Integrand size = 17, antiderivative size = 139 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {120 b \cosh (c+d x)}{d^6}-\frac {6 a \cosh (c+d x)}{d^4}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d} \] Output:
-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*si nh(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
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.66 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {-\left (\left (3 a d^2 \left (2+d^2 x^2\right )+5 b \left (24+12 d^2 x^2+d^4 x^4\right )\right ) \cosh (c+d x)\right )+d x \left (a d^2 \left (6+d^2 x^2\right )+b \left (120+20 d^2 x^2+d^4 x^4\right )\right ) \sinh (c+d x)}{d^6} \] Input:
Integrate[x^3*(a + b*x^2)*Cosh[c + d*x],x]
Output:
(-((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*(120 + 20*d^2*x^2 + d^4*x^4))*Sinh[c + d* x])/d^6
Time = 0.53 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5810, 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.
| \(\displaystyle \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx\) |
| \(\Big \downarrow \) 5810 |
| \(\displaystyle \int \left (a x^3 \cosh (c+d x)+b x^5 \cosh (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 a \cosh (c+d x)}{d^4}+\frac {6 a x \sinh (c+d x)}{d^3}-\frac {3 a x^2 \cosh (c+d x)}{d^2}+\frac {a x^3 \sinh (c+d x)}{d}-\frac {120 b \cosh (c+d x)}{d^6}+\frac {120 b x \sinh (c+d x)}{d^5}-\frac {60 b x^2 \cosh (c+d x)}{d^4}+\frac {20 b x^3 \sinh (c+d x)}{d^3}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {b x^5 \sinh (c+d x)}{d}\) |
Input:
Int[x^3*(a + b*x^2)*Cosh[c + d*x],x]
Output:
(-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 + (1 20*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
Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p _.), x_Symbol] :> Int[ExpandIntegrand[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]
Time = 0.67 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97
| method | result | size |
| parallelrisch | \(\frac {3 \left (\left (\frac {5 b \,x^{2}}{3}+a \right ) d^{2}+20 b \right ) d^{2} x^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \left (x^{2} \left (b \,x^{2}+a \right ) d^{4}+2 \left (10 b \,x^{2}+3 a \right ) d^{2}+120 b \right ) d x \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (5 b \,x^{4}+3 a \,x^{2}\right ) d^{4}+12 \left (5 b \,x^{2}+a \right ) d^{2}+240 b}{d^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(135\) |
| risch | \(\frac {\left (b \,x^{5} d^{5}+a \,d^{5} x^{3}-5 b \,x^{4} d^{4}-3 a \,d^{4} x^{2}+20 b \,d^{3} x^{3}+6 a \,d^{3} x -60 b \,d^{2} x^{2}-6 a \,d^{2}+120 d x b -120 b \right ) {\mathrm e}^{d x +c}}{2 d^{6}}-\frac {\left (b \,x^{5} d^{5}+a \,d^{5} x^{3}+5 b \,x^{4} d^{4}+3 a \,d^{4} x^{2}+20 b \,d^{3} x^{3}+6 a \,d^{3} x +60 b \,d^{2} x^{2}+6 a \,d^{2}+120 d x b +120 b \right ) {\mathrm e}^{-d x -c}}{2 d^{6}}\) | \(175\) |
| orering | \(-\frac {2 \left (5 b^{2} d^{4} x^{6}+8 a b \,d^{4} x^{4}+3 a^{2} d^{4} x^{2}+80 b^{2} d^{2} x^{4}+78 a b \,d^{2} x^{2}+12 d^{2} a^{2}+360 x^{2} b^{2}+240 a b \right ) \cosh \left (d x +c \right )}{d^{6} \left (b \,x^{2}+a \right )}+\frac {\left (b \,x^{4} d^{4}+a \,d^{4} x^{2}+20 b \,d^{2} x^{2}+6 a \,d^{2}+120 b \right ) \left (3 x^{2} \left (b \,x^{2}+a \right ) \cosh \left (d x +c \right )+2 x^{4} b \cosh \left (d x +c \right )+x^{3} \left (b \,x^{2}+a \right ) d \sinh \left (d x +c \right )\right )}{x^{2} d^{6} \left (b \,x^{2}+a \right )}\) | \(196\) |
| 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 | \(\frac {-\frac {b \,c^{5} \sinh \left (d x +c \right )}{d^{2}}+\frac {5 b \,c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {5 b c \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {b \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{2}}-a \,c^{3} \sinh \left (d x +c \right )+3 a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 a c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}\) | \(447\) |
Input:
int(x^3*(b*x^2+a)*cosh(d*x+c),x,method=_RETURNVERBOSE)
Output:
(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*(x^2*(b*x^2+a) *d^4+2*(10*b*x^2+3*a)*d^2+120*b)*d*x*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)
