Integrand size = 19, antiderivative size = 261 \[ \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx=\frac {e^2 e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {(2 c d-b e)^2 e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}}-\frac {e^2 e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {(2 c d-b e)^2 e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}}+\frac {e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c} \]
1/4*e*(-b*e+2*c*d)*sinh(c*x^2+b*x+a)/c^2+1/2*e*(e*x+d)*sinh(c*x^2+b*x+a)/c +1/8*e^2*exp(-a+1/4*b^2/c)*erf(1/2*(2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(3/2)+1/1 6*(-b*e+2*c*d)^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*e^2*exp(a-1/4*b^2/c)*erfi(1/2*(2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(3/2) +1/16*(-b*e+2*c*d)^2*exp(a-1/4*b^2/c)*erfi(1/2*(2*c*x+b)/c^(1/2))*Pi^(1/2) /c^(5/2)
Time = 0.36 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.74 \[ \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx=\frac {\left (4 c^2 d^2+b^2 e^2+2 c e (-2 b d+e)\right ) \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )-\sinh \left (a-\frac {b^2}{4 c}\right )\right )+\left (4 c^2 d^2+b^2 e^2-2 c e (2 b d+e)\right ) \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )+\sinh \left (a-\frac {b^2}{4 c}\right )\right )+4 \sqrt {c} e (4 c d-b e+2 c e x) \sinh (a+x (b+c x))}{16 c^{5/2}} \]
((4*c^2*d^2 + b^2*e^2 + 2*c*e*(-2*b*d + e))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sq rt[c])]*(Cosh[a - b^2/(4*c)] - Sinh[a - b^2/(4*c)]) + (4*c^2*d^2 + b^2*e^2 - 2*c*e*(2*b*d + e))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])]*(Cosh[a - b^2 /(4*c)] + Sinh[a - b^2/(4*c)]) + 4*Sqrt[c]*e*(4*c*d - b*e + 2*c*e*x)*Sinh[ a + x*(b + c*x)])/(16*c^(5/2))
Time = 0.86 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {5910, 5897, 2664, 2633, 2634, 5906, 5898, 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.
| \(\displaystyle \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5910 |
| \(\displaystyle \frac {(2 c d-b e) \int (d+e x) \cosh \left (c x^2+b x+a\right )dx}{2 c}-\frac {e^2 \int \sinh \left (c x^2+b x+a\right )dx}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 5897 |
| \(\displaystyle \frac {(2 c d-b e) \int (d+e x) \cosh \left (c x^2+b x+a\right )dx}{2 c}-\frac {e^2 \left (\frac {1}{2} \int e^{c x^2+b x+a}dx-\frac {1}{2} \int e^{-c x^2-b x-a}dx\right )}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle -\frac {e^2 \left (\frac {1}{2} e^{a-\frac {b^2}{4 c}} \int e^{\frac {(b+2 c x)^2}{4 c}}dx-\frac {1}{2} e^{\frac {b^2}{4 c}-a} \int e^{-\frac {(b+2 c x)^2}{4 c}}dx\right )}{2 c}+\frac {(2 c d-b e) \int (d+e x) \cosh \left (c x^2+b x+a\right )dx}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {e^2 \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{2} e^{\frac {b^2}{4 c}-a} \int e^{-\frac {(b+2 c x)^2}{4 c}}dx\right )}{2 c}+\frac {(2 c d-b e) \int (d+e x) \cosh \left (c x^2+b x+a\right )dx}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {(2 c d-b e) \int (d+e x) \cosh \left (c x^2+b x+a\right )dx}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 5906 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \int \cosh \left (c x^2+b x+a\right )dx}{2 c}+\frac {e \sinh \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 5898 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {1}{2} \int e^{-c x^2-b x-a}dx+\frac {1}{2} \int e^{c x^2+b x+a}dx\right )}{2 c}+\frac {e \sinh \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {1}{2} e^{\frac {b^2}{4 c}-a} \int e^{-\frac {(b+2 c x)^2}{4 c}}dx+\frac {1}{2} e^{a-\frac {b^2}{4 c}} \int e^{\frac {(b+2 c x)^2}{4 c}}dx\right )}{2 c}+\frac {e \sinh \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {1}{2} e^{\frac {b^2}{4 c}-a} \int e^{-\frac {(b+2 c x)^2}{4 c}}dx+\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}+\frac {e \sinh \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}+\frac {e \sinh \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{2 c}+\frac {e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\) |
-1/2*(e^2*(-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*S qrt[c])))/c + (e*(d + e*x)*Sinh[a + b*x + c*x^2])/(2*c) + ((2*c*d - b*e)*( ((2*c*d - b*e)*((E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])]) /(4*Sqrt[c]) + (E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/ (4*Sqrt[c])))/(2*c) + (e*Sinh[a + b*x + c*x^2])/(2*c)))/(2*c)
3.1.28.3.1 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[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[e*(Sinh[a + b*x + c*x^2]/(2*c)), x] - Simp[(b*e - 2*c*d)/(2*c) I nt[Cosh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_))^(m_), x_Sy mbol] :> Simp[e*(d + e*x)^(m - 1)*(Sinh[a + b*x + c*x^2]/(2*c)), x] + (-Sim p[(b*e - 2*c*d)/(2*c) Int[(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2], x], x] - Simp[e^2*((m - 1)/(2*c)) Int[(d + e*x)^(m - 2)*Sinh[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]
Leaf count of result is larger than twice the leaf count of optimal. \(459\) vs. \(2(205)=410\).
