\(\int x^3 (a+b x^2) \cosh (c+d x) \, dx\) [40]

Optimal result

Integrand size = 17, antiderivative size = 139 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {120 b \cosh (c+d x)}{d^6}-\frac {6 a \cosh (c+d x)}{d^4}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d} \]

[Out]

-120*b*cosh(d*x+c)/d^6-6*a*cosh(d*x+c)/d^4-60*b*x^2*cosh(d*x+c)/d^4-3*a*x^2*cosh(d*x+c)/d^2-5*b*x^4*cosh(d*x+c
)/d^2+120*b*x*sinh(d*x+c)/d^5+6*a*x*sinh(d*x+c)/d^3+20*b*x^3*sinh(d*x+c)/d^3+a*x^3*sinh(d*x+c)/d+b*x^5*sinh(d*
x+c)/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5395, 3377, 2718} \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {6 a \cosh (c+d x)}{d^4}+\frac {6 a x \sinh (c+d x)}{d^3}-\frac {3 a x^2 \cosh (c+d x)}{d^2}+\frac {a x^3 \sinh (c+d x)}{d}-\frac {120 b \cosh (c+d x)}{d^6}+\frac {120 b x \sinh (c+d x)}{d^5}-\frac {60 b x^2 \cosh (c+d x)}{d^4}+\frac {20 b x^3 \sinh (c+d x)}{d^3}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {b x^5 \sinh (c+d x)}{d} \]

[In]

Int[x^3*(a + b*x^2)*Cosh[c + d*x],x]

[Out]

(-120*b*Cosh[c + d*x])/d^6 - (6*a*Cosh[c + d*x])/d^4 - (60*b*x^2*Cosh[c + d*x])/d^4 - (3*a*x^2*Cosh[c + d*x])/
d^2 - (5*b*x^4*Cosh[c + d*x])/d^2 + (120*b*x*Sinh[c + d*x])/d^5 + (6*a*x*Sinh[c + d*x])/d^3 + (20*b*x^3*Sinh[c
+ d*x])/d^3 + (a*x^3*Sinh[c + d*x])/d + (b*x^5*Sinh[c + d*x])/d

Rule 2718

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

Rule 3377

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

Rule 5395

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a x^3 \cosh (c+d x)+b x^5 \cosh (c+d x)\right ) \, dx \\ & = a \int x^3 \cosh (c+d x) \, dx+b \int x^5 \cosh (c+d x) \, dx \\ & = \frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}-\frac {(3 a) \int x^2 \sinh (c+d x) \, dx}{d}-\frac {(5 b) \int x^4 \sinh (c+d x) \, dx}{d} \\ & = -\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}+\frac {(6 a) \int x \cosh (c+d x) \, dx}{d^2}+\frac {(20 b) \int x^3 \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}-\frac {(6 a) \int \sinh (c+d x) \, dx}{d^3}-\frac {(60 b) \int x^2 \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {6 a \cosh (c+d x)}{d^4}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}+\frac {(120 b) \int x \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {6 a \cosh (c+d x)}{d^4}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}-\frac {(120 b) \int \sinh (c+d x) \, dx}{d^5} \\ & = -\frac {120 b \cosh (c+d x)}{d^6}-\frac {6 a \cosh (c+d x)}{d^4}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.66 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {-\left (\left (3 a d^2 \left (2+d^2 x^2\right )+5 b \left (24+12 d^2 x^2+d^4 x^4\right )\right ) \cosh (c+d x)\right )+d x \left (a d^2 \left (6+d^2 x^2\right )+b \left (120+20 d^2 x^2+d^4 x^4\right )\right ) \sinh (c+d x)}{d^6} \]

[In]

Integrate[x^3*(a + b*x^2)*Cosh[c + d*x],x]

[Out]

(-((3*a*d^2*(2 + d^2*x^2) + 5*b*(24 + 12*d^2*x^2 + d^4*x^4))*Cosh[c + d*x]) + d*x*(a*d^2*(6 + d^2*x^2) + b*(12
0 + 20*d^2*x^2 + d^4*x^4))*Sinh[c + d*x])/d^6

