\(\int x^3 (a+b x) \cosh (c+d x) \, dx\) [1]

Optimal result

Integrand size = 15, antiderivative size = 124 \[ \int x^3 (a+b x) \cosh (c+d x) \, dx=-\frac {6 a \cosh (c+d x)}{d^4}-\frac {24 b x \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {4 b x^3 \cosh (c+d x)}{d^2}+\frac {24 b \sinh (c+d x)}{d^5}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {12 b x^2 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^4 \sinh (c+d x)}{d} \] Output:

-6*a*cosh(d*x+c)/d^4-24*b*x*cosh(d*x+c)/d^4-3*a*x^2*cosh(d*x+c)/d^2-4*b*x^ 
3*cosh(d*x+c)/d^2+24*b*sinh(d*x+c)/d^5+6*a*x*sinh(d*x+c)/d^3+12*b*x^2*sinh 
(d*x+c)/d^3+a*x^3*sinh(d*x+c)/d+b*x^4*sinh(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.66 \[ \int x^3 (a+b x) \cosh (c+d x) \, dx=\frac {-d \left (3 a \left (2+d^2 x^2\right )+4 b x \left (6+d^2 x^2\right )\right ) \cosh (c+d x)+\left (a d^2 x \left (6+d^2 x^2\right )+b \left (24+12 d^2 x^2+d^4 x^4\right )\right ) \sinh (c+d x)}{d^5} \] Input:

Integrate[x^3*(a + b*x)*Cosh[c + d*x],x]
 

Output:

(-(d*(3*a*(2 + d^2*x^2) + 4*b*x*(6 + d^2*x^2))*Cosh[c + d*x]) + (a*d^2*x*( 
6 + d^2*x^2) + b*(24 + 12*d^2*x^2 + d^4*x^4))*Sinh[c + d*x])/d^5
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {7293, 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.

\(\displaystyle \int x^3 (a+b x) \cosh (c+d x) \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (a x^3 \cosh (c+d x)+b x^4 \cosh (c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 a \cosh (c+d x)}{d^4}+\frac {6 a x \sinh (c+d x)}{d^3}-\frac {3 a x^2 \cosh (c+d x)}{d^2}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {24 b \sinh (c+d x)}{d^5}-\frac {24 b x \cosh (c+d x)}{d^4}+\frac {12 b x^2 \sinh (c+d x)}{d^3}-\frac {4 b x^3 \cosh (c+d x)}{d^2}+\frac {b x^4 \sinh (c+d x)}{d}\)

Input:

Int[x^3*(a + b*x)*Cosh[c + d*x],x]
 

Output:

(-6*a*Cosh[c + d*x])/d^4 - (24*b*x*Cosh[c + d*x])/d^4 - (3*a*x^2*Cosh[c + 
d*x])/d^2 - (4*b*x^3*Cosh[c + d*x])/d^2 + (24*b*Sinh[c + d*x])/d^5 + (6*a* 
x*Sinh[c + d*x])/d^3 + (12*b*x^2*Sinh[c + d*x])/d^3 + (a*x^3*Sinh[c + d*x] 
)/d + (b*x^4*Sinh[c + d*x])/d
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(\frac {3 \left (x \left (\frac {4 b x}{3}+a \right ) d^{2}+8 b \right ) d x \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \left (\left (-b \,x^{4}-a \,x^{3}\right ) d^{4}-6 x \left (2 b x +a \right ) d^{2}-24 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+3 d \left (x^{2} \left (\frac {4 b x}{3}+a \right ) d^{2}+8 b x +4 a \right )}{d^{5} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(119\)
risch	\(\frac {\left (b \,x^{4} d^{4}+a \,d^{4} x^{3}-4 b \,d^{3} x^{3}-3 a \,d^{3} x^{2}+12 b \,d^{2} x^{2}+6 a \,d^{2} x -24 d x b -6 a d +24 b \right ) {\mathrm e}^{d x +c}}{2 d^{5}}-\frac {\left (b \,x^{4} d^{4}+a \,d^{4} x^{3}+4 b \,d^{3} x^{3}+3 a \,d^{3} x^{2}+12 b \,d^{2} x^{2}+6 a \,d^{2} x +24 d x b +6 a d +24 b \right ) {\mathrm e}^{-d x -c}}{2 d^{5}}\)	\(153\)
orering	\(-\frac {2 \left (4 b^{2} d^{4} x^{5}+7 a b \,d^{4} x^{4}+3 a^{2} d^{4} x^{3}+36 b^{2} d^{2} x^{3}+45 a b \,d^{2} x^{2}+12 a^{2} d^{2} x +48 b^{2} x +36 a b \right ) \cosh \left (d x +c \right )}{d^{6} x \left (b x +a \right )}+\frac {\left (b \,x^{4} d^{4}+a \,d^{4} x^{3}+12 b \,d^{2} x^{2}+6 a \,d^{2} x +24 b \right ) \left (3 x^{2} \left (b x +a \right ) \cosh \left (d x +c \right )+x^{3} b \cosh \left (d x +c \right )+x^{3} \left (b x +a \right ) d \sinh \left (d x +c \right )\right )}{d^{6} x^{3} \left (b x +a \right )}\)	\(190\)
meijerg	\(-\frac {16 i b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+15\right ) \sinh \left (d x \right )}{10 \sqrt {\pi }}\right )}{d^{5}}-\frac {16 b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} d^{4} x^{4}+\frac {9}{2} x^{2} d^{2}+9\right ) \cosh \left (d x \right )}{6 \sqrt {\pi }}+\frac {x d \left (\frac {3 x^{2} d^{2}}{2}+9\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {8 a \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} d^{2}}{2}+3\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}+\frac {d x \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}-\frac {8 i a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}\)	\(242\)
parts	\(\frac {b \,x^{4} \sinh \left (d x +c \right )}{d}+\frac {a \,x^{3} \sinh \left (d x +c \right )}{d}-\frac {-\frac {4 b \,c^{3} \cosh \left (d x +c \right )}{d^{3}}+\frac {12 b \,c^{2} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{3}}-\frac {12 b c \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{3}}+\frac {4 b \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{3}}+\frac {3 a \,c^{2} \cosh \left (d x +c \right )}{d^{2}}-\frac {6 a c \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{2}}+\frac {3 a \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{2}}}{d^{2}}\)	\(266\)
derivativedivides	\(\frac {-\frac {4 b \,c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}+\frac {6 b \,c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d}-\frac {4 b c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d}+\frac {b \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d}+\frac {b \,c^{4} \sinh \left (d x +c \right )}{d}-a \,c^{3} \sinh \left (d x +c \right )+3 a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 a c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}\)	\(356\)
default	\(\frac {-\frac {4 b \,c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}+\frac {6 b \,c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d}-\frac {4 b c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d}+\frac {b \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d}+\frac {b \,c^{4} \sinh \left (d x +c \right )}{d}-a \,c^{3} \sinh \left (d x +c \right )+3 a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 a c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}\)	\(356\)