Time = 0.22 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.76
| method | result | size |
| risch | \(\frac {\operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right ) \sqrt {\pi }\, d^{2} {\mathrm e}^{-\frac {4 a c -b^{2}}{4 c}}}{4 \sqrt {c}}-\frac {{\mathrm e}^{-a} e^{2} x \,{\mathrm e}^{-x \left (c x +b \right )}}{4 c}+\frac {{\mathrm e}^{-a} e^{2} b \,{\mathrm e}^{-x \left (c x +b \right )}}{8 c^{2}}+\frac {{\mathrm e}^{-a} e^{2} b^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{16 c^{\frac {5}{2}}}+\frac {{\mathrm e}^{-a} e^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}-\frac {{\mathrm e}^{-a} d e \,{\mathrm e}^{-x \left (c x +b \right )}}{2 c}-\frac {{\mathrm e}^{-a} d e b \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}-\frac {\operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) \sqrt {\pi }\, d^{2} {\mathrm e}^{\frac {4 a c -b^{2}}{4 c}}}{4 \sqrt {-c}}+\frac {{\mathrm e}^{a} e^{2} x \,{\mathrm e}^{x \left (c x +b \right )}}{4 c}-\frac {{\mathrm e}^{a} e^{2} b \,{\mathrm e}^{x \left (c x +b \right )}}{8 c^{2}}-\frac {{\mathrm e}^{a} e^{2} b^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{16 c^{2} \sqrt {-c}}+\frac {{\mathrm e}^{a} e^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{8 c \sqrt {-c}}+\frac {{\mathrm e}^{a} d e \,{\mathrm e}^{x \left (c x +b \right )}}{2 c}+\frac {{\mathrm e}^{a} d e b \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{4 c \sqrt {-c}}\) | \(460\) |
1/4*erf(c^(1/2)*x+1/2*b/c^(1/2))/c^(1/2)*Pi^(1/2)*d^2*exp(-1/4*(4*a*c-b^2) /c)-1/4*exp(-a)*e^2/c*x*exp(-x*(c*x+b))+1/8*exp(-a)*e^2/c^2*b*exp(-x*(c*x+ b))+1/16*exp(-a)*e^2/c^(5/2)*b^2*Pi^(1/2)*exp(1/4*b^2/c)*erf(c^(1/2)*x+1/2 *b/c^(1/2))+1/8*exp(-a)*e^2/c^(3/2)*Pi^(1/2)*exp(1/4*b^2/c)*erf(c^(1/2)*x+ 1/2*b/c^(1/2))-1/2*exp(-a)*d*e/c*exp(-x*(c*x+b))-1/4*exp(-a)*d*e*b/c^(3/2) *Pi^(1/2)*exp(1/4*b^2/c)*erf(c^(1/2)*x+1/2*b/c^(1/2))-1/4*erf(-(-c)^(1/2)* x+1/2*b/(-c)^(1/2))/(-c)^(1/2)*Pi^(1/2)*d^2*exp(1/4*(4*a*c-b^2)/c)+1/4*exp (a)*e^2/c*x*exp(x*(c*x+b))-1/8*exp(a)*e^2/c^2*b*exp(x*(c*x+b))-1/16*exp(a) *e^2/c^2*b^2*Pi^(1/2)*exp(-1/4*b^2/c)/(-c)^(1/2)*erf(-(-c)^(1/2)*x+1/2*b/( -c)^(1/2))+1/8*exp(a)*e^2/c*Pi^(1/2)*exp(-1/4*b^2/c)/(-c)^(1/2)*erf(-(-c)^ (1/2)*x+1/2*b/(-c)^(1/2))+1/2*exp(a)*d*e/c*exp(x*(c*x+b))+1/4*exp(a)*d*e*b /c*Pi^(1/2)*exp(-1/4*b^2/c)/(-c)^(1/2)*erf(-(-c)^(1/2)*x+1/2*b/(-c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (205) = 410\).
