Integrand size = 18, antiderivative size = 119 \[ \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^3} \, dx=-\frac {b \cosh (a+b x)}{8 x}-\frac {3 b \cosh (3 a+3 b x)}{8 x}+\frac {1}{8} b^2 \text {Chi}(b x) \sinh (a)+\frac {9}{8} b^2 \text {Chi}(3 b x) \sinh (3 a)-\frac {\sinh (a+b x)}{8 x^2}-\frac {\sinh (3 a+3 b x)}{8 x^2}+\frac {1}{8} b^2 \cosh (a) \text {Shi}(b x)+\frac {9}{8} b^2 \cosh (3 a) \text {Shi}(3 b x) \] Output:
-1/8*b*cosh(b*x+a)/x-3/8*b*cosh(3*b*x+3*a)/x+1/8*b^2*Chi(b*x)*sinh(a)+9/8* b^2*Chi(3*b*x)*sinh(3*a)-1/8*sinh(b*x+a)/x^2-1/8*sinh(3*b*x+3*a)/x^2+1/8*b ^2*cosh(a)*Shi(b*x)+9/8*b^2*cosh(3*a)*Shi(3*b*x)
Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^3} \, dx=-\frac {b x \cosh (a+b x)+3 b x \cosh (3 (a+b x))-b^2 x^2 \text {Chi}(b x) \sinh (a)-9 b^2 x^2 \text {Chi}(3 b x) \sinh (3 a)+\sinh (a+b x)+\sinh (3 (a+b x))-b^2 x^2 \cosh (a) \text {Shi}(b x)-9 b^2 x^2 \cosh (3 a) \text {Shi}(3 b x)}{8 x^2} \] Input:
Integrate[(Cosh[a + b*x]^2*Sinh[a + b*x])/x^3,x]
Output:
-1/8*(b*x*Cosh[a + b*x] + 3*b*x*Cosh[3*(a + b*x)] - b^2*x^2*CoshIntegral[b *x]*Sinh[a] - 9*b^2*x^2*CoshIntegral[3*b*x]*Sinh[3*a] + Sinh[a + b*x] + Si nh[3*(a + b*x)] - b^2*x^2*Cosh[a]*SinhIntegral[b*x] - 9*b^2*x^2*Cosh[3*a]* SinhIntegral[3*b*x])/x^2
Time = 0.46 (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.
\(\displaystyle \int \frac {\sinh (a+b x) \cosh ^2(a+b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \int \left (\frac {\sinh (a+b x)}{4 x^3}+\frac {\sinh (3 a+3 b x)}{4 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} b^2 \sinh (a) \text {Chi}(b x)+\frac {9}{8} b^2 \sinh (3 a) \text {Chi}(3 b x)+\frac {1}{8} b^2 \cosh (a) \text {Shi}(b x)+\frac {9}{8} b^2 \cosh (3 a) \text {Shi}(3 b x)-\frac {\sinh (a+b x)}{8 x^2}-\frac {\sinh (3 a+3 b x)}{8 x^2}-\frac {b \cosh (a+b x)}{8 x}-\frac {3 b \cosh (3 a+3 b x)}{8 x}\) |
Input:
Int[(Cosh[a + b*x]^2*Sinh[a + b*x])/x^3,x]
Output:
-1/8*(b*Cosh[a + b*x])/x - (3*b*Cosh[3*a + 3*b*x])/(8*x) + (b^2*CoshIntegr al[b*x]*Sinh[a])/8 + (9*b^2*CoshIntegral[3*b*x]*Sinh[3*a])/8 - Sinh[a + b* x]/(8*x^2) - Sinh[3*a + 3*b*x]/(8*x^2) + (b^2*Cosh[a]*SinhIntegral[b*x])/8 + (9*b^2*Cosh[3*a]*SinhIntegral[3*b*x])/8
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 = 4.07 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.34
method | result | size |
risch | \(-\frac {-{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right ) b^{2} x^{2}+{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right ) b^{2} x^{2}-9 \,{\mathrm e}^{-3 a} \operatorname {expIntegral}_{1}\left (3 b x \right ) b^{2} x^{2}+9 \,{\mathrm e}^{3 a} \operatorname {expIntegral}_{1}\left (-3 b x \right ) b^{2} x^{2}+x b \,{\mathrm e}^{b x +a}+b x \,{\mathrm e}^{-b x -a}+3 b x \,{\mathrm e}^{-3 b x -3 a}+3 b x \,{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}-{\mathrm e}^{-b x -a}-{\mathrm e}^{-3 b x -3 a}+{\mathrm e}^{3 b x +3 a}}{16 x^{2}}\) | \(159\) |
Input:
int(cosh(b*x+a)^2*sinh(b*x+a)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/16*(-exp(-a)*Ei(1,b*x)*b^2*x^2+exp(a)*Ei(1,-b*x)*b^2*x^2-9*exp(-3*a)*Ei (1,3*b*x)*b^2*x^2+9*exp(3*a)*Ei(1,-3*b*x)*b^2*x^2+x*b*exp(b*x+a)+b*x*exp(- b*x-a)+3*b*x*exp(-3*b*x-3*a)+3*b*x*exp(3*b*x+3*a)+exp(b*x+a)-exp(-b*x-a)-e xp(-3*b*x-3*a)+exp(3*b*x+3*a))/x^2
