Integrand size = 19, antiderivative size = 234 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {720 b^2 \sinh (c+d x)}{d^7}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d} \] Output:
-720*b^2*x*cosh(d*x+c)/d^6-48*a*b*x*cosh(d*x+c)/d^4-2*a^2*x*cosh(d*x+c)/d^ 2-120*b^2*x^3*cosh(d*x+c)/d^4-8*a*b*x^3*cosh(d*x+c)/d^2-6*b^2*x^5*cosh(d*x +c)/d^2+720*b^2*sinh(d*x+c)/d^7+48*a*b*sinh(d*x+c)/d^5+2*a^2*sinh(d*x+c)/d ^3+360*b^2*x^2*sinh(d*x+c)/d^5+24*a*b*x^2*sinh(d*x+c)/d^3+a^2*x^2*sinh(d*x +c)/d+30*b^2*x^4*sinh(d*x+c)/d^3+2*a*b*x^4*sinh(d*x+c)/d+b^2*x^6*sinh(d*x+ c)/d
Time = 0.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.59 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {-2 d x \left (a^2 d^4+4 a b d^2 \left (6+d^2 x^2\right )+3 b^2 \left (120+20 d^2 x^2+d^4 x^4\right )\right ) \cosh (c+d x)+\left (a^2 d^4 \left (2+d^2 x^2\right )+2 a b d^2 \left (24+12 d^2 x^2+d^4 x^4\right )+b^2 \left (720+360 d^2 x^2+30 d^4 x^4+d^6 x^6\right )\right ) \sinh (c+d x)}{d^7} \] Input:
Integrate[x^2*(a + b*x^2)^2*Cosh[c + d*x],x]
Output:
(-2*d*x*(a^2*d^4 + 4*a*b*d^2*(6 + d^2*x^2) + 3*b^2*(120 + 20*d^2*x^2 + d^4 *x^4))*Cosh[c + d*x] + (a^2*d^4*(2 + d^2*x^2) + 2*a*b*d^2*(24 + 12*d^2*x^2 + d^4*x^4) + b^2*(720 + 360*d^2*x^2 + 30*d^4*x^4 + d^6*x^6))*Sinh[c + d*x ])/d^7
Time = 0.70 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
\(\displaystyle \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx\) |
\(\Big \downarrow \) 5810 |
\(\displaystyle \int \left (a^2 x^2 \cosh (c+d x)+2 a b x^4 \cosh (c+d x)+b^2 x^6 \cosh (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^2 \sinh (c+d x)}{d^3}-\frac {2 a^2 x \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {48 a b \sinh (c+d x)}{d^5}-\frac {48 a b x \cosh (c+d x)}{d^4}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {720 b^2 \sinh (c+d x)}{d^7}-\frac {720 b^2 x \cosh (c+d x)}{d^6}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {b^2 x^6 \sinh (c+d x)}{d}\) |
Input:
Int[x^2*(a + b*x^2)^2*Cosh[c + d*x],x]
Output:
(-720*b^2*x*Cosh[c + d*x])/d^6 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2*a^2*x*C osh[c + d*x])/d^2 - (120*b^2*x^3*Cosh[c + d*x])/d^4 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7 + (48*a*b*Sinh[c + d*x])/d^5 + (2*a^2*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh [c + d*x])/d^5 + (24*a*b*x^2*Sinh[c + d*x])/d^3 + (a^2*x^2*Sinh[c + d*x])/ d + (30*b^2*x^4*Sinh[c + d*x])/d^3 + (2*a*b*x^4*Sinh[c + d*x])/d + (b^2*x^ 6*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 = 0.80 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.55
method | result | size |
parallelrisch | \(\frac {\left (x^{2} \left (b \,x^{2}+a \right )^{2} d^{6}+\left (30 b^{2} x^{4}+24 a b \,x^{2}+2 a^{2}\right ) d^{4}+\left (360 x^{2} b^{2}+48 a b \right ) d^{2}+720 b^{2}\right ) \sinh \left (d x +c \right )-2 \left (\left (3 b \,x^{2}+a \right ) \left (b \,x^{2}+a \right ) d^{4}+\left (60 x^{2} b^{2}+24 a b \right ) d^{2}+360 b^{2}\right ) d \cosh \left (d x +c \right ) x}{d^{7}}\) | \(129\) |