Time = 0.27 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.43 \[ \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx=-\frac {4 \, c^{2} e^{2} x + 8 \, c^{2} d e - 2 \, b c e^{2} - 2 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \cosh \left (c x^{2} + b x + a\right )^{2} + \sqrt {\pi } {\left ({\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} - 2 \, c\right )} e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} - 2 \, c\right )} e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left ({\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} - 2 \, c\right )} e^{2}\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} - 2 \, c\right )} e^{2}\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 (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} + 2 \, c\right )} e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - {\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} + 2 \, c\right )} e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left ({\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} + 2 \, c\right )} e^{2}\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - {\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} + 2 \, c\right )} e^{2}\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} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \sinh \left (c x^{2} + b x + a\right )^{2}}{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*e^2*x + 8*c^2*d*e - 2*b*c*e^2 - 2*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)^2 + sqrt(pi)*((4*c^2*d^2 - 4*b*c*d*e + (b^2 - 2*c)*e^2)*cosh(c*x^2 + b*x + a)*cosh(-1/4*(b^2 - 4*a*c)/c) + (4*c^2*d^2 - 4*b*c*d*e + (b^2 - 2*c)*e^2)*cosh(c*x^2 + b*x + a)*sinh(-1/4*(b^2 - 4*a *c)/c) + ((4*c^2*d^2 - 4*b*c*d*e + (b^2 - 2*c)*e^2)*cosh(-1/4*(b^2 - 4*a*c )/c) + (4*c^2*d^2 - 4*b*c*d*e + (b^2 - 2*c)*e^2)*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)*((4*c^2*d^2 - 4*b*c*d*e + (b^2 + 2*c)*e^2)*cosh(c*x^2 + b*x + a)*cosh( -1/4*(b^2 - 4*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + 2*c)*e^2)*cosh(c*x ^2 + b*x + a)*sinh(-1/4*(b^2 - 4*a*c)/c) + ((4*c^2*d^2 - 4*b*c*d*e + (b^2 + 2*c)*e^2)*cosh(-1/4*(b^2 - 4*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + 2 *c)*e^2)*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*e^2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a) - 2*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*s inh(c*x^2 + b*x + a)^2)/(c^3*cosh(c*x^2 + b*x + a) + c^3*sinh(c*x^2 + b*x + a))
\[ \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right )^{2} \cosh {\left (a + b x + c x^{2} \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (205) = 410\).
Time = 0.36 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.05 \[ \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {-c} x - \frac {b}{2 \, \sqrt {-c}}\right ) e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {-c}} + \frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {c} x + \frac {b}{2 \, \sqrt {c}}\right ) e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} d e e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {5}{2}}} - \frac {4 \, b e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} e^{2} e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{16 \, \sqrt {c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {2 \, c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} d e e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {-c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {5}{2}}} + \frac {4 \, b c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {5}{2}}}\right )} e^{2} e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{16 \, \sqrt {-c}} \]
1/4*sqrt(pi)*d^2*erf(sqrt(-c)*x - 1/2*b/sqrt(-c))*e^(a - 1/4*b^2/c)/sqrt(- c) + 1/4*sqrt(pi)*d^2*erf(sqrt(c)*x + 1/2*b/sqrt(c))*e^(-a + 1/4*b^2/c)/sq rt(c) - 1/4*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/ (sqrt(-(2*c*x + b)^2/c)*c^(3/2)) - 2*e^(1/4*(2*c*x + b)^2/c)/sqrt(c))*d*e* e^(a - 1/4*b^2/c)/sqrt(c) + 1/16*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt(- (2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(5/2)) - 4*b*e^(1/4*(2*c* x + b)^2/c)/c^(3/2) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((- (2*c*x + b)^2/c)^(3/2)*c^(5/2)))*e^2*e^(a - 1/4*b^2/c)/sqrt(c) - 1/4*(sqrt (pi)*(2*c*x + b)*b*(erf(1/2*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^ 2/c)*(-c)^(3/2)) + 2*c*e^(-1/4*(2*c*x + b)^2/c)/(-c)^(3/2))*d*e*e^(-a + 1/ 4*b^2/c)/sqrt(-c) - 1/16*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(5/2)) + 4*b*c*e^(-1/4*(2*c*x + b)^2/c)/(-c)^(5/2) - 4*(2*c*x + b)^3*gamma(3/2, 1/4*(2*c*x + b)^2/c)/(((2* c*x + b)^2/c)^(3/2)*(-c)^(5/2)))*e^2*e^(-a + 1/4*b^2/c)/sqrt(-c)
Time = 0.27 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.87 \[ \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx=-\frac {\frac {\sqrt {\pi } {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2} + 2 \, c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (-c x^{2} - b x - a\right )}}{16 \, c^{2}} - \frac {\frac {\sqrt {\pi } {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2} - 2 \, c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {-c}} - 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (c x^{2} + b x + a\right )}}{16 \, c^{2}} \]
-1/16*(sqrt(pi)*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2 + 2*c*e^2)*erf(-1/2*sqrt( c)*(2*x + b/c))*e^(1/4*(b^2 - 4*a*c)/c)/sqrt(c) + 2*(c*e^2*(2*x + b/c) + 4 *c*d*e - 2*b*e^2)*e^(-c*x^2 - b*x - a))/c^2 - 1/16*(sqrt(pi)*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2 - 2*c*e^2)*erf(-1/2*sqrt(-c)*(2*x + b/c))*e^(-1/4*(b^2 - 4*a*c)/c)/sqrt(-c) - 2*(c*e^2*(2*x + b/c) + 4*c*d*e - 2*b*e^2)*e^(c*x^2 + b*x + a))/c^2
Timed out. \[ \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx=\int \mathrm {cosh}\left (c\,x^2+b\,x+a\right )\,{\left (d+e\,x\right )}^2 \,d x \]