\(\int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx\) [60]

Optimal result

Integrand size = 18, antiderivative size = 154 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx=\frac {\cosh (a+b x)}{12 x^3}+\frac {b^2 \cosh (a+b x)}{24 x}-\frac {\cosh (3 a+3 b x)}{12 x^3}-\frac {3 b^2 \cosh (3 a+3 b x)}{8 x}-\frac {1}{24} b^3 \text {Chi}(b x) \sinh (a)+\frac {9}{8} b^3 \text {Chi}(3 b x) \sinh (3 a)+\frac {b \sinh (a+b x)}{24 x^2}-\frac {b \sinh (3 a+3 b x)}{8 x^2}-\frac {1}{24} b^3 \cosh (a) \text {Shi}(b x)+\frac {9}{8} b^3 \cosh (3 a) \text {Shi}(3 b x) \] Output:

1/12*cosh(b*x+a)/x^3+1/24*b^2*cosh(b*x+a)/x-1/12*cosh(3*b*x+3*a)/x^3-3/8*b 
^2*cosh(3*b*x+3*a)/x-1/24*b^3*Chi(b*x)*sinh(a)+9/8*b^3*Chi(3*b*x)*sinh(3*a 
)+1/24*b*sinh(b*x+a)/x^2-1/8*b*sinh(3*b*x+3*a)/x^2-1/24*b^3*cosh(a)*Shi(b* 
x)+9/8*b^3*cosh(3*a)*Shi(3*b*x)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx=\frac {2 \cosh (a+b x)+b^2 x^2 \cosh (a+b x)-2 \cosh (3 (a+b x))-9 b^2 x^2 \cosh (3 (a+b x))-b^3 x^3 \text {Chi}(b x) \sinh (a)+27 b^3 x^3 \text {Chi}(3 b x) \sinh (3 a)+b x \sinh (a+b x)-3 b x \sinh (3 (a+b x))-b^3 x^3 \cosh (a) \text {Shi}(b x)+27 b^3 x^3 \cosh (3 a) \text {Shi}(3 b x)}{24 x^3} \] Input:

Integrate[(Cosh[a + b*x]*Sinh[a + b*x]^2)/x^4,x]
 

Output:

(2*Cosh[a + b*x] + b^2*x^2*Cosh[a + b*x] - 2*Cosh[3*(a + b*x)] - 9*b^2*x^2 
*Cosh[3*(a + b*x)] - b^3*x^3*CoshIntegral[b*x]*Sinh[a] + 27*b^3*x^3*CoshIn 
tegral[3*b*x]*Sinh[3*a] + b*x*Sinh[a + b*x] - 3*b*x*Sinh[3*(a + b*x)] - b^ 
3*x^3*Cosh[a]*SinhIntegral[b*x] + 27*b^3*x^3*Cosh[3*a]*SinhIntegral[3*b*x] 
)/(24*x^3)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 154, 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.

\(\displaystyle \int \frac {\sinh ^2(a+b x) \cosh (a+b x)}{x^4} \, dx\)

\(\Big \downarrow \) 5971

\(\displaystyle \int \left (\frac {\cosh (3 a+3 b x)}{4 x^4}-\frac {\cosh (a+b x)}{4 x^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{24} b^3 \sinh (a) \text {Chi}(b x)+\frac {9}{8} b^3 \sinh (3 a) \text {Chi}(3 b x)-\frac {1}{24} b^3 \cosh (a) \text {Shi}(b x)+\frac {9}{8} b^3 \cosh (3 a) \text {Shi}(3 b x)+\frac {b^2 \cosh (a+b x)}{24 x}-\frac {3 b^2 \cosh (3 a+3 b x)}{8 x}+\frac {\cosh (a+b x)}{12 x^3}-\frac {\cosh (3 a+3 b x)}{12 x^3}+\frac {b \sinh (a+b x)}{24 x^2}-\frac {b \sinh (3 a+3 b x)}{8 x^2}\)

Input:

Int[(Cosh[a + b*x]*Sinh[a + b*x]^2)/x^4,x]
 

Output:

Cosh[a + b*x]/(12*x^3) + (b^2*Cosh[a + b*x])/(24*x) - Cosh[3*a + 3*b*x]/(1 
2*x^3) - (3*b^2*Cosh[3*a + 3*b*x])/(8*x) - (b^3*CoshIntegral[b*x]*Sinh[a]) 
/24 + (9*b^3*CoshIntegral[3*b*x]*Sinh[3*a])/8 + (b*Sinh[a + b*x])/(24*x^2) 
 - (b*Sinh[3*a + 3*b*x])/(8*x^2) - (b^3*Cosh[a]*SinhIntegral[b*x])/24 + (9 
*b^3*Cosh[3*a]*SinhIntegral[3*b*x])/8
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5971

