\(\int x^2 \cosh (\frac {1}{4}+x+x^2) \, dx\) [11]

Optimal result

Integrand size = 13, antiderivative size = 66 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {3}{16} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right )-\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 x)\right )-\frac {1}{4} \sinh \left (\frac {1}{4}+x+x^2\right )+\frac {1}{2} x \sinh \left (\frac {1}{4}+x+x^2\right ) \] Output:

3/16*Pi^(1/2)*erf(1/2+x)-1/16*Pi^(1/2)*erfi(1/2+x)-1/4*sinh(1/4+x+x^2)+1/2 
*x*sinh(1/4+x+x^2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.09 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{16} \left (3 \sqrt {\pi } \text {erf}\left (\frac {1}{2}+x\right )-\sqrt {\pi } \text {erfi}\left (\frac {1}{2}+x\right )+\frac {2 (-1+2 x) \left (\left (-1+\sqrt {e}\right ) \cosh (x (1+x))+\left (1+\sqrt {e}\right ) \sinh (x (1+x))\right )}{\sqrt [4]{e}}\right ) \] Input:

Integrate[x^2*Cosh[1/4 + x + x^2],x]
 

Output:

(3*Sqrt[Pi]*Erf[1/2 + x] - Sqrt[Pi]*Erfi[1/2 + x] + (2*(-1 + 2*x)*((-1 + S 
qrt[E])*Cosh[x*(1 + x)] + (1 + Sqrt[E])*Sinh[x*(1 + x)]))/E^(1/4))/16
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.80, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, 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.

\(\displaystyle \int x^2 \cosh \left (x^2+x+\frac {1}{4}\right ) \, dx\)

\(\Big \downarrow \) 5910

\(\displaystyle -\frac {1}{2} \int \sinh \left (x^2+x+\frac {1}{4}\right )dx-\frac {1}{2} \int x \cosh \left (x^2+x+\frac {1}{4}\right )dx+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 5897

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int e^{-x^2-x-\frac {1}{4}}dx-\frac {1}{2} \int e^{x^2+x+\frac {1}{4}}dx\right )-\frac {1}{2} \int x \cosh \left (x^2+x+\frac {1}{4}\right )dx+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 2664

\(\displaystyle -\frac {1}{2} \int x \cosh \left (x^2+x+\frac {1}{4}\right )dx+\frac {1}{2} \left (\frac {1}{2} \int e^{-\frac {1}{4} (2 x+1)^2}dx-\frac {1}{2} \int e^{\frac {1}{4} (2 x+1)^2}dx\right )+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int e^{-\frac {1}{4} (2 x+1)^2}dx-\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )-\frac {1}{2} \int x \cosh \left (x^2+x+\frac {1}{4}\right )dx+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {1}{2} \int x \cosh \left (x^2+x+\frac {1}{4}\right )dx+\frac {1}{2} \left (\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (2 x+1)\right )-\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 5906

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \cosh \left (x^2+x+\frac {1}{4}\right )dx-\frac {1}{2} \sinh \left (x^2+x+\frac {1}{4}\right )\right )+\frac {1}{2} \left (\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (2 x+1)\right )-\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 5898

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int e^{-x^2-x-\frac {1}{4}}dx+\frac {1}{2} \int e^{x^2+x+\frac {1}{4}}dx\right )-\frac {1}{2} \sinh \left (x^2+x+\frac {1}{4}\right )\right )+\frac {1}{2} \left (\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (2 x+1)\right )-\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 2664

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int e^{-\frac {1}{4} (2 x+1)^2}dx+\frac {1}{2} \int e^{\frac {1}{4} (2 x+1)^2}dx\right )-\frac {1}{2} \sinh \left (x^2+x+\frac {1}{4}\right )\right )+\frac {1}{2} \left (\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (2 x+1)\right )-\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int e^{-\frac {1}{4} (2 x+1)^2}dx+\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )-\frac {1}{2} \sinh \left (x^2+x+\frac {1}{4}\right )\right )+\frac {1}{2} \left (\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (2 x+1)\right )-\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (2 x+1)\right )+\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )-\frac {1}{2} \sinh \left (x^2+x+\frac {1}{4}\right )\right )+\frac {1}{2} \left (\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (2 x+1)\right )-\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )\right )+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )\)

