Integrand size = 12, antiderivative size = 44 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^2} \, dx=b \text {Chi}(2 b x)-\frac {\sinh (2 b x)}{2 x}-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2 \]
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^2} \, dx=b \text {Chi}(2 b x)-\frac {\sinh (2 b x)}{2 x}-\frac {\cosh (b x) \text {Shi}(b x)}{x}+\frac {1}{2} b \text {Shi}(b x)^2 \]
b*CoshIntegral[2*b*x] - Sinh[2*b*x]/(2*x) - (Cosh[b*x]*SinhIntegral[b*x])/ x + (b*SinhIntegral[b*x]^2)/2
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {7104, 27, 5971, 27, 3042, 26, 3778, 3042, 3782, 7237}
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.
\(\displaystyle \int \frac {\text {Shi}(b x) \cosh (b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 7104 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx+b \int \frac {\cosh (b x) \sinh (b x)}{b x^2}dx-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx+\int \frac {\cosh (b x) \sinh (b x)}{x^2}dx-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx+\int \frac {\sinh (2 b x)}{2 x^2}dx-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx+\frac {1}{2} \int \frac {\sinh (2 b x)}{x^2}dx-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx+\frac {1}{2} \int -\frac {i \sin (2 i b x)}{x^2}dx-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx-\frac {1}{2} i \int \frac {\sin (2 i b x)}{x^2}dx-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx-\frac {1}{2} i \left (2 i b \int \frac {\cosh (2 b x)}{x}dx-\frac {i \sinh (2 b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx-\frac {1}{2} i \left (2 i b \int \frac {\sin \left (2 i b x+\frac {\pi }{2}\right )}{x}dx-\frac {i \sinh (2 b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle b \int \frac {\sinh (b x) \text {Shi}(b x)}{x}dx-\frac {1}{2} i \left (2 i b \text {Chi}(2 b x)-\frac {i \sinh (2 b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle -\frac {1}{2} i \left (2 i b \text {Chi}(2 b x)-\frac {i \sinh (2 b x)}{x}\right )+\frac {1}{2} b \text {Shi}(b x)^2-\frac {\text {Shi}(b x) \cosh (b x)}{x}\) |
(-1/2*I)*((2*I)*b*CoshIntegral[2*b*x] - (I*Sinh[2*b*x])/x) - (Cosh[b*x]*Si nhIntegral[b*x])/x + (b*SinhIntegral[b*x]^2)/2
3.1.47.3.1 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[((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]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> 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]
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[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]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
\[\int \frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )}{x^{2}}d x\]
\[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{2}} \,d x } \]
\[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^2} \, dx=\int \frac {\cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{2}} \,d x } \]
\[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^2} \, dx=\int \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^2} \,d x \]