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97

method	result	size
parallelrisch	\(\frac {3 \left (\left (\frac {5 b \,x^{2}}{3}+a \right ) d^{2}+20 b \right ) d^{2} x^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 d x \left (x^{2} \left (b \,x^{2}+a \right ) d^{4}+2 \left (10 b \,x^{2}+3 a \right ) d^{2}+120 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (5 b \,x^{4}+3 a \,x^{2}\right ) d^{4}+12 \left (5 b \,x^{2}+a \right ) d^{2}+240 b}{d^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(135\)
risch	\(\frac {\left (b \,x^{5} d^{5}+a \,d^{5} x^{3}-5 b \,x^{4} d^{4}-3 a \,d^{4} x^{2}+20 b \,d^{3} x^{3}+6 a \,d^{3} x -60 b \,d^{2} x^{2}-6 a \,d^{2}+120 d x b -120 b \right ) {\mathrm e}^{d x +c}}{2 d^{6}}-\frac {\left (b \,x^{5} d^{5}+a \,d^{5} x^{3}+5 b \,x^{4} d^{4}+3 a \,d^{4} x^{2}+20 b \,d^{3} x^{3}+6 a \,d^{3} x +60 b \,d^{2} x^{2}+6 a \,d^{2}+120 d x b +120 b \right ) {\mathrm e}^{-d x -c}}{2 d^{6}}\)	\(175\)
meijerg	\(-\frac {32 b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {15}{4 \sqrt {\pi }}+\frac {\left (\frac {15}{8} d^{4} x^{4}+\frac {45}{2} x^{2} d^{2}+45\right ) \cosh \left (d x \right )}{12 \sqrt {\pi }}-\frac {x d \left (\frac {3}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+45\right ) \sinh \left (d x \right )}{12 \sqrt {\pi }}\right )}{d^{6}}+\frac {32 i b \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {7}{8} d^{4} x^{4}+\frac {35}{2} x^{2} d^{2}+105\right ) \cosh \left (d x \right )}{28 \sqrt {\pi }}+\frac {i \left (\frac {35}{8} d^{4} x^{4}+\frac {105}{2} x^{2} d^{2}+105\right ) \sinh \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{6}}+\frac {8 a \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} d^{2}}{2}+3\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}+\frac {d x \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}-\frac {8 i a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}\)	\(258\)
parts	\(\frac {b \,x^{5} \sinh \left (d x +c \right )}{d}+\frac {a \,x^{3} \sinh \left (d x +c \right )}{d}-\frac {\frac {5 b \,c^{4} \cosh \left (d x +c \right )}{d^{4}}-\frac {20 b \,c^{3} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{4}}+\frac {30 b \,c^{2} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{4}}-\frac {20 b c \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {5 b \left (\left (d x +c \right )^{4} \cosh \left (d x +c \right )-4 \left (d x +c \right )^{3} \sinh \left (d x +c \right )+12 \left (d x +c \right )^{2} \cosh \left (d x +c \right )-24 \left (d x +c \right ) \sinh \left (d x +c \right )+24 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {3 a \,c^{2} \cosh \left (d x +c \right )}{d^{2}}-\frac {6 a c \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{2}}+\frac {3 a \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{2}}}{d^{2}}\)	\(341\)
derivativedivides	\(\frac {-\frac {b \,c^{5} \sinh \left (d x +c \right )}{d^{2}}+\frac {5 b \,c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {5 b c \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {b \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{2}}-a \,c^{3} \sinh \left (d x +c \right )+3 a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 a c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}\)	\(447\)
default	\(\frac {-\frac {b \,c^{5} \sinh \left (d x +c \right )}{d^{2}}+\frac {5 b \,c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {5 b c \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {b \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{2}}-a \,c^{3} \sinh \left (d x +c \right )+3 a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 a c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}\)	\(447\)

[In]

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

[Out]

(3*((5/3*b*x^2+a)*d^2+20*b)*d^2*x^2*tanh(1/2*d*x+1/2*c)^2-2*d*x*(x^2*(b*x^2+a)*d^4+2*(10*b*x^2+3*a)*d^2+120*b)
*tanh(1/2*d*x+1/2*c)+(5*b*x^4+3*a*x^2)*d^4+12*(5*b*x^2+a)*d^2+240*b)/d^6/(tanh(1/2*d*x+1/2*c)^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.68 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {{\left (5 \, b d^{4} x^{4} + 6 \, a d^{2} + 3 \, {\left (a d^{4} + 20 \, b d^{2}\right )} x^{2} + 120 \, b\right )} \cosh \left (d x + c\right ) - {\left (b d^{5} x^{5} + {\left (a d^{5} + 20 \, b d^{3}\right )} x^{3} + 6 \, {\left (a d^{3} + 20 \, b d\right )} x\right )} \sinh \left (d x + c\right )}{d^{6}} \]

