Integrand size = 20, antiderivative size = 131 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^3} \, dx=\frac {3 b \cosh (2 a+2 b x)}{32 x}-\frac {3 b \cosh (6 a+6 b x)}{32 x}-\frac {3}{16} b^2 \text {Chi}(2 b x) \sinh (2 a)+\frac {9}{16} b^2 \text {Chi}(6 b x) \sinh (6 a)+\frac {3 \sinh (2 a+2 b x)}{64 x^2}-\frac {\sinh (6 a+6 b x)}{64 x^2}-\frac {3}{16} b^2 \cosh (2 a) \text {Shi}(2 b x)+\frac {9}{16} b^2 \cosh (6 a) \text {Shi}(6 b x) \]
3/32*b*cosh(2*b*x+2*a)/x-3/32*b*cosh(6*b*x+6*a)/x-3/16*b^2*cosh(2*a)*Shi(2 *b*x)+9/16*b^2*cosh(6*a)*Shi(6*b*x)-3/16*b^2*Chi(2*b*x)*sinh(2*a)+9/16*b^2 *Chi(6*b*x)*sinh(6*a)+3/64*sinh(2*b*x+2*a)/x^2-1/64*sinh(6*b*x+6*a)/x^2
Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^3} \, dx=-\frac {-6 b x \cosh (2 (a+b x))+6 b x \cosh (6 (a+b x))+12 b^2 x^2 \text {Chi}(2 b x) \sinh (2 a)-36 b^2 x^2 \text {Chi}(6 b x) \sinh (6 a)-3 \sinh (2 (a+b x))+\sinh (6 (a+b x))+12 b^2 x^2 \cosh (2 a) \text {Shi}(2 b x)-36 b^2 x^2 \cosh (6 a) \text {Shi}(6 b x)}{64 x^2} \]
-1/64*(-6*b*x*Cosh[2*(a + b*x)] + 6*b*x*Cosh[6*(a + b*x)] + 12*b^2*x^2*Cos hIntegral[2*b*x]*Sinh[2*a] - 36*b^2*x^2*CoshIntegral[6*b*x]*Sinh[6*a] - 3* Sinh[2*(a + b*x)] + Sinh[6*(a + b*x)] + 12*b^2*x^2*Cosh[2*a]*SinhIntegral[ 2*b*x] - 36*b^2*x^2*Cosh[6*a]*SinhIntegral[6*b*x])/x^2
Time = 0.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5971, 2009}
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.
\(\displaystyle \int \frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \int \left (\frac {\sinh (6 a+6 b x)}{32 x^3}-\frac {3 \sinh (2 a+2 b x)}{32 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3}{16} b^2 \sinh (2 a) \text {Chi}(2 b x)+\frac {9}{16} b^2 \sinh (6 a) \text {Chi}(6 b x)-\frac {3}{16} b^2 \cosh (2 a) \text {Shi}(2 b x)+\frac {9}{16} b^2 \cosh (6 a) \text {Shi}(6 b x)+\frac {3 \sinh (2 a+2 b x)}{64 x^2}-\frac {\sinh (6 a+6 b x)}{64 x^2}+\frac {3 b \cosh (2 a+2 b x)}{32 x}-\frac {3 b \cosh (6 a+6 b x)}{32 x}\) |
(3*b*Cosh[2*a + 2*b*x])/(32*x) - (3*b*Cosh[6*a + 6*b*x])/(32*x) - (3*b^2*C oshIntegral[2*b*x]*Sinh[2*a])/16 + (9*b^2*CoshIntegral[6*b*x]*Sinh[6*a])/1 6 + (3*Sinh[2*a + 2*b*x])/(64*x^2) - Sinh[6*a + 6*b*x]/(64*x^2) - (3*b^2*C osh[2*a]*SinhIntegral[2*b*x])/16 + (9*b^2*Cosh[6*a]*SinhIntegral[6*b*x])/1 6
3.4.32.3.1 Defintions of rubi rules used
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 = 32.46 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.32
method | result | size |
risch | \(-\frac {-36 \,{\mathrm e}^{-6 a} \operatorname {Ei}_{1}\left (6 b x \right ) x^{2} b^{2}+12 \,{\mathrm e}^{-2 a} \operatorname {Ei}_{1}\left (2 b x \right ) x^{2} b^{2}-12 \,{\mathrm e}^{2 a} \operatorname {Ei}_{1}\left (-2 b x \right ) x^{2} b^{2}+36 \,{\mathrm e}^{6 a} \operatorname {Ei}_{1}\left (-6 b x \right ) x^{2} b^{2}+6 \,{\mathrm e}^{-6 b x -6 a} b x -6 \,{\mathrm e}^{-2 b x -2 a} b x -6 \,{\mathrm e}^{2 b x +2 a} b x +6 \,{\mathrm e}^{6 b x +6 a} b x -{\mathrm e}^{-6 b x -6 a}+3 \,{\mathrm e}^{-2 b x -2 a}-3 \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{6 b x +6 a}}{128 x^{2}}\) | \(173\) |
-1/128*(-36*exp(-6*a)*Ei(1,6*b*x)*x^2*b^2+12*exp(-2*a)*Ei(1,2*b*x)*x^2*b^2 -12*exp(2*a)*Ei(1,-2*b*x)*x^2*b^2+36*exp(6*a)*Ei(1,-6*b*x)*x^2*b^2+6*exp(- 6*b*x-6*a)*b*x-6*exp(-2*b*x-2*a)*b*x-6*exp(2*b*x+2*a)*b*x+6*exp(6*b*x+6*a) *b*x-exp(-6*b*x-6*a)+3*exp(-2*b*x-2*a)-3*exp(2*b*x+2*a)+exp(6*b*x+6*a))/x^ 2
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (115) = 230\).
