Integrand size = 18, antiderivative size = 346 \[ \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {240 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {24 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {240 \sinh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {24 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3} \] Output:
240*(d*x+c)^(1/2)*cosh(a+b*(d*x+c)^(1/2))/b^5/d^3-24*c*(d*x+c)^(1/2)*cosh( a+b*(d*x+c)^(1/2))/b^3/d^3+2*c^2*(d*x+c)^(1/2)*cosh(a+b*(d*x+c)^(1/2))/b/d ^3+40*(d*x+c)^(3/2)*cosh(a+b*(d*x+c)^(1/2))/b^3/d^3-4*c*(d*x+c)^(3/2)*cosh (a+b*(d*x+c)^(1/2))/b/d^3+2*(d*x+c)^(5/2)*cosh(a+b*(d*x+c)^(1/2))/b/d^3-24 0*sinh(a+b*(d*x+c)^(1/2))/b^6/d^3+24*c*sinh(a+b*(d*x+c)^(1/2))/b^4/d^3-2*c ^2*sinh(a+b*(d*x+c)^(1/2))/b^2/d^3-120*(d*x+c)*sinh(a+b*(d*x+c)^(1/2))/b^4 /d^3+12*c*(d*x+c)*sinh(a+b*(d*x+c)^(1/2))/b^2/d^3-10*(d*x+c)^2*sinh(a+b*(d *x+c)^(1/2))/b^2/d^3
Time = 0.43 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.60 \[ \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {e^{-a-b \sqrt {c+d x}} \left (120+120 b \sqrt {c+d x}+b^5 d^2 x^2 \sqrt {c+d x}+4 b^3 \sqrt {c+d x} (2 c+5 d x)+12 b^2 (4 c+5 d x)+b^4 d x (4 c+5 d x)+e^{2 \left (a+b \sqrt {c+d x}\right )} \left (-120+120 b \sqrt {c+d x}+b^5 d^2 x^2 \sqrt {c+d x}+4 b^3 \sqrt {c+d x} (2 c+5 d x)-12 b^2 (4 c+5 d x)-b^4 d x (4 c+5 d x)\right )\right )}{b^6 d^3} \] Input:
Integrate[x^2*Sinh[a + b*Sqrt[c + d*x]],x]
Output:
(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^(2*(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 = 0.76 (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 = {5887, 7267, 5809, 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.
| \(\displaystyle \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 5887 |
| \(\displaystyle \frac {\int d^2 x^2 \sinh \left (a+b \sqrt {c+d x}\right )d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int d^2 x^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 5809 |
| \(\displaystyle \frac {2 \int \left (\sinh \left (a+b \sqrt {c+d x}\right ) (c+d x)^{5/2}-2 c \sinh \left (a+b \sqrt {c+d x}\right ) (c+d x)^{3/2}+c^2 \sinh \left (a+b \sqrt {c+d x}\right ) \sqrt {c+d x}\right )d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {120 \sinh \left (a+b \sqrt {c+d x}\right )}{b^6}+\frac {120 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^5}-\frac {60 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^4}+\frac {12 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^4}+\frac {20 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3}-\frac {12 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3}-\frac {c^2 \sinh \left (a+b \sqrt {c+d x}\right )}{b^2}-\frac {5 (c+d x)^2 \sinh \left (a+b \sqrt {c+d x}\right )}{b^2}+\frac {6 c (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2}+\frac {c^2 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b}+\frac {(c+d x)^{5/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b}-\frac {2 c (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b}\right )}{d^3}\) |
Input:
Int[x^2*Sinh[a + b*Sqrt[c + d*x]],x]
Output:
(2*((120*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/b^5 - (12*c*Sqrt[c + d*x ]*Cosh[a + b*Sqrt[c + d*x]])/b^3 + (c^2*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/b + (20*(c + d*x)^(3/2)*Cosh[a + b*Sqrt[c + d*x]])/b^3 - (2*c*(c + d*x)^(3/2)*Cosh[a + b*Sqrt[c + d*x]])/b + ((c + d*x)^(5/2)*Cosh[a + b*Sqrt [c + d*x]])/b - (120*Sinh[a + b*Sqrt[c + d*x]])/b^6 + (12*c*Sinh[a + b*Sqr t[c + d*x]])/b^4 - (c^2*Sinh[a + b*Sqrt[c + d*x]])/b^2 - (60*(c + d*x)*Sin h[a + b*Sqrt[c + d*x]])/b^4 + (6*c*(c + d*x)*Sinh[a + b*Sqrt[c + d*x]])/b^ 2 - (5*(c + d*x)^2*Sinh[a + b*Sqrt[c + d*x]])/b^2))/d^3
