Integrand size = 18, antiderivative size = 346 \[ \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=-\frac {240 \cosh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3} \]
-240*cosh(a+b*(d*x+c)^(1/2))/b^6/d^3+24*c*cosh(a+b*(d*x+c)^(1/2))/b^4/d^3- 2*c^2*cosh(a+b*(d*x+c)^(1/2))/b^2/d^3-120*(d*x+c)*cosh(a+b*(d*x+c)^(1/2))/ b^4/d^3+12*c*(d*x+c)*cosh(a+b*(d*x+c)^(1/2))/b^2/d^3-10*(d*x+c)^2*cosh(a+b *(d*x+c)^(1/2))/b^2/d^3+40*(d*x+c)^(3/2)*sinh(a+b*(d*x+c)^(1/2))/b^3/d^3-4 *c*(d*x+c)^(3/2)*sinh(a+b*(d*x+c)^(1/2))/b/d^3+2*(d*x+c)^(5/2)*sinh(a+b*(d *x+c)^(1/2))/b/d^3+240*sinh(a+b*(d*x+c)^(1/2))*(d*x+c)^(1/2)/b^5/d^3-24*c* sinh(a+b*(d*x+c)^(1/2))*(d*x+c)^(1/2)/b^3/d^3+2*c^2*sinh(a+b*(d*x+c)^(1/2) )*(d*x+c)^(1/2)/b/d^3
Time = 0.32 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.60 \[ \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {e^{a+b \sqrt {c+d x}} \left (-120+120 b \sqrt {c+d x}+b^5 d^2 x^2 \sqrt {c+d x}+4 b^3 \sqrt {c+d x} (2 c+5 d x)-12 b^2 (4 c+5 d x)-b^4 d x (4 c+5 d x)\right )-e^{-a-b \sqrt {c+d x}} \left (120+120 b \sqrt {c+d x}+b^5 d^2 x^2 \sqrt {c+d x}+4 b^3 \sqrt {c+d x} (2 c+5 d x)+12 b^2 (4 c+5 d x)+b^4 d x (4 c+5 d x)\right )}{b^6 d^3} \]
(E^(a + b*Sqrt[c + d*x])*(-120 + 120*b*Sqrt[c + d*x] + b^5*d^2*x^2*Sqrt[c + d*x] + 4*b^3*Sqrt[c + d*x]*(2*c + 5*d*x) - 12*b^2*(4*c + 5*d*x) - b^4*d* x*(4*c + 5*d*x)) - E^(-a - b*Sqrt[c + d*x])*(120 + 120*b*Sqrt[c + d*x] + b ^5*d^2*x^2*Sqrt[c + d*x] + 4*b^3*Sqrt[c + d*x]*(2*c + 5*d*x) + 12*b^2*(4*c + 5*d*x) + b^4*d*x*(4*c + 5*d*x)))/(b^6*d^3)
Time = 0.65 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5888, 7267, 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.
