Integrand size = 10, antiderivative size = 62 \[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\frac {x}{2 b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x \sinh (b x) \text {Shi}(b x)}{b}+\frac {\text {Shi}(2 b x)}{2 b^2} \]
1/2*x/b-cosh(b*x)*Shi(b*x)/b^2+1/2*Shi(2*b*x)/b^2-1/2*cosh(b*x)*sinh(b*x)/ b^2+x*Shi(b*x)*sinh(b*x)/b
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\frac {2 b x-\sinh (2 b x)+4 (-\cosh (b x)+b x \sinh (b x)) \text {Shi}(b x)+2 \text {Shi}(2 b x)}{4 b^2} \]
(2*b*x - Sinh[2*b*x] + 4*(-Cosh[b*x] + b*x*Sinh[b*x])*SinhIntegral[b*x] + 2*SinhIntegral[2*b*x])/(4*b^2)
Time = 0.48 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {7102, 27, 3042, 25, 3115, 24, 7094, 27, 5971, 27, 3042, 26, 3779}
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.
\(\displaystyle \int x \text {Shi}(b x) \cosh (b x) \, dx\) |
\(\Big \downarrow \) 7102 |
\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\int \frac {\sinh ^2(b x)}{b}dx+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\frac {\int \sinh ^2(b x)dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\frac {\int -\sin (i b x)^2dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {\int \sin (i b x)^2dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {\frac {\int 1dx}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 7094 |
\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b x}dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\cosh (b x) \sinh (b x)}{x}dx}{b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{2 x}dx}{b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int -\frac {i \sin (2 i b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}+\frac {i \int \frac {\sin (2 i b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {x \text {Shi}(b x) \sinh (b x)}{b}-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b}}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\) |
(x/2 - (Cosh[b*x]*Sinh[b*x])/(2*b))/b + (x*Sinh[b*x]*SinhIntegral[b*x])/b - ((Cosh[b*x]*SinhIntegral[b*x])/b - SinhIntegral[2*b*x]/(2*b))/b
3.1.50.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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]
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]
Int[Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cosh[a + b*x]*(SinhIntegral[c + d*x]/b), x] - Simp[d/b Int[Cosh[a + b*x]*(Sinh[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
Int[Cosh[(a_.) + (b_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^m*Sinh[a + b*x]*(SinhIntegral[c + d*x]/b), x] + (-Simp[d/b Int[(e + f*x)^m*Sinh[a + b*x]*(Sinh[c + d*x]/( c + d*x)), x], x] - Simp[f*(m/b) Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*Sinh Integral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
Time = 0.80 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {\operatorname {Shi}\left (b x \right ) \left (b x \sinh \left (b x \right )-\cosh \left (b x \right )\right )-\frac {\cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b x}{2}+\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b^{2}}\) | \(46\) |
default | \(\frac {\operatorname {Shi}\left (b x \right ) \left (b x \sinh \left (b x \right )-\cosh \left (b x \right )\right )-\frac {\cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b x}{2}+\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b^{2}}\) | \(46\) |
\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int { x {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int x \cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \]
\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int { x {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int { x {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
Timed out. \[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int x\,\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \]