Integrand size = 20, antiderivative size = 169 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^4} \, dx=\frac {b \cosh (2 a+2 b x)}{32 x^2}-\frac {b \cosh (6 a+6 b x)}{32 x^2}-\frac {1}{8} b^3 \cosh (2 a) \text {Chi}(2 b x)+\frac {9}{8} b^3 \cosh (6 a) \text {Chi}(6 b x)+\frac {\sinh (2 a+2 b x)}{32 x^3}+\frac {b^2 \sinh (2 a+2 b x)}{16 x}-\frac {\sinh (6 a+6 b x)}{96 x^3}-\frac {3 b^2 \sinh (6 a+6 b x)}{16 x}-\frac {1}{8} b^3 \sinh (2 a) \text {Shi}(2 b x)+\frac {9}{8} b^3 \sinh (6 a) \text {Shi}(6 b x) \]
-1/8*b^3*Chi(2*b*x)*cosh(2*a)+9/8*b^3*Chi(6*b*x)*cosh(6*a)+1/32*b*cosh(2*b *x+2*a)/x^2-1/32*b*cosh(6*b*x+6*a)/x^2-1/8*b^3*Shi(2*b*x)*sinh(2*a)+9/8*b^ 3*Shi(6*b*x)*sinh(6*a)+1/32*sinh(2*b*x+2*a)/x^3+1/16*b^2*sinh(2*b*x+2*a)/x -1/96*sinh(6*b*x+6*a)/x^3-3/16*b^2*sinh(6*b*x+6*a)/x
Time = 0.21 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.89 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^4} \, dx=-\frac {-3 b x \cosh (2 (a+b x))+3 b x \cosh (6 (a+b x))+12 b^3 x^3 \cosh (2 a) \text {Chi}(2 b x)-108 b^3 x^3 \cosh (6 a) \text {Chi}(6 b x)-3 \sinh (2 (a+b x))-6 b^2 x^2 \sinh (2 (a+b x))+\sinh (6 (a+b x))+18 b^2 x^2 \sinh (6 (a+b x))+12 b^3 x^3 \sinh (2 a) \text {Shi}(2 b x)-108 b^3 x^3 \sinh (6 a) \text {Shi}(6 b x)}{96 x^3} \]
-1/96*(-3*b*x*Cosh[2*(a + b*x)] + 3*b*x*Cosh[6*(a + b*x)] + 12*b^3*x^3*Cos h[2*a]*CoshIntegral[2*b*x] - 108*b^3*x^3*Cosh[6*a]*CoshIntegral[6*b*x] - 3 *Sinh[2*(a + b*x)] - 6*b^2*x^2*Sinh[2*(a + b*x)] + Sinh[6*(a + b*x)] + 18* b^2*x^2*Sinh[6*(a + b*x)] + 12*b^3*x^3*Sinh[2*a]*SinhIntegral[2*b*x] - 108 *b^3*x^3*Sinh[6*a]*SinhIntegral[6*b*x])/x^3
Time = 0.51 (sec) , antiderivative size = 169, 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.
\(\displaystyle \int \frac {\sinh ^3(a+b x) \cosh ^3(a+b x)}{x^4} \, dx\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \int \left (\frac {\sinh (6 a+6 b x)}{32 x^4}-\frac {3 \sinh (2 a+2 b x)}{32 x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{8} b^3 \cosh (2 a) \text {Chi}(2 b x)+\frac {9}{8} b^3 \cosh (6 a) \text {Chi}(6 b x)-\frac {1}{8} b^3 \sinh (2 a) \text {Shi}(2 b x)+\frac {9}{8} b^3 \sinh (6 a) \text {Shi}(6 b x)+\frac {b^2 \sinh (2 a+2 b x)}{16 x}-\frac {3 b^2 \sinh (6 a+6 b x)}{16 x}+\frac {\sinh (2 a+2 b x)}{32 x^3}-\frac {\sinh (6 a+6 b x)}{96 x^3}+\frac {b \cosh (2 a+2 b x)}{32 x^2}-\frac {b \cosh (6 a+6 b x)}{32 x^2}\) |
(b*Cosh[2*a + 2*b*x])/(32*x^2) - (b*Cosh[6*a + 6*b*x])/(32*x^2) - (b^3*Cos h[2*a]*CoshIntegral[2*b*x])/8 + (9*b^3*Cosh[6*a]*CoshIntegral[6*b*x])/8 + Sinh[2*a + 2*b*x]/(32*x^3) + (b^2*Sinh[2*a + 2*b*x])/(16*x) - Sinh[6*a + 6 *b*x]/(96*x^3) - (3*b^2*Sinh[6*a + 6*b*x])/(16*x) - (b^3*Sinh[2*a]*SinhInt egral[2*b*x])/8 + (9*b^3*Sinh[6*a]*SinhIntegral[6*b*x])/8
3.4.33.3.1 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 = 49.26 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.43
method | result | size |
risch | \(\frac {12 \,{\mathrm e}^{2 a} \operatorname {Ei}_{1}\left (-2 b x \right ) x^{3} b^{3}-108 \,{\mathrm e}^{-6 a} \operatorname {Ei}_{1}\left (6 b x \right ) x^{3} b^{3}+12 \,{\mathrm e}^{-2 a} \operatorname {Ei}_{1}\left (2 b x \right ) x^{3} b^{3}-108 \,{\mathrm e}^{6 a} \operatorname {Ei}_{1}\left (-6 b x \right ) x^{3} b^{3}+6 \,{\mathrm e}^{2 b x +2 a} b^{2} x^{2}+18 \,{\mathrm e}^{-6 b x -6 a} b^{2} x^{2}-6 \,{\mathrm e}^{-2 b x -2 a} b^{2} x^{2}-18 \,{\mathrm e}^{6 b x +6 a} b^{2} x^{2}+3 \,{\mathrm e}^{2 b x +2 a} b x -3 \,{\mathrm e}^{-6 b x -6 a} b x +3 \,{\mathrm e}^{-2 b x -2 a} b x -3 \,{\mathrm e}^{6 b x +6 a} b x +3 \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{-6 b x -6 a}-3 \,{\mathrm e}^{-2 b x -2 a}-{\mathrm e}^{6 b x +6 a}}{192 x^{3}}\) | \(241\) |
1/192*(12*exp(2*a)*Ei(1,-2*b*x)*x^3*b^3-108*exp(-6*a)*Ei(1,6*b*x)*x^3*b^3+ 12*exp(-2*a)*Ei(1,2*b*x)*x^3*b^3-108*exp(6*a)*Ei(1,-6*b*x)*x^3*b^3+6*exp(2 *b*x+2*a)*b^2*x^2+18*exp(-6*b*x-6*a)*b^2*x^2-6*exp(-2*b*x-2*a)*b^2*x^2-18* exp(6*b*x+6*a)*b^2*x^2+3*exp(2*b*x+2*a)*b*x-3*exp(-6*b*x-6*a)*b*x+3*exp(-2 *b*x-2*a)*b*x-3*exp(6*b*x+6*a)*b*x+3*exp(2*b*x+2*a)+exp(-6*b*x-6*a)-3*exp( -2*b*x-2*a)-exp(6*b*x+6*a))/x^3
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (149) = 298\).