| \(\displaystyle \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 5888 |
| \(\displaystyle \frac {\int d^2 x^2 \cosh \left (a+b \sqrt {c+d x}\right )d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int d^2 x^2 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 5810 |
| \(\displaystyle \frac {2 \int \left (\cosh \left (a+b \sqrt {c+d x}\right ) (c+d x)^{5/2}-2 c \cosh \left (a+b \sqrt {c+d x}\right ) (c+d x)^{3/2}+c^2 \cosh \left (a+b \sqrt {c+d x}\right ) \sqrt {c+d x}\right )d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {120 \cosh \left (a+b \sqrt {c+d x}\right )}{b^6}+\frac {120 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5}-\frac {60 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4}+\frac {12 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4}+\frac {20 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3}-\frac {12 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3}-\frac {c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2}-\frac {5 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2}+\frac {6 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2}+\frac {c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b}+\frac {(c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b}-\frac {2 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b}\right )}{d^3}\) |
(2*((-120*Cosh[a + b*Sqrt[c + d*x]])/b^6 + (12*c*Cosh[a + b*Sqrt[c + d*x]] )/b^4 - (c^2*Cosh[a + b*Sqrt[c + d*x]])/b^2 - (60*(c + d*x)*Cosh[a + b*Sqr t[c + d*x]])/b^4 + (6*c*(c + d*x)*Cosh[a + b*Sqrt[c + d*x]])/b^2 - (5*(c + d*x)^2*Cosh[a + b*Sqrt[c + d*x]])/b^2 + (120*Sqrt[c + d*x]*Sinh[a + b*Sqr t[c + d*x]])/b^5 - (12*c*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/b^3 + (c ^2*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/b + (20*(c + d*x)^(3/2)*Sinh[a + b*Sqrt[c + d*x]])/b^3 - (2*c*(c + d*x)^(3/2)*Sinh[a + b*Sqrt[c + d*x]]) /b + ((c + d*x)^(5/2)*Sinh[a + b*Sqrt[c + d*x]])/b))/d^3
3.1.59.3.1 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]
Int[((a_.) + Cosh[(c_.) + (d_.)*(u_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo l] :> Simp[1/Coefficient[u, x, 1]^(m + 1) Subst[Int[(x - Coefficient[u, x , 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p }, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(310)=620\).
Time = 0.18 (sec) , antiderivative size = 831, normalized size of antiderivative = 2.40
| method | result | size |
| derivativedivides | \(\frac {\frac {10 a^{4} \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {2 a^{5} \sinh \left (a +b \sqrt {d x +c}\right )}{b^{4}}-\frac {20 a^{3} \left (\left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}+\frac {20 a^{2} \left (\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {12 a^{2} c \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {4 a^{3} c \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {10 a \left (\left (a +b \sqrt {d x +c}\right )^{4} \sinh \left (a +b \sqrt {d x +c}\right )-4 \left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )+12 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-24 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+24 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}+\frac {12 a c \left (\left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{5} \sinh \left (a +b \sqrt {d x +c}\right )-5 \left (a +b \sqrt {d x +c}\right )^{4} \cosh \left (a +b \sqrt {d x +c}\right )+20 \left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-60 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+120 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-120 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {4 c \left (\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+2 c^{2} \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )-2 c^{2} a \sinh \left (a +b \sqrt {d x +c}\right )}{d^{3} b^{2}}\) | \(831\) |
