\(\int x^2 \cosh ^4(a+b x) \, dx\) [24]

Optimal result

Integrand size = 12, antiderivative size = 134 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {15 x}{64 b^2}+\frac {x^3}{8}-\frac {3 x \cosh ^2(a+b x)}{8 b^2}-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {15 \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^2 \cosh ^3(a+b x) \sinh (a+b x)}{4 b} \] Output:

15/64*x/b^2+1/8*x^3-3/8*x*cosh(b*x+a)^2/b^2-1/8*x*cosh(b*x+a)^4/b^2+15/64* 
cosh(b*x+a)*sinh(b*x+a)/b^3+3/8*x^2*cosh(b*x+a)*sinh(b*x+a)/b+1/32*cosh(b* 
x+a)^3*sinh(b*x+a)/b^3+1/4*x^2*cosh(b*x+a)^3*sinh(b*x+a)/b
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.67 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {32 b^3 x^3-64 b x \cosh (2 (a+b x))-4 b x \cosh (4 (a+b x))+32 \sinh (2 (a+b x))+64 b^2 x^2 \sinh (2 (a+b x))+\sinh (4 (a+b x))+8 b^2 x^2 \sinh (4 (a+b x))}{256 b^3} \] Input:

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

Output:

(32*b^3*x^3 - 64*b*x*Cosh[2*(a + b*x)] - 4*b*x*Cosh[4*(a + b*x)] + 32*Sinh 
[2*(a + b*x)] + 64*b^2*x^2*Sinh[2*(a + b*x)] + Sinh[4*(a + b*x)] + 8*b^2*x 
^2*Sinh[4*(a + b*x)])/(256*b^3)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.35, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3792, 3042, 3115, 3042, 3115, 24, 3792, 15, 3042, 3115, 24}

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.

\(\displaystyle \int x^2 \cosh ^4(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {\int \cosh ^4(a+b x)dx}{8 b^2}+\frac {3}{4} \int x^2 \cosh ^2(a+b x)dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {3}{4} \int \cosh ^2(a+b x)dx+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}\right )+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {3}{4} \left (\frac {\int \cosh ^2(a+b x)dx}{2 b^2}+\frac {\int x^2dx}{2}-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3}{4} \left (\frac {\int \cosh ^2(a+b x)dx}{2 b^2}-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} \left (\frac {\int \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx}{2 b^2}-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {3}{4} \left (\frac {\frac {\int 1dx}{2}+\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b^2}-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {3}{4} \left (-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\)

Input:

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

Output:

-1/8*(x*Cosh[a + b*x]^4)/b^2 + (x^2*Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b) + 
 ((Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b) + (3*(x/2 + (Cosh[a + b*x]*Sinh[a 
+ b*x])/(2*b)))/4)/(8*b^2) + (3*(x^3/6 - (x*Cosh[a + b*x]^2)/(2*b^2) + (x^ 
2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) + (x/2 + (Cosh[a + b*x]*Sinh[a + b*x] 
)/(2*b))/(2*b^2)))/4
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.60

