Integrand size = 16, antiderivative size = 227 \[ \int x^2 \sinh \left (a+b x-c x^2\right ) \, dx=-\frac {b \cosh \left (a+b x-c x^2\right )}{4 c^2}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}-\frac {b^2 e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}}-\frac {e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {b^2 e^{-a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}}-\frac {e^{-a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}} \]
-1/4*b*cosh(-c*x^2+b*x+a)/c^2-1/2*x*cosh(-c*x^2+b*x+a)/c-1/16*b^2*exp(a+1/ 4*b^2/c)*erf(1/2*(-2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(5/2)-1/8*exp(a+1/4*b^2/c) *erf(1/2*(-2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(3/2)+1/16*b^2*exp(-a-1/4*b^2/c)*e rfi(1/2*(-2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(5/2)-1/8*exp(-a-1/4*b^2/c)*erfi(1/ 2*(-2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(3/2)
Time = 0.23 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.67 \[ \int x^2 \sinh \left (a+b x-c x^2\right ) \, dx=\frac {-4 \sqrt {c} (b+2 c x) \cosh (a+x (b-c x))+\left (b^2-2 c\right ) \sqrt {\pi } \text {erfi}\left (\frac {-b+2 c x}{2 \sqrt {c}}\right ) \left (-\cosh \left (a+\frac {b^2}{4 c}\right )+\sinh \left (a+\frac {b^2}{4 c}\right )\right )+\left (b^2+2 c\right ) \sqrt {\pi } \text {erf}\left (\frac {-b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a+\frac {b^2}{4 c}\right )+\sinh \left (a+\frac {b^2}{4 c}\right )\right )}{16 c^{5/2}} \]
(-4*Sqrt[c]*(b + 2*c*x)*Cosh[a + x*(b - c*x)] + (b^2 - 2*c)*Sqrt[Pi]*Erfi[ (-b + 2*c*x)/(2*Sqrt[c])]*(-Cosh[a + b^2/(4*c)] + Sinh[a + b^2/(4*c)]) + ( b^2 + 2*c)*Sqrt[Pi]*Erf[(-b + 2*c*x)/(2*Sqrt[c])]*(Cosh[a + b^2/(4*c)] + S inh[a + b^2/(4*c)]))/(16*c^(5/2))
Time = 0.82 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5909, 5898, 2664, 2633, 2634, 5905, 5897, 2664, 2633, 2634}
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.
| \(\displaystyle \int x^2 \sinh \left (a+b x-c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5909 |
| \(\displaystyle \frac {b \int x \sinh \left (-c x^2+b x+a\right )dx}{2 c}+\frac {\int \cosh \left (-c x^2+b x+a\right )dx}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 5898 |
| \(\displaystyle \frac {\frac {1}{2} \int e^{-c x^2+b x+a}dx+\frac {1}{2} \int e^{c x^2-b x-a}dx}{2 c}+\frac {b \int x \sinh \left (-c x^2+b x+a\right )dx}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {\frac {1}{2} e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}}dx+\frac {1}{2} e^{-a-\frac {b^2}{4 c}} \int e^{\frac {(b-2 c x)^2}{4 c}}dx}{2 c}+\frac {b \int x \sinh \left (-c x^2+b x+a\right )dx}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {\frac {1}{2} e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}}dx-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}}{2 c}+\frac {b \int x \sinh \left (-c x^2+b x+a\right )dx}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {b \int x \sinh \left (-c x^2+b x+a\right )dx}{2 c}+\frac {-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 5905 |
