Integrand size = 17, antiderivative size = 184 \[ \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {120 b^2 \cosh (c+d x)}{d^6}-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {120 b^2 x \sinh (c+d x)}{d^5}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d} \]
-120*b^2*cosh(d*x+c)/d^6-12*a*b*cosh(d*x+c)/d^4-a^2*cosh(d*x+c)/d^2-60*b^2 *x^2*cosh(d*x+c)/d^4-6*a*b*x^2*cosh(d*x+c)/d^2-5*b^2*x^4*cosh(d*x+c)/d^2+1 20*b^2*x*sinh(d*x+c)/d^5+12*a*b*x*sinh(d*x+c)/d^3+a^2*x*sinh(d*x+c)/d+20*b ^2*x^3*sinh(d*x+c)/d^3+2*a*b*x^3*sinh(d*x+c)/d+b^2*x^5*sinh(d*x+c)/d
Time = 0.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.61 \[ \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {-\left (\left (a^2 d^4+6 a b d^2 \left (2+d^2 x^2\right )+5 b^2 \left (24+12 d^2 x^2+d^4 x^4\right )\right ) \cosh (c+d x)\right )+d x \left (a^2 d^4+2 a b d^2 \left (6+d^2 x^2\right )+b^2 \left (120+20 d^2 x^2+d^4 x^4\right )\right ) \sinh (c+d x)}{d^6} \]
(-((a^2*d^4 + 6*a*b*d^2*(2 + d^2*x^2) + 5*b^2*(24 + 12*d^2*x^2 + d^4*x^4)) *Cosh[c + d*x]) + d*x*(a^2*d^4 + 2*a*b*d^2*(6 + d^2*x^2) + b^2*(120 + 20*d ^2*x^2 + d^4*x^4))*Sinh[c + d*x])/d^6
Time = 0.52 (sec) , antiderivative size = 184, 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.
| \(\displaystyle \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx\) |
| \(\Big \downarrow \) 5810 |
| \(\displaystyle \int \left (a^2 x \cosh (c+d x)+2 a b x^3 \cosh (c+d x)+b^2 x^5 \cosh (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}-\frac {12 a b \cosh (c+d x)}{d^4}+\frac {12 a b x \sinh (c+d x)}{d^3}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}+\frac {2 a b x^3 \sinh (c+d x)}{d}-\frac {120 b^2 \cosh (c+d x)}{d^6}+\frac {120 b^2 x \sinh (c+d x)}{d^5}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {b^2 x^5 \sinh (c+d x)}{d}\) |
(-120*b^2*Cosh[c + d*x])/d^6 - (12*a*b*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/d^2 - (60*b^2*x^2*Cosh[c + d*x])/d^4 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (5*b^2*x^4*Cosh[c + d*x])/d^2 + (120*b^2*x*Sinh[c + d*x])/d^5 + (12*a*b *x*Sinh[c + d*x])/d^3 + (a^2*x*Sinh[c + d*x])/d + (20*b^2*x^3*Sinh[c + d*x ])/d^3 + (2*a*b*x^3*Sinh[c + d*x])/d + (b^2*x^5*Sinh[c + d*x])/d
3.1.50.3.1 Defintions of rubi rules used
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 = 0.24 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.84
| method | result | size |
| parallelrisch | \(\frac {6 d^{2} x^{2} \left (\left (\frac {5 b \,x^{2}}{6}+a \right ) d^{2}+10 b \right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 d x \left (\left (b \,x^{2}+a \right )^{2} d^{4}+4 \left (5 x^{2} b^{2}+3 a b \right ) d^{2}+120 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (5 b^{2} x^{4}+6 a b \,x^{2}+2 a^{2}\right ) d^{4}+12 \left (5 x^{2} b^{2}+2 a b \right ) d^{2}+240 b^{2}}{d^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(155\) |
| risch | \(\frac {\left (b^{2} x^{5} d^{5}+2 a b \,d^{5} x^{3}-5 b^{2} x^{4} d^{4}+a^{2} d^{5} x -6 a b \,d^{4} x^{2}+20 b^{2} d^{3} x^{3}-a^{2} d^{4}+12 a b \,d^{3} x -60 x^{2} d^{2} b^{2}-12 a \,d^{2} b +120 b^{2} d x -120 b^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{6}}-\frac {\left (b^{2} x^{5} d^{5}+2 a b \,d^{5} x^{3}+5 b^{2} x^{4} d^{4}+a^{2} d^{5} x +6 a b \,d^{4} x^{2}+20 b^{2} d^{3} x^{3}+a^{2} d^{4}+12 a b \,d^{3} x +60 x^{2} d^{2} b^{2}+12 a \,d^{2} b +120 b^{2} d x +120 b^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{6}}\) | \(240\) |
| meijerg | \(-\frac {32 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {15}{4 \sqrt {\pi }}+\frac {\left (\frac {15}{8} d^{4} x^{4}+\frac {45}{2} x^{2} d^{2}+45\right ) \cosh \left (d x \right )}{12 \sqrt {\pi }}-\frac {x d \left (\frac {3}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+45\right ) \sinh \left (d x \right )}{12 \sqrt {\pi }}\right )}{d^{6}}+\frac {32 i b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {7}{8} d^{4} x^{4}+\frac {35}{2} x^{2} d^{2}+105\right ) \cosh \left (d x \right )}{28 \sqrt {\pi }}+\frac {i \left (\frac {35}{8} d^{4} x^{4}+\frac {105}{2} x^{2} d^{2}+105\right ) \sinh \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{6}}+\frac {16 a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} d^{2}}{2}+3\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}+\frac {d x \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}-\frac {16 i b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}-\frac {2 a^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {a^{2} \sinh \left (c \right ) \left (\cosh \left (d x \right ) x d -\sinh \left (d x \right )\right )}{d^{2}}\) | \(326\) |
