Integrand size = 9, antiderivative size = 25 \[ \int \sinh (b x) \text {Shi}(b x) \, dx=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \] Output:
cosh(b*x)*Shi(b*x)/b-1/2*Shi(2*b*x)/b
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sinh (b x) \text {Shi}(b x) \, dx=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \] Input:
Integrate[Sinh[b*x]*SinhIntegral[b*x],x]
Output:
(Cosh[b*x]*SinhIntegral[b*x])/b - SinhIntegral[2*b*x]/(2*b)
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {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.
\(\displaystyle \int \text {Shi}(b x) \sinh (b x) \, dx\) |
\(\Big \downarrow \) 7094 |
\(\displaystyle \frac {\text {Shi}(b x) \cosh (b x)}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\cosh (b x) \sinh (b x)}{x}dx}{b}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{2 x}dx}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{x}dx}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int -\frac {i \sin (2 i b x)}{x}dx}{2 b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\text {Shi}(b x) \cosh (b x)}{b}+\frac {i \int \frac {\sin (2 i b x)}{x}dx}{2 b}\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b}\) |
Input:
Int[Sinh[b*x]*SinhIntegral[b*x],x]
Output:
(Cosh[b*x]*SinhIntegral[b*x])/b - SinhIntegral[2*b*x]/(2*b)
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[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]
Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )-\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b}\) | \(22\) |
default | \(\frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )-\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b}\) | \(22\) |
Input:
int(sinh(b*x)*Shi(b*x),x,method=_RETURNVERBOSE)
Output:
1/b*(cosh(b*x)*Shi(b*x)-1/2*Shi(2*b*x))
\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \] Input:
integrate(sinh(b*x)*Shi(b*x),x, algorithm="fricas")
Output:
integral(sinh(b*x)*sinh_integral(b*x), x)
\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int \sinh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \] Input:
integrate(sinh(b*x)*Shi(b*x),x)
Output:
Integral(sinh(b*x)*Shi(b*x), x)
\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \] Input:
integrate(sinh(b*x)*Shi(b*x),x, algorithm="maxima")
Output:
integrate(Shi(b*x)*sinh(b*x), x)
\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \] Input:
integrate(sinh(b*x)*Shi(b*x),x, algorithm="giac")
Output:
integrate(Shi(b*x)*sinh(b*x), x)
Timed out. \[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int \mathrm {sinhint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \] Input:
int(sinhint(b*x)*sinh(b*x),x)
Output:
int(sinhint(b*x)*sinh(b*x), x)
\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int \mathit {shi} \left (b x \right ) \sinh \left (b x \right )d x \] Input:
int(sinh(b*x)*Shi(b*x),x)
Output:
int(shi(b*x)*sinh(b*x),x)