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]
 

Maple [A] (verified)

Time = 5.02 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.49

method	result	size
risch	\(-\frac {-27 \,{\mathrm e}^{-3 a} \operatorname {expIntegral}_{1}\left (3 b x \right ) b^{3} x^{3}+{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right ) b^{3} x^{3}-{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right ) b^{3} x^{3}+27 \,{\mathrm e}^{3 a} \operatorname {expIntegral}_{1}\left (-3 b x \right ) b^{3} x^{3}+9 b^{2} x^{2} {\mathrm e}^{-3 b x -3 a}-b^{2} x^{2} {\mathrm e}^{-b x -a}+9 b^{2} x^{2} {\mathrm e}^{3 b x +3 a}-b^{2} x^{2} {\mathrm e}^{b x +a}-3 b x \,{\mathrm e}^{-3 b x -3 a}+b x \,{\mathrm e}^{-b x -a}+3 b x \,{\mathrm e}^{3 b x +3 a}-x b \,{\mathrm e}^{b x +a}+2 \,{\mathrm e}^{-3 b x -3 a}-2 \,{\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{3 b x +3 a}-2 \,{\mathrm e}^{b x +a}}{48 x^{3}}\)	\(229\)

Input:

int(cosh(b*x+a)*sinh(b*x+a)^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/48*(-27*exp(-3*a)*Ei(1,3*b*x)*b^3*x^3+exp(-a)*Ei(1,b*x)*b^3*x^3-exp(a)* 
Ei(1,-b*x)*b^3*x^3+27*exp(3*a)*Ei(1,-3*b*x)*b^3*x^3+9*b^2*x^2*exp(-3*b*x-3 
*a)-b^2*x^2*exp(-b*x-a)+9*b^2*x^2*exp(3*b*x+3*a)-b^2*x^2*exp(b*x+a)-3*b*x* 
exp(-3*b*x-3*a)+b*x*exp(-b*x-a)+3*b*x*exp(3*b*x+3*a)-x*b*exp(b*x+a)+2*exp( 
-3*b*x-3*a)-2*exp(-b*x-a)+2*exp(3*b*x+3*a)-2*exp(b*x+a))/x^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.45 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx=-\frac {6 \, b x \sinh \left (b x + a\right )^{3} + 2 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 6 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) - 27 \, {\left (b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + {\left (b^{3} x^{3} {\rm Ei}\left (b x\right ) - b^{3} x^{3} {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + 2 \, {\left (9 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right ) - 27 \, {\left (b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) + {\left (b^{3} x^{3} {\rm Ei}\left (b x\right ) + b^{3} x^{3} {\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right )}{48 \, x^{3}} \] Input:

integrate(cosh(b*x+a)*sinh(b*x+a)^2/x^4,x, algorithm="fricas")
 

Output:

-1/48*(6*b*x*sinh(b*x + a)^3 + 2*(9*b^2*x^2 + 2)*cosh(b*x + a)^3 + 6*(9*b^ 
2*x^2 + 2)*cosh(b*x + a)*sinh(b*x + a)^2 - 2*(b^2*x^2 + 2)*cosh(b*x + a) - 
 27*(b^3*x^3*Ei(3*b*x) - b^3*x^3*Ei(-3*b*x))*cosh(3*a) + (b^3*x^3*Ei(b*x) 
- b^3*x^3*Ei(-b*x))*cosh(a) + 2*(9*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a 
) - 27*(b^3*x^3*Ei(3*b*x) + b^3*x^3*Ei(-3*b*x))*sinh(3*a) + (b^3*x^3*Ei(b* 
x) + b^3*x^3*Ei(-b*x))*sinh(a))/x^3
 

Sympy [F]

\[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx=\int \frac {\sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{x^{4}}\, dx \] Input:

integrate(cosh(b*x+a)*sinh(b*x+a)**2/x**4,x)
 

Output:

Integral(sinh(a + b*x)**2*cosh(a + b*x)/x**4, x)
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.38 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx=-\frac {27}{8} \, b^{3} e^{\left (-3 \, a\right )} \Gamma \left (-3, 3 \, b x\right ) + \frac {1}{8} \, b^{3} e^{\left (-a\right )} \Gamma \left (-3, b x\right ) - \frac {1}{8} \, b^{3} e^{a} \Gamma \left (-3, -b x\right ) + \frac {27}{8} \, b^{3} e^{\left (3 \, a\right )} \Gamma \left (-3, -3 \, b x\right ) \] Input:

integrate(cosh(b*x+a)*sinh(b*x+a)^2/x^4,x, algorithm="maxima")
 

