\(\int \cosh ^2(a+b x) \text {sech}^4(c+b x) \, dx\) [138]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95 \[ \int \cosh ^2(a+b x) \text {sech}^4(c+b x) \, dx=\frac {\text {sech}(c) \text {sech}^3(c+b x) (3 \sinh (b x)-\sinh (2 a-4 c-3 b x)-3 \sinh (2 a-2 c-b x)-3 \sinh (2 a+b x)+\sinh (2 a+3 b x)+\sinh (2 c+3 b x))}{12 b} \] Input:

Integrate[Cosh[a + b*x]^2*Sech[c + b*x]^4,x]
 

Output:

(Sech[c]*Sech[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.

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

\(\Big \downarrow \) 7299

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

Input:

Int[Cosh[a + b*x]^2*Sech[c + b*x]^4,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.00 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67

method	result	size
parallelrisch	\(\frac {\sinh \left (3 b x +3 c \right )+2 \sinh \left (3 b x +2 a +c \right )+3 \sinh \left (b x +c \right )}{3 b \left (\cosh \left (3 b x +3 c \right )+3 \cosh \left (b x +c \right )\right )}\)	\(56\)
risch	\(-\frac {2 \left (3 \,{\mathrm e}^{4 b x +4 a +4 c}+3 \,{\mathrm e}^{2 b x +4 a +2 c}+3 \,{\mathrm e}^{2 b x +2 a +4 c}+{\mathrm e}^{4 a}+{\mathrm e}^{2 a +2 c}+{\mathrm e}^{4 c}\right ) {\mathrm e}^{4 a -2 c}}{3 \left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{3} b}\)	\(92\)

Input:

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

Output:

1/3/b*(sinh(3*b*x+3*c)+2*sinh(3*b*x+2*a+c)+3*sinh(b*x+c))/(cosh(3*b*x+3*c) 
+3*cosh(b*x+c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (80) = 160\).

Time = 0.07 (sec) , antiderivative size = 493, normalized size of antiderivative = 5.87 \[ \int \cosh ^2(a+b x) \text {sech}^4(c+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(cosh(b*x+a)^2*sech(b*x+c)^4,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]

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

integrate(cosh(b*x+a)**2*sech(b*x+c)**4,x)
 

Output:

Integral(cosh(a + b*x)**2*sech(b*x + c)**4, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (80) = 160\).

Time = 0.04 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.82 \[ \int \cosh ^2(a+b x) \text {sech}^4(c+b x) \, dx=\frac {2 \, {\left (e^{\left (4 \, a + 4 \, c\right )} + e^{\left (2 \, a + 6 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} + e^{\left (2 \, a + 6 \, c\right )}\right )}} + \frac {2 \, e^{\left (-4 \, b x + 4 \, c\right )}}{b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} + e^{\left (2 \, a + 6 \, c\right )}\right )}} + \frac {2 \, e^{\left (4 \, a + 4 \, c\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} + e^{\left (2 \, a + 6 \, c\right )}\right )}} + \frac {2 \, e^{\left (2 \, a + 6 \, c\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} + e^{\left (2 \, a + 6 \, c\right )}\right )}} + \frac {2 \, e^{\left (8 \, c\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (-6 \, b x + 2 \, a\right )} + e^{\left (2 \, a + 6 \, c\right )}\right )}} \] Input:

integrate(cosh(b*x+a)^2*sech(b*x+c)^4,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 = 0.16 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \cosh ^2(a+b x) \text {sech}^4(c+b x) \, dx=-\frac {2 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a + 4 \, c\right )} + 3 \, e^{\left (2 \, b x + 4 \, a + 2 \, c\right )} + 3 \, e^{\left (2 \, b x + 2 \, a + 4 \, c\right )} + e^{\left (4 \, a\right )} + e^{\left (2 \, a + 2 \, c\right )} + e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, a - 2 \, c\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}^{3}} \] Input:

integrate(cosh(b*x+a)^2*sech(b*x+c)^4,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. \[ \int \cosh ^2(a+b x) \text {sech}^4(c+b x) \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\mathrm {cosh}\left (c+b\,x\right )}^4} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.11 \[ \int \cosh ^2(a+b x) \text {sech}^4(c+b x) \, dx=\frac {\frac {2 e^{6 b x +4 a +4 c}}{3}-2 e^{2 b x +2 a +2 c}-\frac {2 e^{2 a}}{3}-\frac {2 e^{2 c}}{3}}{e^{2 a} b \left (e^{6 b x +6 c}+3 e^{4 b x +4 c}+3 e^{2 b x +2 c}+1\right )} \] Input:

int(cosh(b*x+a)^2*sech(b*x+c)^4,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))