| \(\displaystyle \frac {b \left (\frac {b \int \sinh \left (-c x^2+b x+a\right )dx}{2 c}-\frac {\cosh \left (a+b x-c x^2\right )}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 5897 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {1}{2} \int e^{-c x^2+b x+a}dx-\frac {1}{2} \int e^{c x^2-b x-a}dx\right )}{2 c}-\frac {\cosh \left (a+b x-c x^2\right )}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {1}{2} e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}}dx-\frac {1}{2} e^{-a-\frac {b^2}{4 c}} \int e^{\frac {(b-2 c x)^2}{4 c}}dx\right )}{2 c}-\frac {\cosh \left (a+b x-c x^2\right )}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {1}{2} e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}}dx+\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}-\frac {\cosh \left (a+b x-c x^2\right )}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}-\frac {\cosh \left (a+b x-c x^2\right )}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}}{2 c}-\frac {x \cosh \left (a+b x-c x^2\right )}{2 c}\) |
-1/2*(x*Cosh[a + b*x - c*x^2])/c + (-1/4*(E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[( b - 2*c*x)/(2*Sqrt[c])])/Sqrt[c] - (E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c]))/(2*c) + (b*(-1/2*Cosh[a + b*x - c*x^2]/c + (b*(-1/4*(E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/Sqrt [c] + (E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[ c])))/(2*c)))/(2*c)
3.1.6.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ (4*c)) Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
Int[Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[1/2 Int[E^ (a + b*x + c*x^2), x], x] - Simp[1/2 Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]
Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[1/2 Int[E^ (a + b*x + c*x^2), x], x] + Simp[1/2 Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]
Int[((d_.) + (e_.)*(x_))*Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[e*(Cosh[a + b*x + c*x^2]/(2*c)), x] - Simp[(b*e - 2*c*d)/(2*c) I nt[Sinh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[e*(d + e*x)^(m - 1)*(Cosh[a + b*x + c*x^2]/(2*c)), x] + (-Sim p[(b*e - 2*c*d)/(2*c) Int[(d + e*x)^(m - 1)*Sinh[a + b*x + c*x^2], x], x] - Simp[e^2*((m - 1)/(2*c)) Int[(d + e*x)^(m - 2)*Cosh[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && NeQ[b*e - 2*c*d, 0]
Time = 0.38 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(-\frac {x \,{\mathrm e}^{c \,x^{2}-b x -a}}{4 c}-\frac {b \,{\mathrm e}^{c \,x^{2}-b x -a}}{8 c^{2}}-\frac {b^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c +b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{16 c^{2} \sqrt {-c}}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c +b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{8 c \sqrt {-c}}-\frac {x \,{\mathrm e}^{-c \,x^{2}+b x +a}}{4 c}-\frac {b \,{\mathrm e}^{-c \,x^{2}+b x +a}}{8 c^{2}}-\frac {b^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{16 c^{\frac {5}{2}}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}\) | \(244\) |
-1/4/c*x*exp(c*x^2-b*x-a)-1/8*b/c^2*exp(c*x^2-b*x-a)-1/16*b^2/c^2*Pi^(1/2) *exp(-1/4*(4*a*c+b^2)/c)/(-c)^(1/2)*erf((-c)^(1/2)*x+1/2*b/(-c)^(1/2))+1/8 /c*Pi^(1/2)*exp(-1/4*(4*a*c+b^2)/c)/(-c)^(1/2)*erf((-c)^(1/2)*x+1/2*b/(-c) ^(1/2))-1/4/c*x*exp(-c*x^2+b*x+a)-1/8*b/c^2*exp(-c*x^2+b*x+a)-1/16*b^2/c^( 5/2)*Pi^(1/2)*exp(1/4*(4*a*c+b^2)/c)*erf(-c^(1/2)*x+1/2*b/c^(1/2))-1/8/c^( 3/2)*Pi^(1/2)*exp(1/4*(4*a*c+b^2)/c)*erf(-c^(1/2)*x+1/2*b/c^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (179) = 358\).
