3.3.81 \(\int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx\) [281]

3.3.81.1 Optimal result

Integrand size = 18, antiderivative size = 117 \[ \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx=\frac {14 \cosh (a+b x)}{9 b^4}+\frac {2 x^2 \cosh (a+b x)}{3 b^2}-\frac {2 \cosh ^3(a+b x)}{27 b^4}-\frac {4 x \sinh (a+b x)}{3 b^3}-\frac {x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac {2 x \sinh ^3(a+b x)}{9 b^3}+\frac {x^3 \sinh ^3(a+b x)}{3 b} \]

output

14/9*cosh(b*x+a)/b^4+2/3*x^2*cosh(b*x+a)/b^2-2/27*cosh(b*x+a)^3/b^4-4/3*x* 
sinh(b*x+a)/b^3-1/3*x^2*cosh(b*x+a)*sinh(b*x+a)^2/b^2+2/9*x*sinh(b*x+a)^3/ 
b^3+1/3*x^3*sinh(b*x+a)^3/b
 

3.3.81.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx=\frac {81 \left (2+b^2 x^2\right ) \cosh (a+b x)-\left (2+9 b^2 x^2\right ) \cosh (3 (a+b x))+6 b x \left (-26-3 b^2 x^2+\left (2+3 b^2 x^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{108 b^4} \]

input

Integrate[x^3*Cosh[a + b*x]*Sinh[a + b*x]^2,x]
 

output

(81*(2 + b^2*x^2)*Cosh[a + b*x] - (2 + 9*b^2*x^2)*Cosh[3*(a + b*x)] + 6*b* 
x*(-26 - 3*b^2*x^2 + (2 + 3*b^2*x^2)*Cosh[2*(a + b*x)])*Sinh[a + b*x])/(10 
8*b^4)
 

3.3.81.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.68 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.28, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {5895, 3042, 26, 3792, 26, 3042, 26, 3113, 2009, 3777, 3042, 3777, 26, 3042, 26, 3118}

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.

\(\displaystyle \int x^3 \sinh ^2(a+b x) \cosh (a+b x) \, dx\)

\(\Big \downarrow \) 5895

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {\int i x^2 \sin (i a+i b x)^3dx}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \int x^2 \sin (i a+i b x)^3dx}{b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2 \int -i \sinh ^3(a+b x)dx}{9 b^2}+\frac {2}{3} \int i x^2 \sinh (a+b x)dx+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (-\frac {2 i \int \sinh ^3(a+b x)dx}{9 b^2}+\frac {2}{3} i \int x^2 \sinh (a+b x)dx+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (-\frac {2 i \int i \sin (i a+i b x)^3dx}{9 b^2}+\frac {2}{3} i \int -i x^2 \sin (i a+i b x)dx+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2 \int \sin (i a+i b x)^3dx}{9 b^2}+\frac {2}{3} \int x^2 \sin (i a+i b x)dx+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 3113

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2 i \int \left (1-\cosh ^2(a+b x)\right )d\cosh (a+b x)}{9 b^3}+\frac {2}{3} \int x^2 \sin (i a+i b x)dx+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2}{3} \int x^2 \sin (i a+i b x)dx+\frac {2 i \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2}{3} \left (\frac {i x^2 \cosh (a+b x)}{b}-\frac {2 i \int x \cosh (a+b x)dx}{b}\right )+\frac {2 i \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2}{3} \left (\frac {i x^2 \cosh (a+b x)}{b}-\frac {2 i \int x \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )+\frac {2 i \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2}{3} \left (\frac {i x^2 \cosh (a+b x)}{b}-\frac {2 i \left (\frac {x \sinh (a+b x)}{b}-\frac {i \int -i \sinh (a+b x)dx}{b}\right )}{b}\right )+\frac {2 i \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2}{3} \left (\frac {i x^2 \cosh (a+b x)}{b}-\frac {2 i \left (\frac {x \sinh (a+b x)}{b}-\frac {\int \sinh (a+b x)dx}{b}\right )}{b}\right )+\frac {2 i \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2}{3} \left (\frac {i x^2 \cosh (a+b x)}{b}-\frac {2 i \left (\frac {x \sinh (a+b x)}{b}-\frac {\int -i \sin (i a+i b x)dx}{b}\right )}{b}\right )+\frac {2 i \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2}{3} \left (\frac {i x^2 \cosh (a+b x)}{b}-\frac {2 i \left (\frac {x \sinh (a+b x)}{b}+\frac {i \int \sin (i a+i b x)dx}{b}\right )}{b}\right )+\frac {2 i \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {i \left (\frac {2 i \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2}{3} \left (\frac {i x^2 \cosh (a+b x)}{b}-\frac {2 i \left (\frac {x \sinh (a+b x)}{b}-\frac {\cosh (a+b x)}{b^2}\right )}{b}\right )+\frac {2 i x \sinh ^3(a+b x)}{9 b^2}-\frac {i x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{b}\)

