Integrand size = 18, antiderivative size = 119 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^3} \, dx=\frac {\cosh (a+b x)}{8 x^2}-\frac {\cosh (3 a+3 b x)}{8 x^2}-\frac {1}{8} b^2 \cosh (a) \text {Chi}(b x)+\frac {9}{8} b^2 \cosh (3 a) \text {Chi}(3 b x)+\frac {b \sinh (a+b x)}{8 x}-\frac {3 b \sinh (3 a+3 b x)}{8 x}-\frac {1}{8} b^2 \sinh (a) \text {Shi}(b x)+\frac {9}{8} b^2 \sinh (3 a) \text {Shi}(3 b x) \]
-1/8*b^2*Chi(b*x)*cosh(a)+9/8*b^2*Chi(3*b*x)*cosh(3*a)+1/8*cosh(b*x+a)/x^2 -1/8*cosh(3*b*x+3*a)/x^2-1/8*b^2*Shi(b*x)*sinh(a)+9/8*b^2*Shi(3*b*x)*sinh( 3*a)+1/8*b*sinh(b*x+a)/x-3/8*b*sinh(3*b*x+3*a)/x
Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^3} \, dx=\frac {\cosh (a+b x)-\cosh (3 (a+b x))-b^2 x^2 \cosh (a) \text {Chi}(b x)+9 b^2 x^2 \cosh (3 a) \text {Chi}(3 b x)+b x \sinh (a+b x)-3 b x \sinh (3 (a+b x))-b^2 x^2 \sinh (a) \text {Shi}(b x)+9 b^2 x^2 \sinh (3 a) \text {Shi}(3 b x)}{8 x^2} \]
(Cosh[a + b*x] - Cosh[3*(a + b*x)] - b^2*x^2*Cosh[a]*CoshIntegral[b*x] + 9 *b^2*x^2*Cosh[3*a]*CoshIntegral[3*b*x] + b*x*Sinh[a + b*x] - 3*b*x*Sinh[3* (a + b*x)] - b^2*x^2*Sinh[a]*SinhIntegral[b*x] + 9*b^2*x^2*Sinh[3*a]*SinhI ntegral[3*b*x])/(8*x^2)
Time = 0.41 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
\(\displaystyle \int \frac {\sinh ^2(a+b x) \cosh (a+b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \int \left (\frac {\cosh (3 a+3 b x)}{4 x^3}-\frac {\cosh (a+b x)}{4 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{8} b^2 \cosh (a) \text {Chi}(b x)+\frac {9}{8} b^2 \cosh (3 a) \text {Chi}(3 b x)-\frac {1}{8} b^2 \sinh (a) \text {Shi}(b x)+\frac {9}{8} b^2 \sinh (3 a) \text {Shi}(3 b x)+\frac {\cosh (a+b x)}{8 x^2}-\frac {\cosh (3 a+3 b x)}{8 x^2}+\frac {b \sinh (a+b x)}{8 x}-\frac {3 b \sinh (3 a+3 b x)}{8 x}\) |
Cosh[a + b*x]/(8*x^2) - Cosh[3*a + 3*b*x]/(8*x^2) - (b^2*Cosh[a]*CoshInteg ral[b*x])/8 + (9*b^2*Cosh[3*a]*CoshIntegral[3*b*x])/8 + (b*Sinh[a + b*x])/ (8*x) - (3*b*Sinh[3*a + 3*b*x])/(8*x) - (b^2*Sinh[a]*SinhIntegral[b*x])/8 + (9*b^2*Sinh[3*a]*SinhIntegral[3*b*x])/8
3.3.87.3.1 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 = 2.73 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\frac {-9 \,{\mathrm e}^{3 a} \operatorname {Ei}_{1}\left (-3 b x \right ) x^{2} b^{2}-9 \,{\mathrm e}^{-3 a} \operatorname {Ei}_{1}\left (3 b x \right ) x^{2} b^{2}+{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right ) x^{2} b^{2}+{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right ) x^{2} b^{2}+{\mathrm e}^{b x +a} b x -3 \,{\mathrm e}^{3 b x +3 a} b x +3 \,{\mathrm e}^{-3 b x -3 a} b x -{\mathrm e}^{-b x -a} b x +{\mathrm e}^{b x +a}-{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{-3 b x -3 a}+{\mathrm e}^{-b x -a}}{16 x^{2}}\) | \(159\) |
1/16*(-9*exp(3*a)*Ei(1,-3*b*x)*x^2*b^2-9*exp(-3*a)*Ei(1,3*b*x)*x^2*b^2+exp (-a)*Ei(1,b*x)*x^2*b^2+exp(a)*Ei(1,-b*x)*x^2*b^2+exp(b*x+a)*b*x-3*exp(3*b* x+3*a)*b*x+3*exp(-3*b*x-3*a)*b*x-exp(-b*x-a)*b*x+exp(b*x+a)-exp(3*b*x+3*a) -exp(-3*b*x-3*a)+exp(-b*x-a))/x^2
Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.64 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^3} \, dx=-\frac {6 \, b x \sinh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 9 \, {\left (b^{2} x^{2} {\rm Ei}\left (3 \, b x\right ) + b^{2} x^{2} {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + {\left (b^{2} x^{2} {\rm Ei}\left (b x\right ) + b^{2} x^{2} {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + 2 \, {\left (9 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right ) - 9 \, {\left (b^{2} x^{2} {\rm Ei}\left (3 \, b x\right ) - b^{2} x^{2} {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) + {\left (b^{2} x^{2} {\rm Ei}\left (b x\right ) - b^{2} x^{2} {\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right ) - 2 \, \cosh \left (b x + a\right )}{16 \, x^{2}} \]
-1/16*(6*b*x*sinh(b*x + a)^3 + 2*cosh(b*x + a)^3 + 6*cosh(b*x + a)*sinh(b* x + a)^2 - 9*(b^2*x^2*Ei(3*b*x) + b^2*x^2*Ei(-3*b*x))*cosh(3*a) + (b^2*x^2 *Ei(b*x) + b^2*x^2*Ei(-b*x))*cosh(a) + 2*(9*b*x*cosh(b*x + a)^2 - b*x)*sin h(b*x + a) - 9*(b^2*x^2*Ei(3*b*x) - b^2*x^2*Ei(-3*b*x))*sinh(3*a) + (b^2*x ^2*Ei(b*x) - b^2*x^2*Ei(-b*x))*sinh(a) - 2*cosh(b*x + a))/x^2
\[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^3} \, dx=\int \frac {\sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{x^{3}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.49 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^3} \, dx=-\frac {9}{8} \, b^{2} e^{\left (-3 \, a\right )} \Gamma \left (-2, 3 \, b x\right ) + \frac {1}{8} \, b^{2} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) + \frac {1}{8} \, b^{2} e^{a} \Gamma \left (-2, -b x\right ) - \frac {9}{8} \, b^{2} e^{\left (3 \, a\right )} \Gamma \left (-2, -3 \, b x\right ) \]
-9/8*b^2*e^(-3*a)*gamma(-2, 3*b*x) + 1/8*b^2*e^(-a)*gamma(-2, b*x) + 1/8*b ^2*e^a*gamma(-2, -b*x) - 9/8*b^2*e^(3*a)*gamma(-2, -3*b*x)
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.31 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^3} \, dx=\frac {9 \, b^{2} x^{2} {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 9 \, b^{2} x^{2} {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - b^{2} x^{2} {\rm Ei}\left (b x\right ) e^{a} - 3 \, b x e^{\left (3 \, b x + 3 \, a\right )} + b x e^{\left (b x + a\right )} - b x e^{\left (-b x - a\right )} + 3 \, b x e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (3 \, b x + 3 \, a\right )} + e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{16 \, x^{2}} \]
1/16*(9*b^2*x^2*Ei(3*b*x)*e^(3*a) - b^2*x^2*Ei(-b*x)*e^(-a) + 9*b^2*x^2*Ei (-3*b*x)*e^(-3*a) - b^2*x^2*Ei(b*x)*e^a - 3*b*x*e^(3*b*x + 3*a) + b*x*e^(b *x + a) - b*x*e^(-b*x - a) + 3*b*x*e^(-3*b*x - 3*a) - e^(3*b*x + 3*a) + e^ (b*x + a) + e^(-b*x - a) - e^(-3*b*x - 3*a))/x^2
Timed out. \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^3} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^3} \,d x \]