \text {Shi}(2 b x)-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

input

Int[(Cosh[b*x]*SinhIntegral[b*x])/x^3,x]
 

output

$Aborted
 

3.1.46.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

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] - Simp[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 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 3794

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si 
mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 
)))   Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n 
- 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & 
& LtQ[m, -1]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 7098

Int[((e_.) + (f_.)*(x_))^(m_)*Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + 
 (d_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^(m + 1)*Sinh[a + b*x]*(SinhIntegr 
al[c + d*x]/(f*(m + 1))), x] + (-Simp[b/(f*(m + 1))   Int[(e + f*x)^(m + 1) 
*Cosh[a + b*x]*SinhIntegral[c + d*x], x], x] - Simp[d/(f*(m + 1))   Int[(e 
+ f*x)^(m + 1)*Sinh[a + b*x]*(Sinh[c + d*x]/(c + d*x)), x], x]) /; FreeQ[{a 
, b, c, d, e, f}, x] && ILtQ[m, -1]
 

rule 7104

Int[Cosh[(a_.) + (b_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*SinhIntegral[(c_.) 
+ (d_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^(m + 1)*Cosh[a + b*x]*(SinhInteg 
ral[c + d*x]/(f*(m + 1))), x] + (-Simp[b/(f*(m + 1))   Int[(e + f*x)^(m + 1 
)*Sinh[a + b*x]*SinhIntegral[c + d*x], x], x] - Simp[d/(f*(m + 1))   Int[(e 
 + f*x)^(m + 1)*Cosh[a + b*x]*(Sinh[c + d*x]/(c + d*x)), x], x]) /; FreeQ[{ 
a, b, c, d, e, f}, x] && ILtQ[m, -1]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.1.46.4 Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )}{x^{3}}d x\]

input

int(cosh(b*x)*Shi(b*x)/x^3,x)
 

output

int(cosh(b*x)*Shi(b*x)/x^3,x)
 

3.1.46.5 Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]

input

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="fricas")
 

output

integral(cosh(b*x)*sinh_integral(b*x)/x^3, x)
 

3.1.46.6 Sympy [N/A]

Not integrable

Time = 2.73 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{3}}\, dx \]

input

integrate(cosh(b*x)*Shi(b*x)/x**3,x)
 

output

Integral(cosh(b*x)*Shi(b*x)/x**3, x)
 

3.1.46.7 Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]

input

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="maxima")
 

output

integrate(Shi(b*x)*cosh(b*x)/x^3, x)
 

3.1.46.8 Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]

input

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="giac")
 

output

integrate(Shi(b*x)*cosh(b*x)/x^3, x)
 

3.1.46.9 Mupad [N/A]

Not integrable

Time = 5.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^3} \,d x \]

input

int((sinhint(b*x)*cosh(b*x))/x^3,x)
 

output

int((sinhint(b*x)*cosh(b*x))/x^3, x)