input

Int[x^3*Cosh[a + b*x]*Sinh[a + b*x]^2,x]
 

output

(x^3*Sinh[a + b*x]^3)/(3*b) - (I*((((2*I)/9)*(Cosh[a + b*x] - Cosh[a + b*x 
]^3/3))/b^3 - ((I/3)*x^2*Cosh[a + b*x]*Sinh[a + b*x]^2)/b + (((2*I)/9)*x*S 
inh[a + b*x]^3)/b^2 + (2*((I*x^2*Cosh[a + b*x])/b - ((2*I)*(-(Cosh[a + b*x 
]/b^2) + (x*Sinh[a + b*x])/b))/b))/3))/b
 

3.3.81.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

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

rule 3042

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

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

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]
 

rule 5895

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) 
]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 
 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Sinh[a + b*x^n]^ 
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
 

3.3.81.4 Maple [A] (verified)

Time = 3.88 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.21

method	result	size
risch	\(\frac {\left (9 x^{3} b^{3}-9 x^{2} b^{2}+6 b x -2\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{4}}-\frac {\left (x^{3} b^{3}-3 x^{2} b^{2}+6 b x -6\right ) {\mathrm e}^{b x +a}}{8 b^{4}}+\frac {\left (x^{3} b^{3}+3 x^{2} b^{2}+6 b x +6\right ) {\mathrm e}^{-b x -a}}{8 b^{4}}-\frac {\left (9 x^{3} b^{3}+9 x^{2} b^{2}+6 b x +2\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{4}}\)	\(141\)
derivativedivides	\(\frac {-\frac {a^{3} \sinh \left (b x +a \right )^{3}}{3}+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{3}+\frac {2 \cosh \left (b x +a \right )}{9}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{9}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right )^{3}}{3}+\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{9}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{9}-\frac {4 \sinh \left (b x +a \right )}{9}+\frac {2 \sinh \left (b x +a \right )^{3}}{27}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {40 \cosh \left (b x +a \right )}{27}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{9}-\frac {2 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{27}}{b^{4}}\)	\(244\)
default	\(\frac {-\frac {a^{3} \sinh \left (b x +a \right )^{3}}{3}+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{3}+\frac {2 \cosh \left (b x +a \right )}{9}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{9}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right )^{3}}{3}+\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{9}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{9}-\frac {4 \sinh \left (b x +a \right )}{9}+\frac {2 \sinh \left (b x +a \right )^{3}}{27}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {40 \cosh \left (b x +a \right )}{27}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{9}-\frac {2 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{27}}{b^{4}}\)	\(244\)

input

int(x^3*cosh(b*x+a)*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)
 

output

1/216*(9*b^3*x^3-9*b^2*x^2+6*b*x-2)/b^4*exp(3*b*x+3*a)-1/8*(b^3*x^3-3*b^2* 
x^2+6*b*x-6)/b^4*exp(b*x+a)+1/8*(b^3*x^3+3*b^2*x^2+6*b*x+6)/b^4*exp(-b*x-a 
)-1/216*(9*b^3*x^3+9*b^2*x^2+6*b*x+2)/b^4*exp(-3*b*x-3*a)
 