Input:

Int[x^2*Cosh[1/4 + x + x^2],x]
 

Output:

((Sqrt[Pi]*Erf[(1 + 2*x)/2])/4 - (Sqrt[Pi]*Erfi[(1 + 2*x)/2])/4)/2 + (((Sq 
rt[Pi]*Erf[(1 + 2*x)/2])/4 + (Sqrt[Pi]*Erfi[(1 + 2*x)/2])/4)/2 - Sinh[1/4 
+ x + x^2]/2)/2 + (x*Sinh[1/4 + x + x^2])/2
 

Defintions of rubi rules used

rule 2633

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]
 

rule 2634

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]
 

rule 2664

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]
 

rule 5897

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]
 

rule 5898

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]
 

rule 5906

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]
 

rule 5910

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\frac {x \,{\mathrm e}^{-\frac {\left (1+2 x \right )^{2}}{4}}}{4}+\frac {{\mathrm e}^{-\frac {\left (1+2 x \right )^{2}}{4}}}{8}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\frac {1}{2}+x \right )}{16}+\frac {x \,{\mathrm e}^{\frac {\left (1+2 x \right )^{2}}{4}}}{4}-\frac {{\mathrm e}^{\frac {\left (1+2 x \right )^{2}}{4}}}{8}+\frac {i \sqrt {\pi }\, \operatorname {erf}\left (i x +\frac {1}{2} i\right )}{16}\)	\(75\)

Input:

int(x^2*cosh(1/4+x+x^2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*x*exp(-1/4*(1+2*x)^2)+1/8*exp(-1/4*(1+2*x)^2)+3/16*Pi^(1/2)*erf(1/2+x 
)+1/4*x*exp(1/4*(1+2*x)^2)-1/8*exp(1/4*(1+2*x)^2)+1/16*I*Pi^(1/2)*erf(I*x+ 
1/2*I)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (38) = 76\).

Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.95 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {2 \, {\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac {1}{4}\right )^{2} + 4 \, {\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac {1}{4}\right ) \sinh \left (x^{2} + x + \frac {1}{4}\right ) + 2 \, {\left (2 \, x - 1\right )} \sinh \left (x^{2} + x + \frac {1}{4}\right )^{2} + \sqrt {\pi } {\left (3 \, \cosh \left (x^{2} + x + \frac {1}{4}\right ) \operatorname {erf}\left (x + \frac {1}{2}\right ) - \cosh \left (x^{2} + x + \frac {1}{4}\right ) \operatorname {erfi}\left (x + \frac {1}{2}\right ) + {\left (3 \, \operatorname {erf}\left (x + \frac {1}{2}\right ) - \operatorname {erfi}\left (x + \frac {1}{2}\right )\right )} \sinh \left (x^{2} + x + \frac {1}{4}\right )\right )} - 4 \, x + 2}{16 \, {\left (\cosh \left (x^{2} + x + \frac {1}{4}\right ) + \sinh \left (x^{2} + x + \frac {1}{4}\right )\right )}} \] Input:

integrate(x^2*cosh(1/4+x+x^2),x, algorithm="fricas")
 

Output:

1/16*(2*(2*x - 1)*cosh(x^2 + x + 1/4)^2 + 4*(2*x - 1)*cosh(x^2 + x + 1/4)* 
sinh(x^2 + x + 1/4) + 2*(2*x - 1)*sinh(x^2 + x + 1/4)^2 + sqrt(pi)*(3*cosh 
(x^2 + x + 1/4)*erf(x + 1/2) - cosh(x^2 + x + 1/4)*erfi(x + 1/2) + (3*erf( 
x + 1/2) - erfi(x + 1/2))*sinh(x^2 + x + 1/4)) - 4*x + 2)/(cosh(x^2 + x + 
1/4) + sinh(x^2 + x + 1/4))
 

Sympy [F]

\[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\int x^{2} \cosh {\left (x^{2} + x + \frac {1}{4} \right )}\, dx \] Input:

integrate(x**2*cosh(1/4+x+x**2),x)
 

Output:

Integral(x**2*cosh(x**2 + x + 1/4), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (38) = 76\).