| default | \(\frac {\frac {10 a^{4} \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {2 a^{5} \sinh \left (a +b \sqrt {d x +c}\right )}{b^{4}}-\frac {20 a^{3} \left (\left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}+\frac {20 a^{2} \left (\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {12 a^{2} c \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {4 a^{3} c \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {10 a \left (\left (a +b \sqrt {d x +c}\right )^{4} \sinh \left (a +b \sqrt {d x +c}\right )-4 \left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )+12 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-24 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+24 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}+\frac {12 a c \left (\left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{5} \sinh \left (a +b \sqrt {d x +c}\right )-5 \left (a +b \sqrt {d x +c}\right )^{4} \cosh \left (a +b \sqrt {d x +c}\right )+20 \left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-60 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+120 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-120 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {4 c \left (\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+2 c^{2} \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )-2 c^{2} a \sinh \left (a +b \sqrt {d x +c}\right )}{d^{3} b^{2}}\) | \(831\) |
| parts | \(\frac {2 x^{2} \sinh \left (a +b \sqrt {d x +c}\right ) \sqrt {d x +c}}{d b}-\frac {2 x^{2} \cosh \left (a +b \sqrt {d x +c}\right )}{d \,b^{2}}-\frac {8 \left (\frac {6 a^{2} \left (\left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )+2 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {4 a^{3} \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {4 a \left (\left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-6 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {\left (a +b \sqrt {d x +c}\right )^{4} \cosh \left (a +b \sqrt {d x +c}\right )-4 \left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )+12 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-24 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )+24 \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-c \left (\left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )+2 \cosh \left (a +b \sqrt {d x +c}\right )\right )+2 c a \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )+\frac {a^{4} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-a^{2} c \cosh \left (a +b \sqrt {d x +c}\right )-\frac {3 a^{2} \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {a^{3} \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2}}+\frac {3 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}+c \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )-c a \sinh \left (a +b \sqrt {d x +c}\right )\right )}{d^{3} b^{4}}\) | \(850\) |