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sinh[(c_.) + (d_.)*(x _)], x_Symbol] :> Int[ExpandIntegrand[Sinh[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[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbo l] :> Simp[1/Coefficient[u, x, 1]^(m + 1) Subst[Int[(x - Coefficient[u, x , 0])^m*(a + b*Sinh[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 = 0.34 (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 ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {2 a^{5} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{4}}-\frac {20 a^{3} \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^{4}}+\frac {20 a^{2} \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^{4}}-\frac {12 a^{2} c \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^{3} c \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {10 a \left (\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 )\right )}{b^{4}}+\frac {12 a 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 )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{5} \cosh \left (a +b \sqrt {d x +c}\right )-5 \left (a +b \sqrt {d x +c}\right )^{4} \sinh \left (a +b \sqrt {d x +c}\right )+20 \left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )-60 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )+120 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-120 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {4 c \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}}+2 c^{2} \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 )-2 c^{2} a \cosh \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 ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {2 a^{5} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{4}}-\frac {20 a^{3} \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^{4}}+\frac {20 a^{2} \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^{4}}-\frac {12 a^{2} c \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^{3} c \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {10 a \left (\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 )\right )}{b^{4}}+\frac {12 a 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 )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{5} \cosh \left (a +b \sqrt {d x +c}\right )-5 \left (a +b \sqrt {d x +c}\right )^{4} \sinh \left (a +b \sqrt {d x +c}\right )+20 \left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )-60 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )+120 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-120 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{4}}-\frac {4 c \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}}+2 c^{2} \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 )-2 c^{2} a \cosh \left (a +b \sqrt {d x +c}\right )}{d^{3} b^{2}}\) | \(831\) |
| parts | \(\frac {2 x^{2} \sqrt {d x +c}\, \cosh \left (a +b \sqrt {d x +c}\right )}{d b}-\frac {2 x^{2} \sinh \left (a +b \sqrt {d x +c}\right )}{d \,b^{2}}+\frac {-\frac {48 a^{2} \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 {32 a^{3} \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 {32 a \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}}-\frac {8 \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^{2}}+8 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 )-16 c a \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 )-\frac {8 a^{4} \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2}}+8 a^{2} c \sinh \left (a +b \sqrt {d x +c}\right )+\frac {24 a^{2} \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 {8 a^{3} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {24 a \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 {8 \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}}-8 c \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 )+8 c a \cosh \left (a +b \sqrt {d x +c}\right )}{d^{3} b^{4}}\) | \(850\) |
Input:
int(x^2*sinh(a+b*(d*x+c)^(1/2)),x,method=_RETURNVERBOSE)