3.3.81.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.15 \[ \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx=-\frac {{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \sinh \left (b x + a\right )^{3} - 81 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 9 \, {\left (3 \, b^{3} x^{3} - {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} + 18 \, b x\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \]

input

integrate(x^3*cosh(b*x+a)*sinh(b*x+a)^2,x, algorithm="fricas")
 

output

-1/108*((9*b^2*x^2 + 2)*cosh(b*x + a)^3 + 3*(9*b^2*x^2 + 2)*cosh(b*x + a)* 
sinh(b*x + a)^2 - 3*(3*b^3*x^3 + 2*b*x)*sinh(b*x + a)^3 - 81*(b^2*x^2 + 2) 
*cosh(b*x + a) + 9*(3*b^3*x^3 - (3*b^3*x^3 + 2*b*x)*cosh(b*x + a)^2 + 18*b 
*x)*sinh(b*x + a))/b^4
 

3.3.81.6 Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.25 \[ \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx=\begin {cases} \frac {x^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} - \frac {x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} + \frac {2 x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {14 x \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {4 x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {14 \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{4}} + \frac {40 \cosh ^{3}{\left (a + b x \right )}}{27 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{2}{\left (a \right )} \cosh {\left (a \right )}}{4} & \text {otherwise} \end {cases} \]

input

integrate(x**3*cosh(b*x+a)*sinh(b*x+a)**2,x)
 

output

Piecewise((x**3*sinh(a + b*x)**3/(3*b) - x**2*sinh(a + b*x)**2*cosh(a + b* 
x)/b**2 + 2*x**2*cosh(a + b*x)**3/(3*b**2) + 14*x*sinh(a + b*x)**3/(9*b**3 
) - 4*x*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**3) - 14*sinh(a + b*x)**2*cosh 
(a + b*x)/(9*b**4) + 40*cosh(a + b*x)**3/(27*b**4), Ne(b, 0)), (x**4*sinh( 
a)**2*cosh(a)/4, True))
 

3.3.81.7 Maxima [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.37 \[ \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx=\frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{216 \, b^{4}} - \frac {{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} - \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \]

input

integrate(x^3*cosh(b*x+a)*sinh(b*x+a)^2,x, algorithm="maxima")
 

output

1/216*(9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b*x*e^(3*a) - 2*e^(3*a))* 
e^(3*b*x)/b^4 - 1/8*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*e^(b 
*x)/b^4 + 1/8*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(-b*x - a)/b^4 - 1/216*( 
9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4
 

3.3.81.8 Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.20 \[ \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx=\frac {{\left (9 \, b^{3} x^{3} - 9 \, b^{2} x^{2} + 6 \, b x - 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} - \frac {{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} - \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \]

input

integrate(x^3*cosh(b*x+a)*sinh(b*x+a)^2,x, algorithm="giac")
 

output

1/216*(9*b^3*x^3 - 9*b^2*x^2 + 6*b*x - 2)*e^(3*b*x + 3*a)/b^4 - 1/8*(b^3*x 
^3 - 3*b^2*x^2 + 6*b*x - 6)*e^(b*x + a)/b^4 + 1/8*(b^3*x^3 + 3*b^2*x^2 + 6 
*b*x + 6)*e^(-b*x - a)/b^4 - 1/216*(9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^( 
-3*b*x - 3*a)/b^4
 

3.3.81.9 Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx=\frac {\frac {14\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{9}-\frac {4\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{3}}{b^3}+\frac {\frac {2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3}-x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2}+\frac {40\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{27\,b^4}-\frac {14\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{9\,b^4}+\frac {x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b} \]

input

int(x^3*cosh(a + b*x)*sinh(a + b*x)^2,x)
 

output

((14*x*sinh(a + b*x)^3)/9 - (4*x*cosh(a + b*x)^2*sinh(a + b*x))/3)/b^3 + ( 
(2*x^2*cosh(a + b*x)^3)/3 - x^2*cosh(a + b*x)*sinh(a + b*x)^2)/b^2 + (40*c 
osh(a + b*x)^3)/(27*b^4) - (14*cosh(a + b*x)*sinh(a + b*x)^2)/(9*b^4) + (x 
^3*sinh(a + b*x)^3)/(3*b)