risch | \(\frac {\left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{4}-6 b^{2} x^{5} d^{5}+a^{2} d^{6} x^{2}-8 a b \,d^{5} x^{3}+30 b^{2} x^{4} d^{4}-2 a^{2} d^{5} x +24 a b \,d^{4} x^{2}-120 b^{2} d^{3} x^{3}+2 a^{2} d^{4}-48 a b \,d^{3} x +360 b^{2} d^{2} x^{2}+48 a \,d^{2} b -720 b^{2} d x +720 b^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{7}}-\frac {\left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{4}+6 b^{2} x^{5} d^{5}+a^{2} d^{6} x^{2}+8 a b \,d^{5} x^{3}+30 b^{2} x^{4} d^{4}+2 a^{2} d^{5} x +24 a b \,d^{4} x^{2}+120 b^{2} d^{3} x^{3}+2 a^{2} d^{4}+48 a b \,d^{3} x +360 b^{2} d^{2} x^{2}+48 a \,d^{2} b +720 b^{2} d x +720 b^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{7}}\) | \(305\) |
orering | \(-\frac {4 \left (3 b^{3} d^{6} x^{8}+7 a \,b^{2} d^{6} x^{6}+5 a^{2} b \,d^{6} x^{4}+75 b^{3} d^{4} x^{6}+a^{3} d^{6} x^{2}+93 a \,b^{2} d^{4} x^{4}+27 a^{2} b \,d^{4} x^{2}+720 b^{3} d^{2} x^{4}+a^{3} d^{4}+432 a \,b^{2} d^{2} x^{2}+24 a^{2} b \,d^{2}+1080 x^{2} b^{3}+360 b^{2} a \right ) \cosh \left (d x +c \right )}{d^{8} x \left (b \,x^{2}+a \right )}+\frac {\left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{4}+a^{2} d^{6} x^{2}+30 b^{2} x^{4} d^{4}+24 a b \,d^{4} x^{2}+2 a^{2} d^{4}+360 b^{2} d^{2} x^{2}+48 a \,d^{2} b +720 b^{2}\right ) \left (2 x \left (b \,x^{2}+a \right )^{2} \cosh \left (d x +c \right )+4 x^{3} \left (b \,x^{2}+a \right ) \cosh \left (d x +c \right ) b +x^{2} \left (b \,x^{2}+a \right )^{2} d \sinh \left (d x +c \right )\right )}{d^{8} x^{2} \left (b \,x^{2}+a \right )^{2}}\) | \(316\) |
meijerg | \(\frac {64 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {21}{8} d^{4} x^{4}+\frac {105}{2} x^{2} d^{2}+315\right ) \cosh \left (d x \right )}{28 \sqrt {\pi }}-\frac {i \left (\frac {7}{16} x^{6} d^{6}+\frac {105}{8} d^{4} x^{4}+\frac {315}{2} x^{2} d^{2}+315\right ) \sinh \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{7}}+\frac {64 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {45}{4 \sqrt {\pi }}+\frac {\left (\frac {1}{16} x^{6} d^{6}+\frac {15}{8} d^{4} x^{4}+\frac {45}{2} x^{2} d^{2}+45\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}-\frac {x d \left (\frac {3}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+45\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{7}}-\frac {32 i a b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+15\right ) \sinh \left (d x \right )}{10 \sqrt {\pi }}\right )}{d^{5}}-\frac {32 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} d^{4} x^{4}+\frac {9}{2} x^{2} d^{2}+9\right ) \cosh \left (d x \right )}{6 \sqrt {\pi }}+\frac {x d \left (\frac {3 x^{2} d^{2}}{2}+9\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {4 i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {4 a^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}\) | \(392\) |