[In]

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

[Out]

-((5*b*d^4*x^4 + 6*a*d^2 + 3*(a*d^4 + 20*b*d^2)*x^2 + 120*b)*cosh(d*x + c) - (b*d^5*x^5 + (a*d^5 + 20*b*d^3)*x
^3 + 6*(a*d^3 + 20*b*d)*x)*sinh(d*x + c))/d^6

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.21 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\begin {cases} \frac {a x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 a x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 a \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b x^{5} \sinh {\left (c + d x \right )}}{d} - \frac {5 b x^{4} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {20 b x^{3} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {60 b x^{2} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {120 b x \sinh {\left (c + d x \right )}}{d^{5}} - \frac {120 b \cosh {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{6}}{6}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*x**3*sinh(c + d*x)/d - 3*a*x**2*cosh(c + d*x)/d**2 + 6*a*x*sinh(c + d*x)/d**3 - 6*a*cosh(c + d*x)
/d**4 + b*x**5*sinh(c + d*x)/d - 5*b*x**4*cosh(c + d*x)/d**2 + 20*b*x**3*sinh(c + d*x)/d**3 - 60*b*x**2*cosh(c
+ d*x)/d**4 + 120*b*x*sinh(c + d*x)/d**5 - 120*b*cosh(c + d*x)/d**6, Ne(d, 0)), ((a*x**4/4 + b*x**6/6)*cosh(c
), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.80 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {1}{24} \, d {\left (\frac {3 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{5}} + \frac {3 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a e^{\left (-d x - c\right )}}{d^{5}} + \frac {2 \, {\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{7}} + \frac {2 \, {\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b e^{\left (-d x - c\right )}}{d^{7}}\right )} + \frac {1}{12} \, {\left (2 \, b x^{6} + 3 \, a x^{4}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

-1/24*d*(3*(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 + 3*(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 + 2*(d^6*x^6*e^c - 6*d^5*x^5*e^c + 30*d^4*x^4*e^c - 12
0*d^3*x^3*e^c + 360*d^2*x^2*e^c - 720*d*x*e^c + 720*e^c)*b*e^(d*x)/d^7 + 2*(d^6*x^6 + 6*d^5*x^5 + 30*d^4*x^4 +
120*d^3*x^3 + 360*d^2*x^2 + 720*d*x + 720)*b*e^(-d*x - c)/d^7) + 1/12*(2*b*x^6 + 3*a*x^4)*cosh(d*x + c)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.25 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {{\left (b d^{5} x^{5} + a d^{5} x^{3} - 5 \, b d^{4} x^{4} - 3 \, a d^{4} x^{2} + 20 \, b d^{3} x^{3} + 6 \, a d^{3} x - 60 \, b d^{2} x^{2} - 6 \, a d^{2} + 120 \, b d x - 120 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac {{\left (b d^{5} x^{5} + a d^{5} x^{3} + 5 \, b d^{4} x^{4} + 3 \, a d^{4} x^{2} + 20 \, b d^{3} x^{3} + 6 \, a d^{3} x + 60 \, b d^{2} x^{2} + 6 \, a d^{2} + 120 \, b d x + 120 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{6}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {x^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^3}-\frac {3\,x^2\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^4}-\frac {6\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^6}+\frac {6\,x\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^5}-\frac {5\,b\,x^4\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b\,x^5\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]

[In]

int(x^3*cosh(c + d*x)*(a + b*x^2),x)

[Out]

(x^3*sinh(c + d*x)*(20*b + a*d^2))/d^3 - (3*x^2*cosh(c + d*x)*(20*b + a*d^2))/d^4 - (6*cosh(c + d*x)*(20*b + a
*d^2))/d^6 + (6*x*sinh(c + d*x)*(20*b + a*d^2))/d^5 - (5*b*x^4*cosh(c + d*x))/d^2 + (b*x^5*sinh(c + d*x))/d