Time = 0.14 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.77 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{3} \, x^{3} \cosh \left (x^{2} + x + \frac {1}{4}\right ) - \frac {{\left (2 \, x + 1\right )}^{5} \Gamma \left (\frac {5}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{6 \, {\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (2 \, x + 1\right )}^{5} \Gamma \left (\frac {5}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{6 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {5}{2}}} - \frac {{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{8 \, {\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{8 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {3}{2}}} + \frac {1}{48} \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} + \frac {1}{48} \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} + \frac {1}{4} \, \Gamma \left (2, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right ) - \frac {1}{4} \, \Gamma \left (2, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right ) \] Input:

integrate(x^2*cosh(1/4+x+x^2),x, algorithm="maxima")
 

Output:

1/3*x^3*cosh(x^2 + x + 1/4) - 1/6*(2*x + 1)^5*gamma(5/2, 1/4*(2*x + 1)^2)/ 
((2*x + 1)^2)^(5/2) + 1/6*(2*x + 1)^5*gamma(5/2, -1/4*(2*x + 1)^2)/(-(2*x 
+ 1)^2)^(5/2) - 1/8*(2*x + 1)^3*gamma(3/2, 1/4*(2*x + 1)^2)/((2*x + 1)^2)^ 
(3/2) + 1/8*(2*x + 1)^3*gamma(3/2, -1/4*(2*x + 1)^2)/(-(2*x + 1)^2)^(3/2) 
+ 1/48*e^(1/4*(2*x + 1)^2) + 1/48*e^(-1/4*(2*x + 1)^2) + 1/4*gamma(2, 1/4* 
(2*x + 1)^2) - 1/4*gamma(2, -1/4*(2*x + 1)^2)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{8} \, {\left (2 \, x - 1\right )} e^{\left (x^{2} + x + \frac {1}{4}\right )} - \frac {1}{8} \, {\left (2 \, x - 1\right )} e^{\left (-x^{2} - x - \frac {1}{4}\right )} + \frac {3}{16} \, \sqrt {\pi } \operatorname {erf}\left (x + \frac {1}{2}\right ) - \frac {1}{16} i \, \sqrt {\pi } \operatorname {erf}\left (-i \, x - \frac {1}{2} i\right ) \] Input:

integrate(x^2*cosh(1/4+x+x^2),x, algorithm="giac")
 

Output:

1/8*(2*x - 1)*e^(x^2 + x + 1/4) - 1/8*(2*x - 1)*e^(-x^2 - x - 1/4) + 3/16* 
sqrt(pi)*erf(x + 1/2) - 1/16*I*sqrt(pi)*erf(-I*x - 1/2*I)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\int x^2\,\mathrm {cosh}\left (x^2+x+\frac {1}{4}\right ) \,d x \] Input:

int(x^2*cosh(x + x^2 + 1/4),x)
 

Output:

int(x^2*cosh(x + x^2 + 1/4), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.35 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {\sqrt {\pi }\, e^{x^{2}+x +\frac {1}{2}} \mathrm {erf}\left (i x +\frac {1}{2} i \right ) i +3 \sqrt {\pi }\, e^{x^{2}+x +\frac {1}{2}} \mathrm {erf}\left (x +\frac {1}{2}\right )+4 e^{2 x^{2}+2 x +\frac {3}{4}} x -2 e^{2 x^{2}+2 x +\frac {3}{4}}-4 e^{\frac {1}{4}} x +2 e^{\frac {1}{4}}}{16 e^{x^{2}+x +\frac {1}{2}}} \] Input:

int(x^2*cosh(1/4+x+x^2),x)
 

Output:

(sqrt(pi)*e**((2*x**2 + 2*x + 1)/2)*erf((2*i*x + i)/2)*i + 3*sqrt(pi)*e**( 
(2*x**2 + 2*x + 1)/2)*erf((2*x + 1)/2) + 4*e**((8*x**2 + 8*x + 3)/4)*x - 2 
*e**((8*x**2 + 8*x + 3)/4) - 4*e**(1/4)*x + 2*e**(1/4))/(16*e**((2*x**2 + 
2*x + 1)/2))