| parts | \(\frac {b^{2} x^{5} \sinh \left (d x +c \right )}{d}+\frac {2 a b \,x^{3} \sinh \left (d x +c \right )}{d}+\frac {a^{2} x \sinh \left (d x +c \right )}{d}-\frac {\frac {5 b^{2} c^{4} \cosh \left (d x +c \right )}{d^{4}}-\frac {20 b^{2} c^{3} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{4}}+\frac {30 b^{2} c^{2} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {6 b a \,c^{2} \cosh \left (d x +c \right )}{d^{2}}-\frac {20 b^{2} c \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{4}}-\frac {12 b c a \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{2}}+\frac {5 b^{2} \left (\left (d x +c \right )^{4} \cosh \left (d x +c \right )-4 \left (d x +c \right )^{3} \sinh \left (d x +c \right )+12 \left (d x +c \right )^{2} \cosh \left (d x +c \right )-24 \left (d x +c \right ) \sinh \left (d x +c \right )+24 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {6 b a \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{2}}+\cosh \left (d x +c \right ) a^{2}}{d^{2}}\) | \(382\) |
| derivativedivides | \(\frac {\frac {5 b^{2} c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}-\frac {10 b^{2} 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^{4}}+\frac {10 b^{2} 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^{4}}+\frac {6 b \,c^{2} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {5 b^{2} 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^{4}}-\frac {6 b c a \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 {b^{2} \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^{4}}+\frac {2 b 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^{2}}+a^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {b^{2} c^{5} \sinh \left (d x +c \right )}{d^{4}}-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d^{2}}-c \,a^{2} \sinh \left (d x +c \right )}{d^{2}}\) | \(513\) |
| default | \(\frac {\frac {5 b^{2} c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}-\frac {10 b^{2} 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^{4}}+\frac {10 b^{2} 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^{4}}+\frac {6 b \,c^{2} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {5 b^{2} 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^{4}}-\frac {6 b c a \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 {b^{2} \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^{4}}+\frac {2 b 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^{2}}+a^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {b^{2} c^{5} \sinh \left (d x +c \right )}{d^{4}}-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d^{2}}-c \,a^{2} \sinh \left (d x +c \right )}{d^{2}}\) | \(513\) |
(6*d^2*x^2*((5/6*b*x^2+a)*d^2+10*b)*b*tanh(1/2*d*x+1/2*c)^2-2*d*x*((b*x^2+ a)^2*d^4+4*(5*b^2*x^2+3*a*b)*d^2+120*b^2)*tanh(1/2*d*x+1/2*c)+(5*b^2*x^4+6 *a*b*x^2+2*a^2)*d^4+12*(5*b^2*x^2+2*a*b)*d^2+240*b^2)/d^6/(tanh(1/2*d*x+1/ 2*c)^2-1)
Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68 \[ \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {{\left (5 \, b^{2} d^{4} x^{4} + a^{2} d^{4} + 12 \, a b d^{2} + 6 \, {\left (a b d^{4} + 10 \, b^{2} d^{2}\right )} x^{2} + 120 \, b^{2}\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{5} x^{5} + 2 \, {\left (a b d^{5} + 10 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} d^{5} + 12 \, a b d^{3} + 120 \, b^{2} d\right )} x\right )} \sinh \left (d x + c\right )}{d^{6}} \]
-((5*b^2*d^4*x^4 + a^2*d^4 + 12*a*b*d^2 + 6*(a*b*d^4 + 10*b^2*d^2)*x^2 + 1 20*b^2)*cosh(d*x + c) - (b^2*d^5*x^5 + 2*(a*b*d^5 + 10*b^2*d^3)*x^3 + (a^2 *d^5 + 12*a*b*d^3 + 120*b^2*d)*x)*sinh(d*x + c))/d^6