Output:

-27/8*b^3*e^(-3*a)*gamma(-3, 3*b*x) + 1/8*b^3*e^(-a)*gamma(-3, b*x) - 1/8* 
b^3*e^a*gamma(-3, -b*x) + 27/8*b^3*e^(3*a)*gamma(-3, -3*b*x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.44 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx=\frac {27 \, b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 27 \, b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - b^{3} x^{3} {\rm Ei}\left (b x\right ) e^{a} - 9 \, b^{2} x^{2} e^{\left (3 \, b x + 3 \, a\right )} + b^{2} x^{2} e^{\left (b x + a\right )} + b^{2} x^{2} e^{\left (-b x - a\right )} - 9 \, b^{2} x^{2} e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, b x e^{\left (3 \, b x + 3 \, a\right )} + b x e^{\left (b x + a\right )} - b x e^{\left (-b x - a\right )} + 3 \, b x e^{\left (-3 \, b x - 3 \, a\right )} - 2 \, e^{\left (3 \, b x + 3 \, a\right )} + 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, x^{3}} \] Input:

integrate(cosh(b*x+a)*sinh(b*x+a)^2/x^4,x, algorithm="giac")
 

Output:

1/48*(27*b^3*x^3*Ei(3*b*x)*e^(3*a) + b^3*x^3*Ei(-b*x)*e^(-a) - 27*b^3*x^3* 
Ei(-3*b*x)*e^(-3*a) - b^3*x^3*Ei(b*x)*e^a - 9*b^2*x^2*e^(3*b*x + 3*a) + b^ 
2*x^2*e^(b*x + a) + b^2*x^2*e^(-b*x - a) - 9*b^2*x^2*e^(-3*b*x - 3*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) - 2*e^(3*b*x + 3*a) + 2*e^(b*x + a) + 2*e^(-b*x - a) - 2*e^(-3*b*x 
 - 3*a))/x^3
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^4} \,d x \] Input:

int((cosh(a + b*x)*sinh(a + b*x)^2)/x^4,x)
 

Output:

int((cosh(a + b*x)*sinh(a + b*x)^2)/x^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.60 \[ \int \frac {\cosh (a+b x) \sinh ^2(a+b x)}{x^4} \, dx=\frac {e^{3 b x +2 a} \mathit {ei} \left (-b x \right ) b^{3} x^{3}-27 e^{3 b x} \mathit {ei} \left (-3 b x \right ) b^{3} x^{3}-e^{3 b x +4 a} \mathit {ei} \left (b x \right ) b^{3} x^{3}+27 e^{3 b x +6 a} \mathit {ei} \left (3 b x \right ) b^{3} x^{3}-9 e^{6 b x +6 a} b^{2} x^{2}-3 e^{6 b x +6 a} b x -2 e^{6 b x +6 a}+e^{4 b x +4 a} b^{2} x^{2}+e^{4 b x +4 a} b x +2 e^{4 b x +4 a}+e^{2 b x +2 a} b^{2} x^{2}-e^{2 b x +2 a} b x +2 e^{2 b x +2 a}-9 b^{2} x^{2}+3 b x -2}{48 e^{3 b x +3 a} x^{3}} \] Input:

int(cosh(b*x+a)*sinh(b*x+a)^2/x^4,x)
 

Output:

(e**(2*a + 3*b*x)*ei( - b*x)*b**3*x**3 - 27*e**(3*b*x)*ei( - 3*b*x)*b**3*x 
**3 - e**(4*a + 3*b*x)*ei(b*x)*b**3*x**3 + 27*e**(6*a + 3*b*x)*ei(3*b*x)*b 
**3*x**3 - 9*e**(6*a + 6*b*x)*b**2*x**2 - 3*e**(6*a + 6*b*x)*b*x - 2*e**(6 
*a + 6*b*x) + e**(4*a + 4*b*x)*b**2*x**2 + e**(4*a + 4*b*x)*b*x + 2*e**(4* 
a + 4*b*x) + e**(2*a + 2*b*x)*b**2*x**2 - e**(2*a + 2*b*x)*b*x + 2*e**(2*a 
 + 2*b*x) - 9*b**2*x**2 + 3*b*x - 2)/(48*e**(3*a + 3*b*x)*x**3)