parts | \(\frac {b^{2} x^{6} \sinh \left (d x +c \right )}{d}+\frac {2 a b \,x^{4} \sinh \left (d x +c \right )}{d}+\frac {a^{2} x^{2} \sinh \left (d x +c \right )}{d}-\frac {2 \left (\frac {15 b^{2} c^{4} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{5}}-\frac {30 b^{2} c^{3} \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^{5}}+\frac {30 b^{2} c^{2} \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^{5}}+\frac {12 b \,c^{2} a \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{3}}-\frac {15 b^{2} c \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^{5}}-\frac {12 b c 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^{3}}+\frac {3 b^{2} \left (\left (d x +c \right )^{5} \cosh \left (d x +c \right )-5 \left (d x +c \right )^{4} \sinh \left (d x +c \right )+20 \left (d x +c \right )^{3} \cosh \left (d x +c \right )-60 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+120 \left (d x +c \right ) \cosh \left (d x +c \right )-120 \sinh \left (d x +c \right )\right )}{d^{5}}+\frac {4 b a \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 {a^{2} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d}-\frac {3 b^{2} c^{5} \cosh \left (d x +c \right )}{d^{5}}-\frac {4 b \,c^{3} a \cosh \left (d x +c \right )}{d^{3}}-\frac {a^{2} c \cosh \left (d x +c \right )}{d}\right )}{d^{2}}\) | \(570\) |
derivativedivides | \(\frac {a^{2} c^{2} \sinh \left (d x +c \right )+a^{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 )-\frac {8 b \,c^{3} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {12 b \,c^{2} 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 {8 b c 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}}+\frac {b^{2} c^{6} \sinh \left (d x +c \right )}{d^{4}}+\frac {b^{2} \left (\left (d x +c \right )^{6} \sinh \left (d x +c \right )-6 \left (d x +c \right )^{5} \cosh \left (d x +c \right )+30 \left (d x +c \right )^{4} \sinh \left (d x +c \right )-120 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+360 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-720 \left (d x +c \right ) \cosh \left (d x +c \right )+720 \sinh \left (d x +c \right )\right )}{d^{4}}-2 a^{2} c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {6 b^{2} c^{5} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}+\frac {15 b^{2} c^{4} \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 {20 b^{2} c^{3} \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 {15 b^{2} c^{2} \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 {2 b \,c^{4} a \sinh \left (d x +c \right )}{d^{2}}-\frac {6 b^{2} c \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 )^{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}}}{d^{3}}\) | \(738\) |
default | \(\frac {a^{2} c^{2} \sinh \left (d x +c \right )+a^{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 )-\frac {8 b \,c^{3} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {12 b \,c^{2} 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 {8 b c 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}}+\frac {b^{2} c^{6} \sinh \left (d x +c \right )}{d^{4}}+\frac {b^{2} \left (\left (d x +c \right )^{6} \sinh \left (d x +c \right )-6 \left (d x +c \right )^{5} \cosh \left (d x +c \right )+30 \left (d x +c \right )^{4} \sinh \left (d x +c \right )-120 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+360 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-720 \left (d x +c \right ) \cosh \left (d x +c \right )+720 \sinh \left (d x +c \right )\right )}{d^{4}}-2 a^{2} c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {6 b^{2} c^{5} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}+\frac {15 b^{2} c^{4} \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 {20 b^{2} c^{3} \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 {15 b^{2} c^{2} \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 {2 b \,c^{4} a \sinh \left (d x +c \right )}{d^{2}}-\frac {6 b^{2} c \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 )^{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}}}{d^{3}}\) | \(738\) |
Input:
int(x^2*(b*x^2+a)^2*cosh(d*x+c),x,method=_RETURNVERBOSE)
Output:
((x^2*(b*x^2+a)^2*d^6+(30*b^2*x^4+24*a*b*x^2+2*a^2)*d^4+(360*b^2*x^2+48*a* b)*d^2+720*b^2)*sinh(d*x+c)-2*((3*b*x^2+a)*(b*x^2+a)*d^4+(60*b^2*x^2+24*a* b)*d^2+360*b^2)*d*cosh(d*x+c)*x)/d^7
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.66 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {2 \, {\left (3 \, b^{2} d^{5} x^{5} + 4 \, {\left (a b d^{5} + 15 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} d^{5} + 24 \, a b d^{3} + 360 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{6} x^{6} + 2 \, a^{2} d^{4} + 2 \, {\left (a b d^{6} + 15 \, b^{2} d^{4}\right )} x^{4} + 48 \, a b d^{2} + {\left (a^{2} d^{6} + 24 \, a b d^{4} + 360 \, b^{2} d^{2}\right )} x^{2} + 720 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{7}} \] Input:
integrate(x^2*(b*x^2+a)^2*cosh(d*x+c),x, algorithm="fricas")
Output:
-(2*(3*b^2*d^5*x^5 + 4*(a*b*d^5 + 15*b^2*d^3)*x^3 + (a^2*d^5 + 24*a*b*d^3 + 360*b^2*d)*x)*cosh(d*x + c) - (b^2*d^6*x^6 + 2*a^2*d^4 + 2*(a*b*d^6 + 15 *b^2*d^4)*x^4 + 48*a*b*d^2 + (a^2*d^6 + 24*a*b*d^4 + 360*b^2*d^2)*x^2 + 72 0*b^2)*sinh(d*x + c))/d^7
Time = 0.69 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.22 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\begin {cases} \frac {a^{2} x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a^{2} x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a^{2} \sinh {\left (c + d x \right )}}{d^{3}} + \frac {2 a b x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {8 a b x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {24 a b x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {48 a b x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {48 a b \sinh {\left (c + d x \right )}}{d^{5}} + \frac {b^{2} x^{6} \sinh {\left (c + d x \right )}}{d} - \frac {6 b^{2} x^{5} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {360 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{5}} - \frac {720 b^{2} x \cosh {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \sinh {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(b*x**2+a)**2*cosh(d*x+c),x)
Output:
Piecewise((a**2*x**2*sinh(c + d*x)/d - 2*a**2*x*cosh(c + d*x)/d**2 + 2*a** 2*sinh(c + d*x)/d**3 + 2*a*b*x**4*sinh(c + d*x)/d - 8*a*b*x**3*cosh(c + d* x)/d**2 + 24*a*b*x**2*sinh(c + d*x)/d**3 - 48*a*b*x*cosh(c + d*x)/d**4 + 4 8*a*b*sinh(c + d*x)/d**5 + b**2*x**6*sinh(c + d*x)/d - 6*b**2*x**5*cosh(c + d*x)/d**2 + 30*b**2*x**4*sinh(c + d*x)/d**3 - 120*b**2*x**3*cosh(c + d*x )/d**4 + 360*b**2*x**2*sinh(c + d*x)/d**5 - 720*b**2*x*cosh(c + d*x)/d**6 + 720*b**2*sinh(c + d*x)/d**7, Ne(d, 0)), ((a**2*x**3/3 + 2*a*b*x**5/5 + b **2*x**7/7)*cosh(c), True))
Time = 0.05 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.64 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {1}{210} \, d {\left (\frac {35 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{4}} + \frac {35 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a^{2} e^{\left (-d x - c\right )}}{d^{4}} + \frac {42 \, {\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{6}} + \frac {42 \, {\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} a b e^{\left (-d x - c\right )}}{d^{6}} + \frac {15 \, {\left (d^{7} x^{7} e^{c} - 7 \, d^{6} x^{6} e^{c} + 42 \, d^{5} x^{5} e^{c} - 210 \, d^{4} x^{4} e^{c} + 840 \, d^{3} x^{3} e^{c} - 2520 \, d^{2} x^{2} e^{c} + 5040 \, d x e^{c} - 5040 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{8}} + \frac {15 \, {\left (d^{7} x^{7} + 7 \, d^{6} x^{6} + 42 \, d^{5} x^{5} + 210 \, d^{4} x^{4} + 840 \, d^{3} x^{3} + 2520 \, d^{2} x^{2} + 5040 \, d x + 5040\right )} b^{2} e^{\left (-d x - c\right )}}{d^{8}}\right )} + \frac {1}{105} \, {\left (15 \, b^{2} x^{7} + 42 \, a b x^{5} + 35 \, a^{2} x^{3}\right )} \cosh \left (d x + c\right ) \] Input:
integrate(x^2*(b*x^2+a)^2*cosh(d*x+c),x, algorithm="maxima")
Output:
-1/210*d*(35*(d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*a^2*e^(d*x) /d^4 + 35*(d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6)*a^2*e^(-d*x - c)/d^4 + 42*(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)*a*b*e^(d*x)/d^6 + 42*(d^5*x^5 + 5*d^4*x^4 + 20*d^3*x^3 + 60*d^2* x^2 + 120*d*x + 120)*a*b*e^(-d*x - c)/d^6 + 15*(d^7*x^7*e^c - 7*d^6*x^6*e^ c + 42*d^5*x^5*e^c - 210*d^4*x^4*e^c + 840*d^3*x^3*e^c - 2520*d^2*x^2*e^c + 5040*d*x*e^c - 5040*e^c)*b^2*e^(d*x)/d^8 + 15*(d^7*x^7 + 7*d^6*x^6 + 42* d^5*x^5 + 210*d^4*x^4 + 840*d^3*x^3 + 2520*d^2*x^2 + 5040*d*x + 5040)*b^2* e^(-d*x - c)/d^8) + 1/105*(15*b^2*x^7 + 42*a*b*x^5 + 35*a^2*x^3)*cosh(d*x + c)
Time = 0.12 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.30 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} - 6 \, b^{2} d^{5} x^{5} + a^{2} d^{6} x^{2} - 8 \, a b d^{5} x^{3} + 30 \, b^{2} d^{4} x^{4} - 2 \, a^{2} d^{5} x + 24 \, a b d^{4} x^{2} - 120 \, b^{2} d^{3} x^{3} + 2 \, a^{2} d^{4} - 48 \, a b d^{3} x + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} - 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} + 6 \, b^{2} d^{5} x^{5} + a^{2} d^{6} x^{2} + 8 \, a b d^{5} x^{3} + 30 \, b^{2} d^{4} x^{4} + 2 \, a^{2} d^{5} x + 24 \, a b d^{4} x^{2} + 120 \, b^{2} d^{3} x^{3} + 2 \, a^{2} d^{4} + 48 \, a b d^{3} x + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} + 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \] Input:
integrate(x^2*(b*x^2+a)^2*cosh(d*x+c),x, algorithm="giac")
Output:
1/2*(b^2*d^6*x^6 + 2*a*b*d^6*x^4 - 6*b^2*d^5*x^5 + a^2*d^6*x^2 - 8*a*b*d^5 *x^3 + 30*b^2*d^4*x^4 - 2*a^2*d^5*x + 24*a*b*d^4*x^2 - 120*b^2*d^3*x^3 + 2 *a^2*d^4 - 48*a*b*d^3*x + 360*b^2*d^2*x^2 + 48*a*b*d^2 - 720*b^2*d*x + 720 *b^2)*e^(d*x + c)/d^7 - 1/2*(b^2*d^6*x^6 + 2*a*b*d^6*x^4 + 6*b^2*d^5*x^5 + a^2*d^6*x^2 + 8*a*b*d^5*x^3 + 30*b^2*d^4*x^4 + 2*a^2*d^5*x + 24*a*b*d^4*x ^2 + 120*b^2*d^3*x^3 + 2*a^2*d^4 + 48*a*b*d^3*x + 360*b^2*d^2*x^2 + 48*a*b *d^2 + 720*b^2*d*x + 720*b^2)*e^(-d*x - c)/d^7
Time = 2.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.78 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^7}-\frac {6\,b^2\,x^5\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b^2\,x^6\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {2\,x\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^6}+\frac {x^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^5}-\frac {8\,x^3\,\mathrm {cosh}\left (c+d\,x\right )\,\left (15\,b^2+a\,b\,d^2\right )}{d^4}+\frac {2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )\,\left (15\,b^2+a\,b\,d^2\right )}{d^3} \] Input:
int(x^2*cosh(c + d*x)*(a + b*x^2)^2,x)
Output:
(2*sinh(c + d*x)*(360*b^2 + a^2*d^4 + 24*a*b*d^2))/d^7 - (6*b^2*x^5*cosh(c + d*x))/d^2 + (b^2*x^6*sinh(c + d*x))/d - (2*x*cosh(c + d*x)*(360*b^2 + a ^2*d^4 + 24*a*b*d^2))/d^6 + (x^2*sinh(c + d*x)*(360*b^2 + a^2*d^4 + 24*a*b *d^2))/d^5 - (8*x^3*cosh(c + d*x)*(15*b^2 + a*b*d^2))/d^4 + (2*x^4*sinh(c + d*x)*(15*b^2 + a*b*d^2))/d^3
Time = 0.15 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {-2 \cosh \left (d x +c \right ) a^{2} d^{5} x -8 \cosh \left (d x +c \right ) a b \,d^{5} x^{3}-48 \cosh \left (d x +c \right ) a b \,d^{3} x -6 \cosh \left (d x +c \right ) b^{2} d^{5} x^{5}-120 \cosh \left (d x +c \right ) b^{2} d^{3} x^{3}-720 \cosh \left (d x +c \right ) b^{2} d x +\sinh \left (d x +c \right ) a^{2} d^{6} x^{2}+2 \sinh \left (d x +c \right ) a^{2} d^{4}+2 \sinh \left (d x +c \right ) a b \,d^{6} x^{4}+24 \sinh \left (d x +c \right ) a b \,d^{4} x^{2}+48 \sinh \left (d x +c \right ) a b \,d^{2}+\sinh \left (d x +c \right ) b^{2} d^{6} x^{6}+30 \sinh \left (d x +c \right ) b^{2} d^{4} x^{4}+360 \sinh \left (d x +c \right ) b^{2} d^{2} x^{2}+720 \sinh \left (d x +c \right ) b^{2}}{d^{7}} \] Input:
int(x^2*(b*x^2+a)^2*cosh(d*x+c),x)
Output:
( - 2*cosh(c + d*x)*a**2*d**5*x - 8*cosh(c + d*x)*a*b*d**5*x**3 - 48*cosh( c + d*x)*a*b*d**3*x - 6*cosh(c + d*x)*b**2*d**5*x**5 - 120*cosh(c + d*x)*b **2*d**3*x**3 - 720*cosh(c + d*x)*b**2*d*x + sinh(c + d*x)*a**2*d**6*x**2 + 2*sinh(c + d*x)*a**2*d**4 + 2*sinh(c + d*x)*a*b*d**6*x**4 + 24*sinh(c + d*x)*a*b*d**4*x**2 + 48*sinh(c + d*x)*a*b*d**2 + sinh(c + d*x)*b**2*d**6*x **6 + 30*sinh(c + d*x)*b**2*d**4*x**4 + 360*sinh(c + d*x)*b**2*d**2*x**2 + 720*sinh(c + d*x)*b**2)/d**7