Time = 0.44 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.23 \[ \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\begin {cases} \frac {a^{2} x \sinh {\left (c + d x \right )}}{d} - \frac {a^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {6 a b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 a b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {12 a b \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b^{2} x^{5} \sinh {\left (c + d x \right )}}{d} - \frac {5 b^{2} x^{4} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {20 b^{2} x^{3} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {60 b^{2} x^{2} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {120 b^{2} x \sinh {\left (c + d x \right )}}{d^{5}} - \frac {120 b^{2} \cosh {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**2*x*sinh(c + d*x)/d - a**2*cosh(c + d*x)/d**2 + 2*a*b*x**3*s inh(c + d*x)/d - 6*a*b*x**2*cosh(c + d*x)/d**2 + 12*a*b*x*sinh(c + d*x)/d* *3 - 12*a*b*cosh(c + d*x)/d**4 + b**2*x**5*sinh(c + d*x)/d - 5*b**2*x**4*c osh(c + d*x)/d**2 + 20*b**2*x**3*sinh(c + d*x)/d**3 - 60*b**2*x**2*cosh(c + d*x)/d**4 + 120*b**2*x*sinh(c + d*x)/d**5 - 120*b**2*cosh(c + d*x)/d**6, Ne(d, 0)), ((a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6)*cosh(c), True))
Time = 0.19 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.92 \[ \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {{\left (b x^{2} + a\right )}^{3} \cosh \left (d x + c\right )}{6 \, b} - \frac {{\left (\frac {a^{3} e^{\left (d x + c\right )}}{d} + \frac {a^{3} e^{\left (-d x - c\right )}}{d} + \frac {3 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a^{2} b e^{\left (d x\right )}}{d^{3}} + \frac {3 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} a^{2} b e^{\left (-d x - c\right )}}{d^{3}} + \frac {3 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a b^{2} e^{\left (d x\right )}}{d^{5}} + \frac {3 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a b^{2} e^{\left (-d x - c\right )}}{d^{5}} + \frac {{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b^{3} e^{\left (d x\right )}}{d^{7}} + \frac {{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b^{3} e^{\left (-d x - c\right )}}{d^{7}}\right )} d}{12 \, b} \]
1/6*(b*x^2 + a)^3*cosh(d*x + c)/b - 1/12*(a^3*e^(d*x + c)/d + a^3*e^(-d*x - c)/d + 3*(d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*a^2*b*e^(d*x)/d^3 + 3*(d^2*x^ 2 + 2*d*x + 2)*a^2*b*e^(-d*x - c)/d^3 + 3*(d^4*x^4*e^c - 4*d^3*x^3*e^c + 1 2*d^2*x^2*e^c - 24*d*x*e^c + 24*e^c)*a*b^2*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*b^2*e^(-d*x - c)/d^5 + (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^3*e^(d*x)/d^7 + (d^6*x^6 + 6*d^5*x^5 + 30*d^4*x^4 + 12 0*d^3*x^3 + 360*d^2*x^2 + 720*d*x + 720)*b^3*e^(-d*x - c)/d^7)*d/b
Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.30 \[ \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{3} - 5 \, b^{2} d^{4} x^{4} + a^{2} d^{5} x - 6 \, a b d^{4} x^{2} + 20 \, b^{2} d^{3} x^{3} - a^{2} d^{4} + 12 \, a b d^{3} x - 60 \, b^{2} d^{2} x^{2} - 12 \, a b d^{2} + 120 \, b^{2} d x - 120 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac {{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{3} + 5 \, b^{2} d^{4} x^{4} + a^{2} d^{5} x + 6 \, a b d^{4} x^{2} + 20 \, b^{2} d^{3} x^{3} + a^{2} d^{4} + 12 \, a b d^{3} x + 60 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} + 120 \, b^{2} d x + 120 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{6}} \]
1/2*(b^2*d^5*x^5 + 2*a*b*d^5*x^3 - 5*b^2*d^4*x^4 + a^2*d^5*x - 6*a*b*d^4*x ^2 + 20*b^2*d^3*x^3 - a^2*d^4 + 12*a*b*d^3*x - 60*b^2*d^2*x^2 - 12*a*b*d^2 + 120*b^2*d*x - 120*b^2)*e^(d*x + c)/d^6 - 1/2*(b^2*d^5*x^5 + 2*a*b*d^5*x ^3 + 5*b^2*d^4*x^4 + a^2*d^5*x + 6*a*b*d^4*x^2 + 20*b^2*d^3*x^3 + a^2*d^4 + 12*a*b*d^3*x + 60*b^2*d^2*x^2 + 12*a*b*d^2 + 120*b^2*d*x + 120*b^2)*e^(- d*x - c)/d^6
Time = 1.73 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.80 \[ \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {b^2\,x^5\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {5\,b^2\,x^4\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}-\frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a^2\,d^4+12\,a\,b\,d^2+120\,b^2\right )}{d^6}+\frac {x\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+12\,a\,b\,d^2+120\,b^2\right )}{d^5}-\frac {6\,x^2\,\mathrm {cosh}\left (c+d\,x\right )\,\left (10\,b^2+a\,b\,d^2\right )}{d^4}+\frac {2\,x^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (10\,b^2+a\,b\,d^2\right )}{d^3} \]