Integrand size = 16, antiderivative size = 60 \[ \int \frac {\cosh (a+b x) \sinh (a+b x)}{x^3} \, dx=-\frac {b \cosh (2 a+2 b x)}{2 x}+b^2 \text {Chi}(2 b x) \sinh (2 a)-\frac {\sinh (2 a+2 b x)}{4 x^2}+b^2 \cosh (2 a) \text {Shi}(2 b x) \]
-1/2*b*cosh(2*b*x+2*a)/x+b^2*cosh(2*a)*Shi(2*b*x)+b^2*Chi(2*b*x)*sinh(2*a) -1/4*sinh(2*b*x+2*a)/x^2
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02 \[ \int \frac {\cosh (a+b x) \sinh (a+b x)}{x^3} \, dx=\frac {1}{2} \left (2 b^2 \text {Chi}(2 b x) \sinh (2 a)-\frac {2 b x \cosh (2 (a+b x))+\sinh (2 (a+b x))}{2 x^2}+2 b^2 \cosh (2 a) \text {Shi}(2 b x)\right ) \]
(2*b^2*CoshIntegral[2*b*x]*Sinh[2*a] - (2*b*x*Cosh[2*(a + b*x)] + Sinh[2*( a + b*x)])/(2*x^2) + 2*b^2*Cosh[2*a]*SinhIntegral[2*b*x])/2
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.28, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5971, 27, 3042, 26, 3778, 3042, 3778, 26, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}
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.
\(\displaystyle \int \frac {\sinh (a+b x) \cosh (a+b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \int \frac {\sinh (2 a+2 b x)}{2 x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int -\frac {i \sin (2 i a+2 i b x)}{x^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {1}{2} i \int \frac {\sin (2 i a+2 i b x)}{x^3}dx\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle -\frac {1}{2} i \left (i b \int \frac {\cosh (2 a+2 b x)}{x^2}dx-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} i \left (i b \int \frac {\sin \left (2 i a+2 i b x+\frac {\pi }{2}\right )}{x^2}dx-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}+2 i b \int -\frac {i \sinh (2 a+2 b x)}{x}dx\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (2 b \int \frac {\sinh (2 a+2 b x)}{x}dx-\frac {\cosh (2 a+2 b x)}{x}\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}+2 b \int -\frac {i \sin (2 i a+2 i b x)}{x}dx\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}-2 i b \int \frac {\sin (2 i a+2 i b x)}{x}dx\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}-2 i b \left (i \sinh (2 a) \int \frac {\cosh (2 b x)}{x}dx+\cosh (2 a) \int \frac {i \sinh (2 b x)}{x}dx\right )\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}-2 i b \left (i \sinh (2 a) \int \frac {\cosh (2 b x)}{x}dx+i \cosh (2 a) \int \frac {\sinh (2 b x)}{x}dx\right )\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}-2 i b \left (i \sinh (2 a) \int \frac {\sin \left (2 i b x+\frac {\pi }{2}\right )}{x}dx+i \cosh (2 a) \int -\frac {i \sin (2 i b x)}{x}dx\right )\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}-2 i b \left (i \sinh (2 a) \int \frac {\sin \left (2 i b x+\frac {\pi }{2}\right )}{x}dx+\cosh (2 a) \int \frac {\sin (2 i b x)}{x}dx\right )\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}-2 i b \left (i \sinh (2 a) \int \frac {\sin \left (2 i b x+\frac {\pi }{2}\right )}{x}dx+i \cosh (2 a) \text {Shi}(2 b x)\right )\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle -\frac {1}{2} i \left (i b \left (-\frac {\cosh (2 a+2 b x)}{x}-2 i b (i \sinh (2 a) \text {Chi}(2 b x)+i \cosh (2 a) \text {Shi}(2 b x))\right )-\frac {i \sinh (2 a+2 b x)}{2 x^2}\right )\) |
(-1/2*I)*(((-1/2*I)*Sinh[2*a + 2*b*x])/x^2 + I*b*(-(Cosh[2*a + 2*b*x]/x) - (2*I)*b*(I*CoshIntegral[2*b*x]*Sinh[2*a] + I*Cosh[2*a]*SinhIntegral[2*b*x ])))
3.3.57.3.1 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[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[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 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]
Time = 1.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.48
method | result | size |
risch | \(-\frac {-4 \,{\mathrm e}^{-2 a} \operatorname {Ei}_{1}\left (2 b x \right ) x^{2} b^{2}+4 \,{\mathrm e}^{2 a} \operatorname {Ei}_{1}\left (-2 b x \right ) x^{2} b^{2}+2 \,{\mathrm e}^{-2 b x -2 a} b x +2 \,{\mathrm e}^{2 b x +2 a} b x -{\mathrm e}^{-2 b x -2 a}+{\mathrm e}^{2 b x +2 a}}{8 x^{2}}\) | \(89\) |
-1/8*(-4*exp(-2*a)*Ei(1,2*b*x)*x^2*b^2+4*exp(2*a)*Ei(1,-2*b*x)*x^2*b^2+2*e xp(-2*b*x-2*a)*b*x+2*exp(2*b*x+2*a)*b*x-exp(-2*b*x-2*a)+exp(2*b*x+2*a))/x^ 2
Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.73 \[ \int \frac {\cosh (a+b x) \sinh (a+b x)}{x^3} \, dx=-\frac {b x \cosh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{2} - {\left (b^{2} x^{2} {\rm Ei}\left (2 \, b x\right ) - b^{2} x^{2} {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} x^{2} {\rm Ei}\left (2 \, b x\right ) + b^{2} x^{2} {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{2 \, x^{2}} \]
-1/2*(b*x*cosh(b*x + a)^2 + b*x*sinh(b*x + a)^2 - (b^2*x^2*Ei(2*b*x) - b^2 *x^2*Ei(-2*b*x))*cosh(2*a) + cosh(b*x + a)*sinh(b*x + a) - (b^2*x^2*Ei(2*b *x) + b^2*x^2*Ei(-2*b*x))*sinh(2*a))/x^2
\[ \int \frac {\cosh (a+b x) \sinh (a+b x)}{x^3} \, dx=\int \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{x^{3}}\, dx \]
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.50 \[ \int \frac {\cosh (a+b x) \sinh (a+b x)}{x^3} \, dx=b^{2} e^{\left (-2 \, a\right )} \Gamma \left (-2, 2 \, b x\right ) - b^{2} e^{\left (2 \, a\right )} \Gamma \left (-2, -2 \, b x\right ) \]
Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.43 \[ \int \frac {\cosh (a+b x) \sinh (a+b x)}{x^3} \, dx=\frac {4 \, b^{2} x^{2} {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - 4 \, b^{2} x^{2} {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - 2 \, b x e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b x e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, x^{2}} \]
1/8*(4*b^2*x^2*Ei(2*b*x)*e^(2*a) - 4*b^2*x^2*Ei(-2*b*x)*e^(-2*a) - 2*b*x*e ^(2*b*x + 2*a) - 2*b*x*e^(-2*b*x - 2*a) - e^(2*b*x + 2*a) + e^(-2*b*x - 2* a))/x^2
Timed out. \[ \int \frac {\cosh (a+b x) \sinh (a+b x)}{x^3} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )}{x^3} \,d x \]