method	result	size
parallelrisch	\(\frac {\left (64 x^{2} b^{2}+32\right ) \sinh \left (2 b x +2 a \right )+\left (8 x^{2} b^{2}+1\right ) \sinh \left (4 b x +4 a \right )+32 \left (x^{2} b^{2}-2 \cosh \left (2 b x +2 a \right )-\frac {\cosh \left (4 b x +4 a \right )}{8}\right ) x b}{256 b^{3}}\)	\(81\)
risch	\(\frac {x^{3}}{8}+\frac {\left (8 x^{2} b^{2}-4 b x +1\right ) {\mathrm e}^{4 b x +4 a}}{512 b^{3}}+\frac {\left (2 x^{2} b^{2}-2 b x +1\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{3}}-\frac {\left (2 x^{2} b^{2}+2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{3}}-\frac {\left (8 x^{2} b^{2}+4 b x +1\right ) {\mathrm e}^{-4 b x -4 a}}{512 b^{3}}\)	\(119\)
derivativedivides	\(\frac {a^{2} \left (\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}\right )-2 a \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}-\frac {3 \cosh \left (b x +a \right )^{2}}{16}\right )+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{8}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}+\frac {15 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {15 b x}{64}+\frac {15 a}{64}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}}{b^{3}}\)	\(237\)
default	\(\frac {a^{2} \left (\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}\right )-2 a \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}-\frac {3 \cosh \left (b x +a \right )^{2}}{16}\right )+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{8}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}+\frac {15 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {15 b x}{64}+\frac {15 a}{64}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}}{b^{3}}\)	\(237\)
orering	\(\frac {\left (32 x^{6} b^{6}-120 x^{4} b^{4}-30 x^{2} b^{2}+135\right ) \cosh \left (b x +a \right )^{4}}{96 x^{3} b^{6}}+\frac {5 \left (112 x^{4} b^{4}+6 x^{2} b^{2}-171\right ) \left (2 x \cosh \left (b x +a \right )^{4}+4 x^{2} \cosh \left (b x +a \right )^{3} b \sinh \left (b x +a \right )\right )}{768 x^{4} b^{6}}-\frac {5 \left (16 x^{4} b^{4}-18 x^{2} b^{2}-63\right ) \left (2 \cosh \left (b x +a \right )^{4}+16 x \cosh \left (b x +a \right )^{3} b \sinh \left (b x +a \right )+12 x^{2} \cosh \left (b x +a \right )^{2} b^{2} \sinh \left (b x +a \right )^{2}+4 x^{2} \cosh \left (b x +a \right )^{4} b^{2}\right )}{768 x^{3} b^{6}}-\frac {\left (88 x^{2} b^{2}+135\right ) \left (24 \cosh \left (b x +a \right )^{3} b \sinh \left (b x +a \right )+72 x \cosh \left (b x +a \right )^{2} b^{2} \sinh \left (b x +a \right )^{2}+24 x \cosh \left (b x +a \right )^{4} b^{2}+24 x^{2} \cosh \left (b x +a \right ) b^{3} \sinh \left (b x +a \right )^{3}+40 x^{2} \cosh \left (b x +a \right )^{3} b^{3} \sinh \left (b x +a \right )\right )}{1536 x^{2} b^{6}}+\frac {\left (8 x^{2} b^{2}+15\right ) \left (144 \cosh \left (b x +a \right )^{2} b^{2} \sinh \left (b x +a \right )^{2}+48 \cosh \left (b x +a \right )^{4} b^{2}+192 x \cosh \left (b x +a \right ) b^{3} \sinh \left (b x +a \right )^{3}+320 x \cosh \left (b x +a \right )^{3} b^{3} \sinh \left (b x +a \right )+24 x^{2} b^{4} \sinh \left (b x +a \right )^{4}+192 x^{2} \cosh \left (b x +a \right )^{2} b^{4} \sinh \left (b x +a \right )^{2}+40 x^{2} \cosh \left (b x +a \right )^{4} b^{4}\right )}{1536 x \,b^{6}}\)	\(462\)

Input:

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

Output:

1/256*((64*b^2*x^2+32)*sinh(2*b*x+2*a)+(8*b^2*x^2+1)*sinh(4*b*x+4*a)+32*(x 
^2*b^2-2*cosh(2*b*x+2*a)-1/8*cosh(4*b*x+4*a))*x*b)/b^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.10 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {8 \, b^{3} x^{3} - b x \cosh \left (b x + a\right )^{4} - b x \sinh \left (b x + a\right )^{4} + {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - 16 \, b x \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} + 8 \, b x\right )} \sinh \left (b x + a\right )^{2} + {\left ({\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} + 16 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{64 \, b^{3}} \] Input:

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

Output:

1/64*(8*b^3*x^3 - b*x*cosh(b*x + a)^4 - b*x*sinh(b*x + a)^4 + (8*b^2*x^2 + 
 1)*cosh(b*x + a)*sinh(b*x + a)^3 - 16*b*x*cosh(b*x + a)^2 - 2*(3*b*x*cosh 
(b*x + a)^2 + 8*b*x)*sinh(b*x + a)^2 + ((8*b^2*x^2 + 1)*cosh(b*x + a)^3 + 
16*(2*b^2*x^2 + 1)*cosh(b*x + a))*sinh(b*x + a))/b^3
 