Output:
2/d^3/b^2*(5/b^4*a^4*((a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1/2))-sinh(a+b *(d*x+c)^(1/2)))-1/b^4*a^5*cosh(a+b*(d*x+c)^(1/2))-10/b^4*a^3*((a+b*(d*x+c )^(1/2))^2*cosh(a+b*(d*x+c)^(1/2))-2*(a+b*(d*x+c)^(1/2))*sinh(a+b*(d*x+c)^ (1/2))+2*cosh(a+b*(d*x+c)^(1/2)))+10/b^4*a^2*((a+b*(d*x+c)^(1/2))^3*cosh(a +b*(d*x+c)^(1/2))-3*(a+b*(d*x+c)^(1/2))^2*sinh(a+b*(d*x+c)^(1/2))+6*(a+b*( d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1/2))-6*sinh(a+b*(d*x+c)^(1/2)))-6/b^2*a^2 *c*((a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1/2))-sinh(a+b*(d*x+c)^(1/2)))+2 /b^2*a^3*c*cosh(a+b*(d*x+c)^(1/2))-5/b^4*a*((a+b*(d*x+c)^(1/2))^4*cosh(a+b *(d*x+c)^(1/2))-4*(a+b*(d*x+c)^(1/2))^3*sinh(a+b*(d*x+c)^(1/2))+12*(a+b*(d *x+c)^(1/2))^2*cosh(a+b*(d*x+c)^(1/2))-24*(a+b*(d*x+c)^(1/2))*sinh(a+b*(d* x+c)^(1/2))+24*cosh(a+b*(d*x+c)^(1/2)))+6/b^2*a*c*((a+b*(d*x+c)^(1/2))^2*c osh(a+b*(d*x+c)^(1/2))-2*(a+b*(d*x+c)^(1/2))*sinh(a+b*(d*x+c)^(1/2))+2*cos h(a+b*(d*x+c)^(1/2)))+1/b^4*((a+b*(d*x+c)^(1/2))^5*cosh(a+b*(d*x+c)^(1/2)) -5*(a+b*(d*x+c)^(1/2))^4*sinh(a+b*(d*x+c)^(1/2))+20*(a+b*(d*x+c)^(1/2))^3* cosh(a+b*(d*x+c)^(1/2))-60*(a+b*(d*x+c)^(1/2))^2*sinh(a+b*(d*x+c)^(1/2))+1 20*(a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1/2))-120*sinh(a+b*(d*x+c)^(1/2)) )-2/b^2*c*((a+b*(d*x+c)^(1/2))^3*cosh(a+b*(d*x+c)^(1/2))-3*(a+b*(d*x+c)^(1 /2))^2*sinh(a+b*(d*x+c)^(1/2))+6*(a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1/2 ))-6*sinh(a+b*(d*x+c)^(1/2)))+c^2*((a+b*(d*x+c)^(1/2))*cosh(a+b*(d*x+c)^(1 /2))-sinh(a+b*(d*x+c)^(1/2)))-c^2*a*cosh(a+b*(d*x+c)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.30 \[ \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left ({\left (b^{5} d^{2} x^{2} + 20 \, b^{3} d x + 8 \, b^{3} c + 120 \, b\right )} \sqrt {d x + c} \cosh \left (\sqrt {d x + c} b + a\right ) - {\left (5 \, b^{4} d^{2} x^{2} + 48 \, b^{2} c + 4 \, {\left (b^{4} c + 15 \, b^{2}\right )} d x + 120\right )} \sinh \left (\sqrt {d x + c} b + a\right )\right )}}{b^{6} d^{3}} \] Input:
integrate(x^2*sinh(a+b*(d*x+c)^(1/2)),x, algorithm="fricas")
Output:
2*((b^5*d^2*x^2 + 20*b^3*d*x + 8*b^3*c + 120*b)*sqrt(d*x + c)*cosh(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)* sinh(sqrt(d*x + c)*b + a))/(b^6*d^3)
Time = 0.35 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.78 \[ \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\begin {cases} \frac {x^{3} \sinh {\left (a \right )}}{3} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{3} \sinh {\left (a + b \sqrt {c} \right )}}{3} & \text {for}\: d = 0 \\\frac {2 x^{2} \sqrt {c + d x} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b d} - \frac {8 c x \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} - \frac {10 x^{2} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} + \frac {16 c \sqrt {c + d x} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{3}} + \frac {40 x \sqrt {c + d x} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {96 c \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{3}} - \frac {120 x \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} + \frac {240 \sqrt {c + d x} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{5} d^{3}} - \frac {240 \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{6} d^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*sinh(a+b*(d*x+c)**(1/2)),x)
Output:
Piecewise((x**3*sinh(a)/3, Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x**3*sinh(a + b*sqrt(c))/3, Eq(d, 0)), (2*x**2*sqrt(c + d*x)*cosh(a + b*sqrt(c + d*x) )/(b*d) - 8*c*x*sinh(a + b*sqrt(c + d*x))/(b**2*d**2) - 10*x**2*sinh(a + b *sqrt(c + d*x))/(b**2*d) + 16*c*sqrt(c + d*x)*cosh(a + b*sqrt(c + d*x))/(b **3*d**3) + 40*x*sqrt(c + d*x)*cosh(a + b*sqrt(c + d*x))/(b**3*d**2) - 96* c*sinh(a + b*sqrt(c + d*x))/(b**4*d**3) - 120*x*sinh(a + b*sqrt(c + d*x))/ (b**4*d**2) + 240*sqrt(c + d*x)*cosh(a + b*sqrt(c + d*x))/(b**5*d**3) - 24 0*sinh(a + b*sqrt(c + d*x))/(b**6*d**3), True))