2/d^3/b^2*(5/b^4*a^4*((a+b*(d*x+c)^(1/2))*sinh(a+b*(d*x+c)^(1/2))-cosh(a+b *(d*x+c)^(1/2)))-1/b^4*a^5*sinh(a+b*(d*x+c)^(1/2))-10/b^4*a^3*((a+b*(d*x+c )^(1/2))^2*sinh(a+b*(d*x+c)^(1/2))-2*(a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^ (1/2))+2*sinh(a+b*(d*x+c)^(1/2)))+10/b^4*a^2*((a+b*(d*x+c)^(1/2))^3*sinh(a +b*(d*x+c)^(1/2))-3*(a+b*(d*x+c)^(1/2))^2*cosh(a+b*(d*x+c)^(1/2))+6*(a+b*( d*x+c)^(1/2))*sinh(a+b*(d*x+c)^(1/2))-6*cosh(a+b*(d*x+c)^(1/2)))-6/b^2*a^2 *c*((a+b*(d*x+c)^(1/2))*sinh(a+b*(d*x+c)^(1/2))-cosh(a+b*(d*x+c)^(1/2)))+2 /b^2*a^3*c*sinh(a+b*(d*x+c)^(1/2))-5/b^4*a*((a+b*(d*x+c)^(1/2))^4*sinh(a+b *(d*x+c)^(1/2))-4*(a+b*(d*x+c)^(1/2))^3*cosh(a+b*(d*x+c)^(1/2))+12*(a+b*(d *x+c)^(1/2))^2*sinh(a+b*(d*x+c)^(1/2))-24*(a+b*(d*x+c)^(1/2))*cosh(a+b*(d* x+c)^(1/2))+24*sinh(a+b*(d*x+c)^(1/2)))+6/b^2*a*c*((a+b*(d*x+c)^(1/2))^2*s inh(a+b*(d*x+c)^(1/2))-2*(a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1/2))+2*sin h(a+b*(d*x+c)^(1/2)))+1/b^4*((a+b*(d*x+c)^(1/2))^5*sinh(a+b*(d*x+c)^(1/2)) -5*(a+b*(d*x+c)^(1/2))^4*cosh(a+b*(d*x+c)^(1/2))+20*(a+b*(d*x+c)^(1/2))^3* sinh(a+b*(d*x+c)^(1/2))-60*(a+b*(d*x+c)^(1/2))^2*cosh(a+b*(d*x+c)^(1/2))+1 20*(a+b*(d*x+c)^(1/2))*sinh(a+b*(d*x+c)^(1/2))-120*cosh(a+b*(d*x+c)^(1/2)) )-2/b^2*c*((a+b*(d*x+c)^(1/2))^3*sinh(a+b*(d*x+c)^(1/2))-3*(a+b*(d*x+c)^(1 /2))^2*cosh(a+b*(d*x+c)^(1/2))+6*(a+b*(d*x+c)^(1/2))*sinh(a+b*(d*x+c)^(1/2 ))-6*cosh(a+b*(d*x+c)^(1/2)))+c^2*((a+b*(d*x+c)^(1/2))*sinh(a+b*(d*x+c)^(1 /2))-cosh(a+b*(d*x+c)^(1/2)))-c^2*a*sinh(a+b*(d*x+c)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.30 \[ \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left ({\left (b^{5} d^{2} x^{2} + 20 \, b^{3} d x + 8 \, b^{3} c + 120 \, b\right )} \sqrt {d x + c} \sinh \left (\sqrt {d x + c} b + a\right ) - {\left (5 \, b^{4} d^{2} x^{2} + 48 \, b^{2} c + 4 \, {\left (b^{4} c + 15 \, b^{2}\right )} d x + 120\right )} \cosh \left (\sqrt {d x + c} b + a\right )\right )}}{b^{6} d^{3}} \]
2*((b^5*d^2*x^2 + 20*b^3*d*x + 8*b^3*c + 120*b)*sqrt(d*x + c)*sinh(sqrt(d* x + c)*b + a) - (5*b^4*d^2*x^2 + 48*b^2*c + 4*(b^4*c + 15*b^2)*d*x + 120)* cosh(sqrt(d*x + c)*b + a))/(b^6*d^3)
Time = 0.33 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.78 \[ \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\begin {cases} \frac {x^{3} \cosh {\left (a \right )}}{3} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{3} \cosh {\left (a + b \sqrt {c} \right )}}{3} & \text {for}\: d = 0 \\\frac {2 x^{2} \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b d} - \frac {8 c x \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} - \frac {10 x^{2} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} + \frac {16 c \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{3}} + \frac {40 x \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {96 c \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{3}} - \frac {120 x \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} + \frac {240 \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{5} d^{3}} - \frac {240 \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{6} d^{3}} & \text {otherwise} \end {cases} \]
Piecewise((x**3*cosh(a)/3, Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x**3*cosh(a + b*sqrt(c))/3, Eq(d, 0)), (2*x**2*sqrt(c + d*x)*sinh(a + b*sqrt(c + d*x) )/(b*d) - 8*c*x*cosh(a + b*sqrt(c + d*x))/(b**2*d**2) - 10*x**2*cosh(a + b *sqrt(c + d*x))/(b**2*d) + 16*c*sqrt(c + d*x)*sinh(a + b*sqrt(c + d*x))/(b **3*d**3) + 40*x*sqrt(c + d*x)*sinh(a + b*sqrt(c + d*x))/(b**3*d**2) - 96* c*cosh(a + b*sqrt(c + d*x))/(b**4*d**3) - 120*x*cosh(a + b*sqrt(c + d*x))/ (b**4*d**2) + 240*sqrt(c + d*x)*sinh(a + b*sqrt(c + d*x))/(b**5*d**3) - 24 0*cosh(a + b*sqrt(c + d*x))/(b**6*d**3), True))