Time = 0.26 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.09 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^3} \, dx=-\frac {3 \, b x \cosh \left (b x + a\right )^{6} + 45 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + 3 \, b x \sinh \left (b x + a\right )^{6} + 10 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - 3 \, b x \cosh \left (b x + a\right )^{2} + 3 \, {\left (15 \, b x \cosh \left (b x + a\right )^{4} - b x\right )} \sinh \left (b x + a\right )^{2} - 9 \, {\left (b^{2} x^{2} {\rm Ei}\left (6 \, b x\right ) - b^{2} x^{2} {\rm Ei}\left (-6 \, b x\right )\right )} \cosh \left (6 \, a\right ) + 3 \, {\left (b^{2} x^{2} {\rm Ei}\left (2 \, b x\right ) - b^{2} x^{2} {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + 3 \, {\left (\cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 9 \, {\left (b^{2} x^{2} {\rm Ei}\left (6 \, b x\right ) + b^{2} x^{2} {\rm Ei}\left (-6 \, b x\right )\right )} \sinh \left (6 \, a\right ) + 3 \, {\left (b^{2} x^{2} {\rm Ei}\left (2 \, b x\right ) + b^{2} x^{2} {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{32 \, x^{2}} \]
-1/32*(3*b*x*cosh(b*x + a)^6 + 45*b*x*cosh(b*x + a)^2*sinh(b*x + a)^4 + 3* b*x*sinh(b*x + a)^6 + 10*cosh(b*x + a)^3*sinh(b*x + a)^3 + 3*cosh(b*x + a) *sinh(b*x + a)^5 - 3*b*x*cosh(b*x + a)^2 + 3*(15*b*x*cosh(b*x + a)^4 - b*x )*sinh(b*x + a)^2 - 9*(b^2*x^2*Ei(6*b*x) - b^2*x^2*Ei(-6*b*x))*cosh(6*a) + 3*(b^2*x^2*Ei(2*b*x) - b^2*x^2*Ei(-2*b*x))*cosh(2*a) + 3*(cosh(b*x + a)^5 - cosh(b*x + a))*sinh(b*x + a) - 9*(b^2*x^2*Ei(6*b*x) + b^2*x^2*Ei(-6*b*x ))*sinh(6*a) + 3*(b^2*x^2*Ei(2*b*x) + b^2*x^2*Ei(-2*b*x))*sinh(2*a))/x^2
\[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^3} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x^{3}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.47 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^3} \, dx=\frac {9}{16} \, b^{2} e^{\left (-6 \, a\right )} \Gamma \left (-2, 6 \, b x\right ) - \frac {3}{16} \, b^{2} e^{\left (-2 \, a\right )} \Gamma \left (-2, 2 \, b x\right ) + \frac {3}{16} \, b^{2} e^{\left (2 \, a\right )} \Gamma \left (-2, -2 \, b x\right ) - \frac {9}{16} \, b^{2} e^{\left (6 \, a\right )} \Gamma \left (-2, -6 \, b x\right ) \]
9/16*b^2*e^(-6*a)*gamma(-2, 6*b*x) - 3/16*b^2*e^(-2*a)*gamma(-2, 2*b*x) + 3/16*b^2*e^(2*a)*gamma(-2, -2*b*x) - 9/16*b^2*e^(6*a)*gamma(-2, -6*b*x)
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.28 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^3} \, dx=\frac {36 \, b^{2} x^{2} {\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - 12 \, b^{2} x^{2} {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + 12 \, b^{2} x^{2} {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - 36 \, b^{2} x^{2} {\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} - 6 \, b x e^{\left (6 \, b x + 6 \, a\right )} + 6 \, b x e^{\left (2 \, b x + 2 \, a\right )} + 6 \, b x e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, b x e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (6 \, b x + 6 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 3 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{128 \, x^{2}} \]
1/128*(36*b^2*x^2*Ei(6*b*x)*e^(6*a) - 12*b^2*x^2*Ei(2*b*x)*e^(2*a) + 12*b^ 2*x^2*Ei(-2*b*x)*e^(-2*a) - 36*b^2*x^2*Ei(-6*b*x)*e^(-6*a) - 6*b*x*e^(6*b* x + 6*a) + 6*b*x*e^(2*b*x + 2*a) + 6*b*x*e^(-2*b*x - 2*a) - 6*b*x*e^(-6*b* x - 6*a) - e^(6*b*x + 6*a) + 3*e^(2*b*x + 2*a) - 3*e^(-2*b*x - 2*a) + e^(- 6*b*x - 6*a))/x^2
Timed out. \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^3} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{x^3} \,d x \]