Time = 0.25 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.86 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^4} \, dx=-\frac {3 \, b x \cosh \left (b x + a\right )^{6} + 45 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + 3 \, b x \sinh \left (b x + a\right )^{6} + 20 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 6 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - 3 \, b x \cosh \left (b x + a\right )^{2} + 3 \, {\left (15 \, b x \cosh \left (b x + a\right )^{4} - b x\right )} \sinh \left (b x + a\right )^{2} - 54 \, {\left (b^{3} x^{3} {\rm Ei}\left (6 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-6 \, b x\right )\right )} \cosh \left (6 \, a\right ) + 6 \, {\left (b^{3} x^{3} {\rm Ei}\left (2 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + 6 \, {\left ({\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{5} - {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 54 \, {\left (b^{3} x^{3} {\rm Ei}\left (6 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-6 \, b x\right )\right )} \sinh \left (6 \, a\right ) + 6 \, {\left (b^{3} x^{3} {\rm Ei}\left (2 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{96 \, x^{3}} \]
-1/96*(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 + 20*(18*b^2*x^2 + 1)*cosh(b*x + a)^3*sinh(b*x + a)^3 + 6*(18*b^2*x^2 + 1)*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 - 54*(b^3*x^3*Ei(6*b*x ) + b^3*x^3*Ei(-6*b*x))*cosh(6*a) + 6*(b^3*x^3*Ei(2*b*x) + b^3*x^3*Ei(-2*b *x))*cosh(2*a) + 6*((18*b^2*x^2 + 1)*cosh(b*x + a)^5 - (2*b^2*x^2 + 1)*cos h(b*x + a))*sinh(b*x + a) - 54*(b^3*x^3*Ei(6*b*x) - b^3*x^3*Ei(-6*b*x))*si nh(6*a) + 6*(b^3*x^3*Ei(2*b*x) - b^3*x^3*Ei(-2*b*x))*sinh(2*a))/x^3
\[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^4} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x^{4}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.36 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^4} \, dx=\frac {27}{8} \, b^{3} e^{\left (-6 \, a\right )} \Gamma \left (-3, 6 \, b x\right ) - \frac {3}{8} \, b^{3} e^{\left (-2 \, a\right )} \Gamma \left (-3, 2 \, b x\right ) - \frac {3}{8} \, b^{3} e^{\left (2 \, a\right )} \Gamma \left (-3, -2 \, b x\right ) + \frac {27}{8} \, b^{3} e^{\left (6 \, a\right )} \Gamma \left (-3, -6 \, b x\right ) \]
27/8*b^3*e^(-6*a)*gamma(-3, 6*b*x) - 3/8*b^3*e^(-2*a)*gamma(-3, 2*b*x) - 3 /8*b^3*e^(2*a)*gamma(-3, -2*b*x) + 27/8*b^3*e^(6*a)*gamma(-3, -6*b*x)
Time = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.40 \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^4} \, dx=\frac {108 \, b^{3} x^{3} {\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - 12 \, b^{3} x^{3} {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - 12 \, b^{3} x^{3} {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} + 108 \, b^{3} x^{3} {\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} - 18 \, b^{2} x^{2} e^{\left (6 \, b x + 6 \, a\right )} + 6 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (-2 \, b x - 2 \, a\right )} + 18 \, b^{2} x^{2} e^{\left (-6 \, b x - 6 \, a\right )} - 3 \, b x e^{\left (6 \, b x + 6 \, a\right )} + 3 \, b x e^{\left (2 \, b x + 2 \, a\right )} + 3 \, b x e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, b x e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (6 \, b x + 6 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 3 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{192 \, x^{3}} \]
1/192*(108*b^3*x^3*Ei(6*b*x)*e^(6*a) - 12*b^3*x^3*Ei(2*b*x)*e^(2*a) - 12*b ^3*x^3*Ei(-2*b*x)*e^(-2*a) + 108*b^3*x^3*Ei(-6*b*x)*e^(-6*a) - 18*b^2*x^2* e^(6*b*x + 6*a) + 6*b^2*x^2*e^(2*b*x + 2*a) - 6*b^2*x^2*e^(-2*b*x - 2*a) + 18*b^2*x^2*e^(-6*b*x - 6*a) - 3*b*x*e^(6*b*x + 6*a) + 3*b*x*e^(2*b*x + 2* a) + 3*b*x*e^(-2*b*x - 2*a) - 3*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^3
Timed out. \[ \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x^4} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{x^4} \,d x \]