Integrand size = 12, antiderivative size = 12 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=-\frac {b \cosh (2 b x)}{4 x}-\frac {b \sinh ^2(b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \text {Int}\left (\frac {\cosh (b x) \text {Shi}(b x)}{x},x\right ) \]
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
Not integrable
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \]
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 |
\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \text {Shi}(2 b x)-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\) |
3.1.46.3.1 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[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[((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]
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[((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]
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]
Not integrable
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00
\[\int \frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )}{x^{3}}d x\]
Not integrable
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]
Not integrable
Time = 2.73 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{3}}\, dx \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]
Not integrable
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \]
Not integrable
Time = 5.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^3} \,d x \]