Time = 0.21 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.40 \[ \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, d^{3} x^{3} \cosh \left (\sqrt {d x + c} b + a\right ) + {\left (\frac {c^{3} e^{\left (\sqrt {d x + c} b + a\right )}}{b} + \frac {c^{3} e^{\left (-\sqrt {d x + c} b - a\right )}}{b} - \frac {3 \, {\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt {d x + c} b e^{a} + 2 \, e^{a}\right )} c^{2} e^{\left (\sqrt {d x + c} b\right )}}{b^{3}} - \frac {3 \, {\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt {d x + c} b + 2\right )} c^{2} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{4} e^{a} - 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 12 \, {\left (d x + c\right )} b^{2} e^{a} - 24 \, \sqrt {d x + c} b e^{a} + 24 \, e^{a}\right )} c e^{\left (\sqrt {d x + c} b\right )}}{b^{5}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{4} + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 12 \, {\left (d x + c\right )} b^{2} + 24 \, \sqrt {d x + c} b + 24\right )} c e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{6} e^{a} - 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} e^{a} + 30 \, {\left (d x + c\right )}^{2} b^{4} e^{a} - 120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 360 \, {\left (d x + c\right )} b^{2} e^{a} - 720 \, \sqrt {d x + c} b e^{a} + 720 \, e^{a}\right )} e^{\left (\sqrt {d x + c} b\right )}}{b^{7}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{6} + 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} + 30 \, {\left (d x + c\right )}^{2} b^{4} + 120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 360 \, {\left (d x + c\right )} b^{2} + 720 \, \sqrt {d x + c} b + 720\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{7}}\right )} b}{6 \, d^{3}} \]
1/6*(2*d^3*x^3*cosh(sqrt(d*x + c)*b + a) + (c^3*e^(sqrt(d*x + c)*b + a)/b + c^3*e^(-sqrt(d*x + c)*b - a)/b - 3*((d*x + c)*b^2*e^a - 2*sqrt(d*x + c)* b*e^a + 2*e^a)*c^2*e^(sqrt(d*x + c)*b)/b^3 - 3*((d*x + c)*b^2 + 2*sqrt(d*x + c)*b + 2)*c^2*e^(-sqrt(d*x + c)*b - a)/b^3 + 3*((d*x + c)^2*b^4*e^a - 4 *(d*x + c)^(3/2)*b^3*e^a + 12*(d*x + c)*b^2*e^a - 24*sqrt(d*x + c)*b*e^a + 24*e^a)*c*e^(sqrt(d*x + c)*b)/b^5 + 3*((d*x + c)^2*b^4 + 4*(d*x + c)^(3/2 )*b^3 + 12*(d*x + c)*b^2 + 24*sqrt(d*x + c)*b + 24)*c*e^(-sqrt(d*x + c)*b - a)/b^5 - ((d*x + c)^3*b^6*e^a - 6*(d*x + c)^(5/2)*b^5*e^a + 30*(d*x + c) ^2*b^4*e^a - 120*(d*x + c)^(3/2)*b^3*e^a + 360*(d*x + c)*b^2*e^a - 720*sqr t(d*x + c)*b*e^a + 720*e^a)*e^(sqrt(d*x + c)*b)/b^7 - ((d*x + c)^3*b^6 + 6 *(d*x + c)^(5/2)*b^5 + 30*(d*x + c)^2*b^4 + 120*(d*x + c)^(3/2)*b^3 + 360* (d*x + c)*b^2 + 720*sqrt(d*x + c)*b + 720)*e^(-sqrt(d*x + c)*b - a)/b^7)*b )/d^3
Leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (310) = 620\).
Time = 0.29 (sec) , antiderivative size = 915, normalized size of antiderivative = 2.64 \[ \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {\frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{4} c^{2} - a b^{4} c^{2} - 2 \, {\left (\sqrt {d x + c} b + a\right )}^{3} b^{2} c + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} b^{2} c + 2 \, a^{3} b^{2} c - b^{4} c^{2} + {\left (\sqrt {d x + c} b + a\right )}^{5} - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} a + 10 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a^{2} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{3} + 5 \, {\left (\sqrt {d x + c} b + a\right )} a^{4} - a^{5} + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} b^{2} c - 12 \, {\left (\sqrt {d x + c} b + a\right )} a b^{2} c + 6 \, a^{2} b^{2} c - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a - 30 