Time = 0.25 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.06 \[ \int x^2 \sinh \left (a+b x-c x^2\right ) \, dx=-\frac {4 \, c^{2} x + 2 \, {\left (2 \, c^{2} x + b c\right )} \cosh \left (c x^{2} - b x - a\right )^{2} - \sqrt {\pi } {\left ({\left (b^{2} - 2 \, c\right )} \cosh \left (c x^{2} - b x - a\right ) \cosh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) - {\left (b^{2} - 2 \, c\right )} \cosh \left (c x^{2} - b x - a\right ) \sinh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + {\left ({\left (b^{2} - 2 \, c\right )} \cosh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) - {\left (b^{2} - 2 \, c\right )} \sinh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {{\left (2 \, c x - b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {\pi } {\left ({\left (b^{2} + 2 \, c\right )} \cosh \left (c x^{2} - b x - a\right ) \cosh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + {\left (b^{2} + 2 \, c\right )} \cosh \left (c x^{2} - b x - a\right ) \sinh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + {\left ({\left (b^{2} + 2 \, c\right )} \cosh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + {\left (b^{2} + 2 \, c\right )} \sinh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x - b}{2 \, \sqrt {c}}\right ) + 4 \, {\left (2 \, c^{2} x + b c\right )} \cosh \left (c x^{2} - b x - a\right ) \sinh \left (c x^{2} - b x - a\right ) + 2 \, {\left (2 \, c^{2} x + b c\right )} \sinh \left (c x^{2} - b x - a\right )^{2} + 2 \, b c}{16 \, {\left (c^{3} \cosh \left (c x^{2} - b x - a\right ) + c^{3} \sinh \left (c x^{2} - b x - a\right )\right )}} \]
-1/16*(4*c^2*x + 2*(2*c^2*x + b*c)*cosh(c*x^2 - b*x - a)^2 - sqrt(pi)*((b^ 2 - 2*c)*cosh(c*x^2 - b*x - a)*cosh(1/4*(b^2 + 4*a*c)/c) - (b^2 - 2*c)*cos h(c*x^2 - b*x - a)*sinh(1/4*(b^2 + 4*a*c)/c) + ((b^2 - 2*c)*cosh(1/4*(b^2 + 4*a*c)/c) - (b^2 - 2*c)*sinh(1/4*(b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x - a) )*sqrt(-c)*erf(1/2*(2*c*x - b)*sqrt(-c)/c) - sqrt(pi)*((b^2 + 2*c)*cosh(c* x^2 - b*x - a)*cosh(1/4*(b^2 + 4*a*c)/c) + (b^2 + 2*c)*cosh(c*x^2 - b*x - a)*sinh(1/4*(b^2 + 4*a*c)/c) + ((b^2 + 2*c)*cosh(1/4*(b^2 + 4*a*c)/c) + (b ^2 + 2*c)*sinh(1/4*(b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x - a))*sqrt(c)*erf(1/ 2*(2*c*x - b)/sqrt(c)) + 4*(2*c^2*x + b*c)*cosh(c*x^2 - b*x - a)*sinh(c*x^ 2 - b*x - a) + 2*(2*c^2*x + b*c)*sinh(c*x^2 - b*x - a)^2 + 2*b*c)/(c^3*cos h(c*x^2 - b*x - a) + c^3*sinh(c*x^2 - b*x - a))
\[ \int x^2 \sinh \left (a+b x-c x^2\right ) \, dx=\int x^{2} \sinh {\left (a + b x - c x^{2} \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 834 vs. \(2 (179) = 358\).