Time = 0.05 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.40 \[ \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, d^{3} x^{3} \sinh \left (\sqrt {d x + c} b + a\right ) + {\left (\frac {c^{3} e^{\left (\sqrt {d x + c} b + a\right )}}{b} - \frac {c^{3} e^{\left (-\sqrt {d x + c} b - a\right )}}{b} - \frac {3 \, {\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt {d x + c} b e^{a} + 2 \, e^{a}\right )} c^{2} e^{\left (\sqrt {d x + c} b\right )}}{b^{3}} + \frac {3 \, {\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt {d x + c} b + 2\right )} c^{2} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{4} e^{a} - 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 12 \, {\left (d x + c\right )} b^{2} e^{a} - 24 \, \sqrt {d x + c} b e^{a} + 24 \, e^{a}\right )} c e^{\left (\sqrt {d x + c} b\right )}}{b^{5}} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{4} + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 12 \, {\left (d x + c\right )} b^{2} + 24 \, \sqrt {d x + c} b + 24\right )} c e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{6} e^{a} - 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} e^{a} + 30 \, {\left (d x + c\right )}^{2} b^{4} e^{a} - 120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 360 \, {\left (d x + c\right )} b^{2} e^{a} - 720 \, \sqrt {d x + c} b e^{a} + 720 \, e^{a}\right )} e^{\left (\sqrt {d x + c} b\right )}}{b^{7}} + \frac {{\left ({\left (d x + c\right )}^{3} b^{6} + 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} + 30 \, {\left (d x + c\right )}^{2} b^{4} + 120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 360 \, {\left (d x + c\right )} b^{2} + 720 \, \sqrt {d x + c} b + 720\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{7}}\right )} b}{6 \, d^{3}} \] Input:
integrate(x^2*sinh(a+b*(d*x+c)^(1/2)),x, algorithm="maxima")
Output:
1/6*(2*d^3*x^3*sinh(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. 914 vs. \(2 (310) = 620\).
Time = 0.14 (sec) , antiderivative size = 914, normalized size of antiderivative = 2.64 \[ \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx =\text {Too large to display} \] Input:
integrate(x^2*sinh(a+b*(d*x+c)^(1/2)),x, algorithm="giac")
Output:
(((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. \[ \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\int x^2\,\mathrm {sinh}\left (a+b\,\sqrt {c+d\,x}\right ) \,d x \] Input:
int(x^2*sinh(a + b*(c + d*x)^(1/2)),x)
Output:
int(x^2*sinh(a + b*(c + d*x)^(1/2)), x)
Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.55 \[ \int x^2 \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \sqrt {d x +c}\, \cosh \left (\sqrt {d x +c}\, b +a \right ) b^{5} d^{2} x^{2}+16 \sqrt {d x +c}\, \cosh \left (\sqrt {d x +c}\, b +a \right ) b^{3} c +40 \sqrt {d x +c}\, \cosh \left (\sqrt {d x +c}\, b +a \right ) b^{3} d x +240 \sqrt {d x +c}\, \cosh \left (\sqrt {d x +c}\, b +a \right ) b -8 \sinh \left (\sqrt {d x +c}\, b +a \right ) b^{4} c d x -10 \sinh \left (\sqrt {d x +c}\, b +a \right ) b^{4} d^{2} x^{2}-96 \sinh \left (\sqrt {d x +c}\, b +a \right ) b^{2} c -120 \sinh \left (\sqrt {d x +c}\, b +a \right ) b^{2} d x -240 \sinh \left (\sqrt {d x +c}\, b +a \right )}{b^{6} d^{3}} \] Input:
int(x^2*sinh(a+b*(d*x+c)^(1/2)),x)
Output:
(2*(sqrt(c + d*x)*cosh(sqrt(c + d*x)*b + a)*b**5*d**2*x**2 + 8*sqrt(c + d* x)*cosh(sqrt(c + d*x)*b + a)*b**3*c + 20*sqrt(c + d*x)*cosh(sqrt(c + d*x)* b + a)*b**3*d*x + 120*sqrt(c + d*x)*cosh(sqrt(c + d*x)*b + a)*b - 4*sinh(s qrt(c + d*x)*b + a)*b**4*c*d*x - 5*sinh(sqrt(c + d*x)*b + a)*b**4*d**2*x** 2 - 48*sinh(sqrt(c + d*x)*b + a)*b**2*c - 60*sinh(sqrt(c + d*x)*b + a)*b** 2*d*x - 120*sinh(sqrt(c + d*x)*b + a)))/(b**6*d**3)