\(\int \text {csch}^4(c+b x) \sinh ^2(a+b x) \, dx\) [82]

Optimal result

Integrand size = 17, antiderivative size = 84 \[ \int \text {csch}^4(c+b x) \sinh ^2(a+b x) \, dx=-\frac {\cosh (2 (a-c)) \coth (c+b x)}{b}+\frac {\coth (c+b x) \sinh ^2(a-c)}{b}-\frac {\coth ^3(c+b x) \sinh ^2(a-c)}{3 b}-\frac {\text {csch}^2(c+b x) \sinh (2 (a-c))}{2 b} \] Output:

-cosh(2*a-2*c)*coth(b*x+c)/b+coth(b*x+c)*sinh(a-c)^2/b-1/3*coth(b*x+c)^3*s 
inh(a-c)^2/b-1/2*csch(b*x+c)^2*sinh(2*a-2*c)/b
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95 \[ \int \text {csch}^4(c+b x) \sinh ^2(a+b x) \, dx=\frac {\text {csch}(c) \text {csch}^3(c+b x) (-3 \sinh (b x)-\sinh (2 a-4 c-3 b x)+3 \sinh (2 a-2 c-b x)-3 \sinh (2 a+b x)+\sinh (2 a+3 b x)+\sinh (2 c+3 b x))}{12 b} \] Input:

Integrate[Csch[c + b*x]^4*Sinh[a + b*x]^2,x]
 

Output:

(Csch[c]*Csch[c + b*x]^3*(-3*Sinh[b*x] - Sinh[2*a - 4*c - 3*b*x] + 3*Sinh[ 
2*a - 2*c - b*x] - 3*Sinh[2*a + b*x] + Sinh[2*a + 3*b*x] + Sinh[2*c + 3*b* 
x]))/(12*b)
 

Rubi [F]

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.

\(\displaystyle \int \sinh ^2(a+b x) \text {csch}^4(b x+c) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sinh ^2(a+b x) \text {csch}^4(b x+c)dx\)

Input:

Int[Csch[c + b*x]^4*Sinh[a + b*x]^2,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71

method	result	size
parallelrisch	\(\frac {\operatorname {sech}\left (\frac {b x}{2}+\frac {c}{2}\right )^{3} \operatorname {csch}\left (\frac {b x}{2}+\frac {c}{2}\right )^{3} \left (-\cosh \left (3 b x +3 c \right )-2 \cosh \left (3 b x +2 a +c \right )+3 \cosh \left (b x +c \right )\right )}{96 b}\)	\(60\)
risch	\(\frac {2 \left (3 \,{\mathrm e}^{4 b x +4 a +4 c}-3 \,{\mathrm e}^{2 b x +4 a +2 c}-3 \,{\mathrm e}^{2 b x +2 a +4 c}+{\mathrm e}^{4 a}+{\mathrm e}^{2 a +2 c}+{\mathrm e}^{4 c}\right ) {\mathrm e}^{4 a -2 c}}{3 \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{3} b}\)	\(94\)

Input:

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

Output:

1/96*sech(1/2*b*x+1/2*c)^3*csch(1/2*b*x+1/2*c)^3*(-cosh(3*b*x+3*c)-2*cosh( 
3*b*x+2*a+c)+3*cosh(b*x+c))/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (80) = 160\).

Time = 0.09 (sec) , antiderivative size = 497, normalized size of antiderivative = 5.92 \[ \int \text {csch}^4(c+b x) \sinh ^2(a+b x) \, dx=-\frac {2 \, {\left ({\left (5 \, \cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )^{2} + {\left (5 \, \cosh \left (-a + c\right )^{2} - 6 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + 5 \, \sinh \left (-a + c\right )^{2} + 1\right )} \sinh \left (b x + c\right )^{2} + {\left (5 \, \cosh \left (b x + c\right )^{2} - 3\right )} \sinh \left (-a + c\right )^{2} - 3 \, \cosh \left (-a + c\right )^{2} - 2 \, {\left (6 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right )\right )} \sinh \left (b x + c\right ) - 6 \, {\left (\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) - \cosh \left (-a + c\right )\right )} \sinh \left (-a + c\right ) - 3\right )}}{3 \, {\left (b \cosh \left (b x + c\right )^{4} \cosh \left (-a + c\right )^{2} - 4 \, b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} + {\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{4} + 4 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{3} + 3 \, b \cosh \left (-a + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} - 2 \, b \cosh \left (-a + c\right )^{2} - {\left (3 \, b \cosh \left (b x + c\right )^{2} - 2 \, b\right )} \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} - {\left (b \cosh \left (b x + c\right )^{4} - 4 \, b \cosh \left (b x + c\right )^{2} + 3 \, b\right )} \sinh \left (-a + c\right )^{2} + 4 \, {\left (b \cosh \left (b x + c\right )^{3} \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - {\left (b \cosh \left (b x + c\right )^{3} - b \cosh \left (b x + c\right )\right )} \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )\right )}} \] Input:

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

Output:

-2/3*((5*cosh(-a + c)^2 + 1)*cosh(b*x + c)^2 + (5*cosh(-a + c)^2 - 6*cosh( 
-a + c)*sinh(-a + c) + 5*sinh(-a + c)^2 + 1)*sinh(b*x + c)^2 + (5*cosh(b*x 
 + c)^2 - 3)*sinh(-a + c)^2 - 3*cosh(-a + c)^2 - 2*(6*cosh(b*x + c)*cosh(- 
a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)* 
cosh(b*x + c))*sinh(b*x + c) - 6*(cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + 
 c))*sinh(-a + c) - 3)/(b*cosh(b*x + c)^4*cosh(-a + c)^2 - 4*b*cosh(b*x + 
c)^2*cosh(-a + c)^2 + (b*cosh(-a + c)^2 - b*sinh(-a + c)^2)*sinh(b*x + c)^ 
4 + 4*(b*cosh(b*x + c)*cosh(-a + c)^2 - b*cosh(b*x + c)*sinh(-a + c)^2)*si 
nh(b*x + c)^3 + 3*b*cosh(-a + c)^2 + 2*(3*b*cosh(b*x + c)^2*cosh(-a + c)^2 
 - 2*b*cosh(-a + c)^2 - (3*b*cosh(b*x + c)^2 - 2*b)*sinh(-a + c)^2)*sinh(b 
*x + c)^2 - (b*cosh(b*x + c)^4 - 4*b*cosh(b*x + c)^2 + 3*b)*sinh(-a + c)^2 
 + 4*(b*cosh(b*x + c)^3*cosh(-a + c)^2 - b*cosh(b*x + c)*cosh(-a + c)^2 - 
(b*cosh(b*x + c)^3 - b*cosh(b*x + c))*sinh(-a + c)^2)*sinh(b*x + c))
 

Sympy [F]

\[ \int \text {csch}^4(c+b x) \sinh ^2(a+b x) \, dx=\int \sinh ^{2}{\left (a + b x \right )} \operatorname {csch}^{4}{\left (b x + c \right )}\, dx \] Input:

integrate(csch(b*x+c)**4*sinh(b*x+a)**2,x)
 

Output:

Integral(sinh(a + b*x)**2*csch(b*x + c)**4, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (80) = 160\).

Time = 0.04 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.94 \[ \int \text {csch}^4(c+b x) \sinh ^2(a+b x) \, dx=-\frac {2 \, {\left (e^{\left (4 \, a + 4 \, c\right )} + e^{\left (2 \, a + 6 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} - e^{\left (2 \, a + 6 \, c\right )}\right )}} + \frac {2 \, e^{\left (-4 \, b x + 4 \, c\right )}}{b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} - e^{\left (2 \, a + 6 \, c\right )}\right )}} + \frac {2 \, e^{\left (4 \, a + 4 \, c\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} - e^{\left (2 \, a + 6 \, c\right )}\right )}} + \frac {2 \, e^{\left (2 \, a + 6 \, c\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} - e^{\left (2 \, a + 6 \, c\right )}\right )}} + \frac {2 \, e^{\left (8 \, c\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} - e^{\left (2 \, a + 6 \, c\right )}\right )}} \] Input:

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

Output:

-2*(e^(4*a + 4*c) + e^(2*a + 6*c))*e^(-2*b*x - 2*a)/(b*(3*e^(-2*b*x + 2*a 
+ 4*c) - 3*e^(-4*b*x + 2*a + 2*c) + e^(-6*b*x + 2*a) - e^(2*a + 6*c))) + 2 
*e^(-4*b*x + 4*c)/(b*(3*e^(-2*b*x + 2*a + 4*c) - 3*e^(-4*b*x + 2*a + 2*c) 
+ e^(-6*b*x + 2*a) - e^(2*a + 6*c))) + 2/3*e^(4*a + 4*c)/(b*(3*e^(-2*b*x + 
 2*a + 4*c) - 3*e^(-4*b*x + 2*a + 2*c) + e^(-6*b*x + 2*a) - e^(2*a + 6*c)) 
) + 2/3*e^(2*a + 6*c)/(b*(3*e^(-2*b*x + 2*a + 4*c) - 3*e^(-4*b*x + 2*a + 2 
*c) + e^(-6*b*x + 2*a) - e^(2*a + 6*c))) + 2/3*e^(8*c)/(b*(3*e^(-2*b*x + 2 
*a + 4*c) - 3*e^(-4*b*x + 2*a + 2*c) + e^(-6*b*x + 2*a) - e^(2*a + 6*c)))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \text {csch}^4(c+b x) \sinh ^2(a+b x) \, dx=-\frac {2 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a + 4 \, c\right )} - 3 \, e^{\left (2 \, b x + 4 \, a + 2 \, c\right )} - 3 \, e^{\left (2 \, b x + 2 \, a + 4 \, c\right )} + e^{\left (4 \, a\right )} + e^{\left (2 \, a + 2 \, c\right )} + e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, a - 2 \, c\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, c\right )} - 1\right )}^{3}} \] Input:

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

Output:

-2/3*(3*e^(4*b*x + 4*a + 4*c) - 3*e^(2*b*x + 4*a + 2*c) - 3*e^(2*b*x + 2*a 
 + 4*c) + e^(4*a) + e^(2*a + 2*c) + e^(4*c))*e^(-2*a - 2*c)/(b*(e^(2*b*x + 
 2*c) - 1)^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.13 \[ \int \text {csch}^4(c+b x) \sinh ^2(a+b x) \, dx=\frac {-\frac {2 e^{6 b x +4 a +4 c}}{3}+2 e^{2 b x +2 a +2 c}-\frac {2 e^{2 a}}{3}-\frac {2 e^{2 c}}{3}}{e^{2 a} b \left (e^{6 b x +6 c}-3 e^{4 b x +4 c}+3 e^{2 b x +2 c}-1\right )} \] Input:

int(csch(b*x+c)^4*sinh(b*x+a)^2,x)
 

Output:

(2*( - e**(4*a + 6*b*x + 4*c) + 3*e**(2*a + 2*b*x + 2*c) - e**(2*a) - e**( 
2*c)))/(3*e**(2*a)*b*(e**(6*b*x + 6*c) - 3*e**(4*b*x + 4*c) + 3*e**(2*b*x 
+ 2*c) - 1))