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.68 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {{\left (5 \, b d^{4} x^{4} + 6 \, a d^{2} + 3 \, {\left (a d^{4} + 20 \, b d^{2}\right )} x^{2} + 120 \, b\right )} \cosh \left (d x + c\right ) - {\left (b d^{5} x^{5} + {\left (a d^{5} + 20 \, b d^{3}\right )} x^{3} + 6 \, {\left (a d^{3} + 20 \, b d\right )} x\right )} \sinh \left (d x + c\right )}{d^{6}} \] Input:
integrate(x^3*(b*x^2+a)*cosh(d*x+c),x, algorithm="fricas")
Output:
-((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
Time = 0.56 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.21 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\begin {cases} \frac {a x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 a x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 a \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b x^{5} \sinh {\left (c + d x \right )}}{d} - \frac {5 b x^{4} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {20 b x^{3} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {60 b x^{2} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {120 b x \sinh {\left (c + d x \right )}}{d^{5}} - \frac {120 b \cosh {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{6}}{6}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(b*x**2+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**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))
Time = 0.04 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.80 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {1}{24} \, d {\left (\frac {3 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{5}} + \frac {3 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a e^{\left (-d x - c\right )}}{d^{5}} + \frac {2 \, {\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{7}} + \frac {2 \, {\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b e^{\left (-d x - c\right )}}{d^{7}}\right )} + \frac {1}{12} \, {\left (2 \, b x^{6} + 3 \, a x^{4}\right )} \cosh \left (d x + c\right ) \] Input:
integrate(x^3*(b*x^2+a)*cosh(d*x+c),x, algorithm="maxima")
Output:
-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 - 120* 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 + 7 20)*b*e^(-d*x - c)/d^7) + 1/12*(2*b*x^6 + 3*a*x^4)*cosh(d*x + c)
Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.25 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {{\left (b d^{5} x^{5} + a d^{5} x^{3} - 5 \, b d^{4} x^{4} - 3 \, a d^{4} x^{2} + 20 \, b d^{3} x^{3} + 6 \, a d^{3} x - 60 \, b d^{2} x^{2} - 6 \, a d^{2} + 120 \, b d x - 120 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac {{\left (b d^{5} x^{5} + a d^{5} x^{3} + 5 \, b d^{4} x^{4} + 3 \, a d^{4} x^{2} + 20 \, b d^{3} x^{3} + 6 \, a d^{3} x + 60 \, b d^{2} x^{2} + 6 \, a d^{2} + 120 \, b d x + 120 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{6}} \] Input:
integrate(x^3*(b*x^2+a)*cosh(d*x+c),x, algorithm="giac")
Output:
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 - 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
Time = 1.95 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {x^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^3}-\frac {3\,x^2\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^4}-\frac {6\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^6}+\frac {6\,x\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^5}-\frac {5\,b\,x^4\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b\,x^5\,\mathrm {sinh}\left (c+d\,x\right )}{d} \] Input:
int(x^3*cosh(c + d*x)*(a + b*x^2),x)
Output:
(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
Time = 0.15 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99 \[ \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {-3 \cosh \left (d x +c \right ) a \,d^{4} x^{2}-6 \cosh \left (d x +c \right ) a \,d^{2}-5 \cosh \left (d x +c \right ) b \,d^{4} x^{4}-60 \cosh \left (d x +c \right ) b \,d^{2} x^{2}-120 \cosh \left (d x +c \right ) b +\sinh \left (d x +c \right ) a \,d^{5} x^{3}+6 \sinh \left (d x +c \right ) a \,d^{3} x +\sinh \left (d x +c \right ) b \,d^{5} x^{5}+20 \sinh \left (d x +c \right ) b \,d^{3} x^{3}+120 \sinh \left (d x +c \right ) b d x}{d^{6}} \] Input:
int(x^3*(b*x^2+a)*cosh(d*x+c),x)
Output:
( - 3*cosh(c + d*x)*a*d**4*x**2 - 6*cosh(c + d*x)*a*d**2 - 5*cosh(c + d*x) *b*d**4*x**4 - 60*cosh(c + d*x)*b*d**2*x**2 - 120*cosh(c + d*x)*b + sinh(c + d*x)*a*d**5*x**3 + 6*sinh(c + d*x)*a*d**3*x + sinh(c + d*x)*b*d**5*x**5 + 20*sinh(c + d*x)*b*d**3*x**3 + 120*sinh(c + d*x)*b*d*x)/d**6