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.56 \[ \int x^2 \cosh ^4(a+b x) \, dx=\begin {cases} \frac {x^{3} \sinh ^{4}{\left (a + b x \right )}}{8} - \frac {x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac {x^{3} \cosh ^{4}{\left (a + b x \right )}}{8} - \frac {3 x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {5 x^{2} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} + \frac {15 x \sinh ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32 b^{2}} - \frac {17 x \cosh ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {15 \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{3}} + \frac {17 \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \cosh ^{4}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((x**3*sinh(a + b*x)**4/8 - x**3*sinh(a + b*x)**2*cosh(a + b*x)** 
2/4 + x**3*cosh(a + b*x)**4/8 - 3*x**2*sinh(a + b*x)**3*cosh(a + b*x)/(8*b 
) + 5*x**2*sinh(a + b*x)*cosh(a + b*x)**3/(8*b) + 15*x*sinh(a + b*x)**4/(6 
4*b**2) - 3*x*sinh(a + b*x)**2*cosh(a + b*x)**2/(32*b**2) - 17*x*cosh(a + 
b*x)**4/(64*b**2) - 15*sinh(a + b*x)**3*cosh(a + b*x)/(64*b**3) + 17*sinh( 
a + b*x)*cosh(a + b*x)**3/(64*b**3), Ne(b, 0)), (x**3*cosh(a)**4/3, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {1}{8} \, x^{3} + \frac {{\left (8 \, b^{2} x^{2} e^{\left (4 \, a\right )} - 4 \, b x e^{\left (4 \, a\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{512 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{16 \, b^{3}} - \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} - \frac {{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \] Input:

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

Output:

1/8*x^3 + 1/512*(8*b^2*x^2*e^(4*a) - 4*b*x*e^(4*a) + e^(4*a))*e^(4*b*x)/b^ 
3 + 1/16*(2*b^2*x^2*e^(2*a) - 2*b*x*e^(2*a) + e^(2*a))*e^(2*b*x)/b^3 - 1/1 
6*(2*b^2*x^2 + 2*b*x + 1)*e^(-2*b*x - 2*a)/b^3 - 1/512*(8*b^2*x^2 + 4*b*x 
+ 1)*e^(-4*b*x - 4*a)/b^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {1}{8} \, x^{3} + \frac {{\left (8 \, b^{2} x^{2} - 4 \, b x + 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{512 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{3}} - \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} - \frac {{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \] Input:

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

Output:

1/8*x^3 + 1/512*(8*b^2*x^2 - 4*b*x + 1)*e^(4*b*x + 4*a)/b^3 + 1/16*(2*b^2* 
x^2 - 2*b*x + 1)*e^(2*b*x + 2*a)/b^3 - 1/16*(2*b^2*x^2 + 2*b*x + 1)*e^(-2* 
b*x - 2*a)/b^3 - 1/512*(8*b^2*x^2 + 4*b*x + 1)*e^(-4*b*x - 4*a)/b^3
 

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.70 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8}+\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{256}-b\,\left (\frac {x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4}+\frac {x\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{64}\right )+b^2\,\left (\frac {x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4}+\frac {x^2\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}\right )}{b^3}+\frac {x^3}{8} \] Input:

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

Output:

(sinh(2*a + 2*b*x)/8 + sinh(4*a + 4*b*x)/256 - b*((x*cosh(2*a + 2*b*x))/4 
+ (x*cosh(4*a + 4*b*x))/64) + b^2*((x^2*sinh(2*a + 2*b*x))/4 + (x^2*sinh(4 
*a + 4*b*x))/32))/b^3 + x^3/8
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.34 \[ \int x^2 \cosh ^4(a+b x) \, dx=\frac {8 e^{8 b x +8 a} b^{2} x^{2}-4 e^{8 b x +8 a} b x +e^{8 b x +8 a}+64 e^{6 b x +6 a} b^{2} x^{2}-64 e^{6 b x +6 a} b x +32 e^{6 b x +6 a}+64 e^{4 b x +4 a} b^{3} x^{3}-64 e^{2 b x +2 a} b^{2} x^{2}-64 e^{2 b x +2 a} b x -32 e^{2 b x +2 a}-8 b^{2} x^{2}-4 b x -1}{512 e^{4 b x +4 a} b^{3}} \] Input:

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

Output:

(8*e**(8*a + 8*b*x)*b**2*x**2 - 4*e**(8*a + 8*b*x)*b*x + e**(8*a + 8*b*x) 
+ 64*e**(6*a + 6*b*x)*b**2*x**2 - 64*e**(6*a + 6*b*x)*b*x + 32*e**(6*a + 6 
*b*x) + 64*e**(4*a + 4*b*x)*b**3*x**3 - 64*e**(2*a + 2*b*x)*b**2*x**2 - 64 
*e**(2*a + 2*b*x)*b*x - 32*e**(2*a + 2*b*x) - 8*b**2*x**2 - 4*b*x - 1)/(51 
2*e**(4*a + 4*b*x)*b**3)