Time = 0.44 (sec) , antiderivative size = 834, normalized size of antiderivative = 3.67 \[ \int x^2 \sinh \left (a+b x-c x^2\right ) \, dx=\text {Too large to display} \]
-1/3*x^3*sinh(c*x^2 - b*x - a) - 1/96*(sqrt(pi)*(2*c*x - b)*b^3*(erf(1/2*s qrt((2*c*x - b)^2/c)) - 1)/(sqrt((2*c*x - b)^2/c)*(-c)^(7/2)) - 6*b^2*c*e^ (-1/4*(2*c*x - b)^2/c)/(-c)^(7/2) - 12*(2*c*x - b)^3*b*gamma(3/2, 1/4*(2*c *x - b)^2/c)/(((2*c*x - b)^2/c)^(3/2)*(-c)^(7/2)) - 8*c^2*gamma(2, 1/4*(2* c*x - b)^2/c)/(-c)^(7/2))*b*e^(a + 1/4*b^2/c)/sqrt(-c) - 1/96*(sqrt(pi)*(2 *c*x - b)*b^4*(erf(1/2*sqrt((2*c*x - b)^2/c)) - 1)/(sqrt((2*c*x - b)^2/c)* (-c)^(9/2)) - 8*b^3*c*e^(-1/4*(2*c*x - b)^2/c)/(-c)^(9/2) - 24*(2*c*x - b) ^3*b^2*gamma(3/2, 1/4*(2*c*x - b)^2/c)/(((2*c*x - b)^2/c)^(3/2)*(-c)^(9/2) ) - 32*b*c^2*gamma(2, 1/4*(2*c*x - b)^2/c)/(-c)^(9/2) - 16*(2*c*x - b)^5*g amma(5/2, 1/4*(2*c*x - b)^2/c)/(((2*c*x - b)^2/c)^(5/2)*(-c)^(9/2)))*c*e^( a + 1/4*b^2/c)/sqrt(-c) - 1/96*(sqrt(pi)*(2*c*x - b)*b^3*(erf(1/2*sqrt(-(2 *c*x - b)^2/c)) - 1)/(sqrt(-(2*c*x - b)^2/c)*c^(7/2)) + 6*b^2*e^(1/4*(2*c* x - b)^2/c)/c^(5/2) - 12*(2*c*x - b)^3*b*gamma(3/2, -1/4*(2*c*x - b)^2/c)/ ((-(2*c*x - b)^2/c)^(3/2)*c^(7/2)) - 8*gamma(2, -1/4*(2*c*x - b)^2/c)/c^(3 /2))*b*e^(-a - 1/4*b^2/c)/sqrt(c) + 1/96*(sqrt(pi)*(2*c*x - b)*b^4*(erf(1/ 2*sqrt(-(2*c*x - b)^2/c)) - 1)/(sqrt(-(2*c*x - b)^2/c)*c^(9/2)) + 8*b^3*e^ (1/4*(2*c*x - b)^2/c)/c^(7/2) - 24*(2*c*x - b)^3*b^2*gamma(3/2, -1/4*(2*c* x - b)^2/c)/((-(2*c*x - b)^2/c)^(3/2)*c^(9/2)) - 32*b*gamma(2, -1/4*(2*c*x - b)^2/c)/c^(5/2) - 16*(2*c*x - b)^5*gamma(5/2, -1/4*(2*c*x - b)^2/c)/((- (2*c*x - b)^2/c)^(5/2)*c^(9/2)))*sqrt(c)*e^(-a - 1/4*b^2/c)
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.74 \[ \int x^2 \sinh \left (a+b x-c x^2\right ) \, dx=-\frac {\frac {\sqrt {\pi } {\left (b^{2} + 2 \, c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c {\left (2 \, x - \frac {b}{c}\right )} + 2 \, b\right )} e^{\left (-c x^{2} + b x + a\right )}}{16 \, c^{2}} + \frac {\frac {\sqrt {\pi } {\left (b^{2} - 2 \, c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{\sqrt {-c}} - 2 \, {\left (c {\left (2 \, x - \frac {b}{c}\right )} + 2 \, b\right )} e^{\left (c x^{2} - b x - a\right )}}{16 \, c^{2}} \]
-1/16*(sqrt(pi)*(b^2 + 2*c)*erf(-1/2*sqrt(c)*(2*x - b/c))*e^(1/4*(b^2 + 4* a*c)/c)/sqrt(c) + 2*(c*(2*x - b/c) + 2*b)*e^(-c*x^2 + b*x + a))/c^2 + 1/16 *(sqrt(pi)*(b^2 - 2*c)*erf(-1/2*sqrt(-c)*(2*x - b/c))*e^(-1/4*(b^2 + 4*a*c )/c)/sqrt(-c) - 2*(c*(2*x - b/c) + 2*b)*e^(c*x^2 - b*x - a))/c^2
Timed out. \[ \int x^2 \sinh \left (a+b x-c x^2\right ) \, dx=\int x^2\,\mathrm {sinh}\left (-c\,x^2+b\,x+a\right ) \,d x \]