Time = 0.10 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.65 \[ \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^3} \, dx=-\frac {6 \, b x \cosh \left (b x + a\right )^{3} + 18 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )^{3} - 9 \, {\left (b^{2} x^{2} {\rm Ei}\left (3 \, b x\right ) - b^{2} x^{2} {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) - {\left (b^{2} x^{2} {\rm Ei}\left (b x\right ) - b^{2} x^{2} {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) - 9 \, {\left (b^{2} x^{2} {\rm Ei}\left (3 \, b x\right ) + b^{2} x^{2} {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) - {\left (b^{2} x^{2} {\rm Ei}\left (b x\right ) + b^{2} x^{2} {\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right )}{16 \, x^{2}} \] Input:
integrate(cosh(b*x+a)^2*sinh(b*x+a)/x^3,x, algorithm="fricas")
Output:
-1/16*(6*b*x*cosh(b*x + a)^3 + 18*b*x*cosh(b*x + a)*sinh(b*x + a)^2 + 2*b* x*cosh(b*x + a) + 2*sinh(b*x + a)^3 - 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*(3*cos h(b*x + a)^2 + 1)*sinh(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))/x^2
\[ \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^3} \, dx=\int \frac {\sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x^{3}}\, dx \] Input:
integrate(cosh(b*x+a)**2*sinh(b*x+a)/x**3,x)
Output:
Integral(sinh(a + b*x)*cosh(a + b*x)**2/x**3, x)
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.49 \[ \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^3} \, dx=\frac {9}{8} \, b^{2} e^{\left (-3 \, a\right )} \Gamma \left (-2, 3 \, b x\right ) + \frac {1}{8} \, b^{2} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - \frac {1}{8} \, b^{2} e^{a} \Gamma \left (-2, -b x\right ) - \frac {9}{8} \, b^{2} e^{\left (3 \, a\right )} \Gamma \left (-2, -3 \, b x\right ) \] Input:
integrate(cosh(b*x+a)^2*sinh(b*x+a)/x^3,x, algorithm="maxima")
Output:
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 = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.31 \[ \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^3} \, dx=\frac {9 \, b^{2} x^{2} {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, b^{2} x^{2} {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + b^{2} x^{2} {\rm Ei}\left (b x\right ) e^{a} - 3 \, b x e^{\left (3 \, b x + 3 \, a\right )} - b x e^{\left (b x + a\right )} - b x e^{\left (-b x - a\right )} - 3 \, b x e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{16 \, x^{2}} \] Input:
integrate(cosh(b*x+a)^2*sinh(b*x+a)/x^3,x, algorithm="giac")
Output:
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. \[ \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^3} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{x^3} \,d x \] Input:
int((cosh(a + b*x)^2*sinh(a + b*x))/x^3,x)
Output:
int((cosh(a + b*x)^2*sinh(a + b*x))/x^3, x)
Time = 0.21 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.55 \[ \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^3} \, dx=\frac {-e^{3 b x +2 a} \mathit {ei} \left (-b x \right ) b^{2} x^{2}-9 e^{3 b x} \mathit {ei} \left (-3 b x \right ) b^{2} x^{2}+e^{3 b x +4 a} \mathit {ei} \left (b x \right ) b^{2} x^{2}+9 e^{3 b x +6 a} \mathit {ei} \left (3 b x \right ) b^{2} x^{2}-3 e^{6 b x +6 a} b x -e^{6 b x +6 a}-e^{4 b x +4 a} b x -e^{4 b x +4 a}-e^{2 b x +2 a} b x +e^{2 b x +2 a}-3 b x +1}{16 e^{3 b x +3 a} x^{2}} \] Input:
int(cosh(b*x+a)^2*sinh(b*x+a)/x^3,x)
Output:
( - e**(2*a + 3*b*x)*ei( - b*x)*b**2*x**2 - 9*e**(3*b*x)*ei( - 3*b*x)*b**2 *x**2 + e**(4*a + 3*b*x)*ei(b*x)*b**2*x**2 + 9*e**(6*a + 3*b*x)*ei(3*b*x)* b**2*x**2 - 3*e**(6*a + 6*b*x)*b*x - e**(6*a + 6*b*x) - e**(4*a + 4*b*x)*b *x - e**(4*a + 4*b*x) - e**(2*a + 2*b*x)*b*x + e**(2*a + 2*b*x) - 3*b*x + 1)/(16*e**(3*a + 3*b*x)*x**2)