Input:

int(x^3*(b*x+a)*cosh(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

(3*(x*(4/3*b*x+a)*d^2+8*b)*d*x*tanh(1/2*d*x+1/2*c)^2+2*((-b*x^4-a*x^3)*d^4 
-6*x*(2*b*x+a)*d^2-24*b)*tanh(1/2*d*x+1/2*c)+3*d*(x^2*(4/3*b*x+a)*d^2+8*b* 
x+4*a))/d^5/(tanh(1/2*d*x+1/2*c)^2-1)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69 \[ \int x^3 (a+b x) \cosh (c+d x) \, dx=-\frac {{\left (4 \, b d^{3} x^{3} + 3 \, a d^{3} x^{2} + 24 \, b d x + 6 \, a d\right )} \cosh \left (d x + c\right ) - {\left (b d^{4} x^{4} + a d^{4} x^{3} + 12 \, b d^{2} x^{2} + 6 \, a d^{2} x + 24 \, b\right )} \sinh \left (d x + c\right )}{d^{5}} \] Input:

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

Output:

-((4*b*d^3*x^3 + 3*a*d^3*x^2 + 24*b*d*x + 6*a*d)*cosh(d*x + c) - (b*d^4*x^ 
4 + a*d^4*x^3 + 12*b*d^2*x^2 + 6*a*d^2*x + 24*b)*sinh(d*x + c))/d^5
 

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22 \[ \int x^3 (a+b x) \cosh (c+d x) \, dx=\begin {cases} \frac {a x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 a x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 a \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {4 b x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 b x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {24 b x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {24 b \sinh {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{5}}{5}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \] Input:

integrate(x**3*(b*x+a)*cosh(d*x+c),x)
 

Output:

Piecewise((a*x**3*sinh(c + d*x)/d - 3*a*x**2*cosh(c + d*x)/d**2 + 6*a*x*si 
nh(c + d*x)/d**3 - 6*a*cosh(c + d*x)/d**4 + b*x**4*sinh(c + d*x)/d - 4*b*x 
**3*cosh(c + d*x)/d**2 + 12*b*x**2*sinh(c + d*x)/d**3 - 24*b*x*cosh(c + d* 
x)/d**4 + 24*b*sinh(c + d*x)/d**5, Ne(d, 0)), ((a*x**4/4 + b*x**5/5)*cosh( 
c), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.87 \[ \int x^3 (a+b x) \cosh (c+d x) \, dx=-\frac {1}{40} \, d {\left (\frac {5 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{5}} + \frac {5 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a e^{\left (-d x - c\right )}}{d^{5}} + \frac {4 \, {\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{6}} + \frac {4 \, {\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b e^{\left (-d x - c\right )}}{d^{6}}\right )} + \frac {1}{20} \, {\left (4 \, b x^{5} + 5 \, a x^{4}\right )} \cosh \left (d x + c\right ) \] Input:

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

Output:

-1/40*d*(5*(d^4*x^4*e^c - 4*d^3*x^3*e^c + 12*d^2*x^2*e^c - 24*d*x*e^c + 24 
*e^c)*a*e^(d*x)/d^5 + 5*(d^4*x^4 + 4*d^3*x^3 + 12*d^2*x^2 + 24*d*x + 24)*a 
*e^(-d*x - c)/d^5 + 4*(d^5*x^5*e^c - 5*d^4*x^4*e^c + 20*d^3*x^3*e^c - 60*d 
^2*x^2*e^c + 120*d*x*e^c - 120*e^c)*b*e^(d*x)/d^6 + 4*(d^5*x^5 + 5*d^4*x^4 
 + 20*d^3*x^3 + 60*d^2*x^2 + 120*d*x + 120)*b*e^(-d*x - c)/d^6) + 1/20*(4* 
b*x^5 + 5*a*x^4)*cosh(d*x + c)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.23 \[ \int x^3 (a+b x) \cosh (c+d x) \, dx=\frac {{\left (b d^{4} x^{4} + a d^{4} x^{3} - 4 \, b d^{3} x^{3} - 3 \, a d^{3} x^{2} + 12 \, b d^{2} x^{2} + 6 \, a d^{2} x - 24 \, b d x - 6 \, a d + 24 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac {{\left (b d^{4} x^{4} + a d^{4} x^{3} + 4 \, b d^{3} x^{3} + 3 \, a d^{3} x^{2} + 12 \, b d^{2} x^{2} + 6 \, a d^{2} x + 24 \, b d x + 6 \, a d + 24 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \] Input:

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

Output:

1/2*(b*d^4*x^4 + a*d^4*x^3 - 4*b*d^3*x^3 - 3*a*d^3*x^2 + 12*b*d^2*x^2 + 6* 
a*d^2*x - 24*b*d*x - 6*a*d + 24*b)*e^(d*x + c)/d^5 - 1/2*(b*d^4*x^4 + a*d^ 
4*x^3 + 4*b*d^3*x^3 + 3*a*d^3*x^2 + 12*b*d^2*x^2 + 6*a*d^2*x + 24*b*d*x + 
6*a*d + 24*b)*e^(-d*x - c)/d^5
 

Mupad [B] (verification not implemented)

Time = 2.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98 \[ \int x^3 (a+b x) \cosh (c+d x) \, dx=\frac {12\,b\,x^2\,\mathrm {sinh}\left (c+d\,x\right )+6\,a\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^3}-\frac {6\,a\,\mathrm {cosh}\left (c+d\,x\right )+24\,b\,x\,\mathrm {cosh}\left (c+d\,x\right )}{d^4}-\frac {3\,a\,x^2\,\mathrm {cosh}\left (c+d\,x\right )+4\,b\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {a\,x^3\,\mathrm {sinh}\left (c+d\,x\right )+b\,x^4\,\mathrm {sinh}\left (c+d\,x\right )}{d}+\frac {24\,b\,\mathrm {sinh}\left (c+d\,x\right )}{d^5} \] Input:

int(x^3*cosh(c + d*x)*(a + b*x),x)
 

Output:

(12*b*x^2*sinh(c + d*x) + 6*a*x*sinh(c + d*x))/d^3 - (6*a*cosh(c + d*x) + 
24*b*x*cosh(c + d*x))/d^4 - (3*a*x^2*cosh(c + d*x) + 4*b*x^3*cosh(c + d*x) 
)/d^2 + (a*x^3*sinh(c + d*x) + b*x^4*sinh(c + d*x))/d + (24*b*sinh(c + d*x 
))/d^5
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int x^3 (a+b x) \cosh (c+d x) \, dx=\frac {-3 \cosh \left (d x +c \right ) a \,d^{3} x^{2}-6 \cosh \left (d x +c \right ) a d -4 \cosh \left (d x +c \right ) b \,d^{3} x^{3}-24 \cosh \left (d x +c \right ) b d x +\sinh \left (d x +c \right ) a \,d^{4} x^{3}+6 \sinh \left (d x +c \right ) a \,d^{2} x +\sinh \left (d x +c \right ) b \,d^{4} x^{4}+12 \sinh \left (d x +c \right ) b \,d^{2} x^{2}+24 b \sinh \left (d x +c \right )}{d^{5}} \] Input:

int(x^3*(b*x+a)*cosh(d*x+c),x)
 

Output:

( - 3*cosh(c + d*x)*a*d**3*x**2 - 6*cosh(c + d*x)*a*d - 4*cosh(c + d*x)*b* 
d**3*x**3 - 24*cosh(c + d*x)*b*d*x + sinh(c + d*x)*a*d**4*x**3 + 6*sinh(c 
+ d*x)*a*d**2*x + sinh(c + d*x)*b*d**4*x**4 + 12*sinh(c + d*x)*b*d**2*x**2 
 + 24*sinh(c + d*x)*b)/d**5