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{2} + 20 \, {\left (\sqrt {d x + c} b + a\right )} a^{3} - 5 \, a^{4} - 12 \, {\left (\sqrt {d x + c} b + a\right )} b^{2} c + 12 \, a b^{2} c + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a + 60 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} - 20 \, a^{3} + 12 \, b^{2} c - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 120 \, {\left (\sqrt {d x + c} b + a\right )} a - 60 \, a^{2} + 120 \, \sqrt {d x + c} b - 120\right )} e^{\left (\sqrt {d x + c} b + a\right )}}{b^{5} d^{2}} - \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{4} c^{2} - a b^{4} c^{2} - 2 \, {\left (\sqrt {d x + c} b + a\right )}^{3} b^{2} c + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} b^{2} c + 2 \, a^{3} b^{2} c + b^{4} c^{2} + {\left (\sqrt {d x + c} b + a\right )}^{5} - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} a + 10 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a^{2} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{3} + 5 \, {\left (\sqrt {d x + c} b + a\right )} a^{4} - a^{5} - 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} b^{2} c + 12 \, {\left (\sqrt {d x + c} b + a\right )} a b^{2} c - 6 \, a^{2} b^{2} c + 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} - 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a + 30 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{2} - 20 \, {\left (\sqrt {d x + c} b + a\right )} a^{3} + 5 \, a^{4} - 12 \, {\left (\sqrt {d x + c} b + a\right )} b^{2} c + 12 \, a b^{2} c + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a + 60 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} - 20 \, a^{3} - 12 \, b^{2} c + 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} - 120 \, {\left (\sqrt {d x + c} b + a\right )} a + 60 \, a^{2} + 120 \, \sqrt {d x + c} b + 120\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5} d^{2}}}{b d} \]
(((sqrt(d*x + c)*b + a)*b^4*c^2 - a*b^4*c^2 - 2*(sqrt(d*x + c)*b + a)^3*b^ 2*c + 6*(sqrt(d*x + c)*b + a)^2*a*b^2*c - 6*(sqrt(d*x + c)*b + a)*a^2*b^2* c + 2*a^3*b^2*c - b^4*c^2 + (sqrt(d*x + c)*b + a)^5 - 5*(sqrt(d*x + c)*b + a)^4*a + 10*(sqrt(d*x + c)*b + a)^3*a^2 - 10*(sqrt(d*x + c)*b + a)^2*a^3 + 5*(sqrt(d*x + c)*b + a)*a^4 - a^5 + 6*(sqrt(d*x + c)*b + a)^2*b^2*c - 12 *(sqrt(d*x + c)*b + a)*a*b^2*c + 6*a^2*b^2*c - 5*(sqrt(d*x + c)*b + a)^4 + 20*(sqrt(d*x + c)*b + a)^3*a - 30*(sqrt(d*x + c)*b + a)^2*a^2 + 20*(sqrt( d*x + c)*b + a)*a^3 - 5*a^4 - 12*(sqrt(d*x + c)*b + a)*b^2*c + 12*a*b^2*c + 20*(sqrt(d*x + c)*b + a)^3 - 60*(sqrt(d*x + c)*b + a)^2*a + 60*(sqrt(d*x + c)*b + a)*a^2 - 20*a^3 + 12*b^2*c - 60*(sqrt(d*x + c)*b + a)^2 + 120*(s qrt(d*x + c)*b + a)*a - 60*a^2 + 120*sqrt(d*x + c)*b - 120)*e^(sqrt(d*x + c)*b + a)/(b^5*d^2) - ((sqrt(d*x + c)*b + a)*b^4*c^2 - a*b^4*c^2 - 2*(sqrt (d*x + c)*b + a)^3*b^2*c + 6*(sqrt(d*x + c)*b + a)^2*a*b^2*c - 6*(sqrt(d*x + c)*b + a)*a^2*b^2*c + 2*a^3*b^2*c + b^4*c^2 + (sqrt(d*x + c)*b + a)^5 - 5*(sqrt(d*x + c)*b + a)^4*a + 10*(sqrt(d*x + c)*b + a)^3*a^2 - 10*(sqrt(d *x + c)*b + a)^2*a^3 + 5*(sqrt(d*x + c)*b + a)*a^4 - a^5 - 6*(sqrt(d*x + c )*b + a)^2*b^2*c + 12*(sqrt(d*x + c)*b + a)*a*b^2*c - 6*a^2*b^2*c + 5*(sqr t(d*x + c)*b + a)^4 - 20*(sqrt(d*x + c)*b + a)^3*a + 30*(sqrt(d*x + c)*b + a)^2*a^2 - 20*(sqrt(d*x + c)*b + a)*a^3 + 5*a^4 - 12*(sqrt(d*x + c)*b + a )*b^2*c + 12*a*b^2*c + 20*(sqrt(d*x + c)*b + a)^3 - 60*(sqrt(d*x + c)*b...
Timed out. \[ \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\int x^2\,\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right ) \,d x \]