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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5495, 5491, 5483, 2266, 2235, 2236, 5482} \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {3}{16} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-2 x-1)\right )-\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )+\frac {1}{2} x \sinh \left (x^2+x+\frac {1}{4}\right )-\frac {1}{4} \sinh \left (x^2+x+\frac {1}{4}\right ) \]

[In]

Int[x^2*Cosh[1/4 + x + x^2],x]

[Out]

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

Rule 2235

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 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 5482

Int[Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] - Dist[1/2
, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]

Rule 5483

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] + Dist[1/2
, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]

Rule 5491

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] - Dist[(b*e - 2*c*d)/(2*c), Int[Cosh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*
e - 2*c*d, 0]

Rule 5495

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*(
Sinh[a + b*x + c*x^2]/(2*c)), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2], x]
, x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sinh \left (\frac {1}{4}+x+x^2\right )-\frac {1}{2} \int x \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx-\frac {1}{2} \int \sinh \left (\frac {1}{4}+x+x^2\right ) \, dx \\ & = -\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 )+\frac {1}{4} \int e^{-\frac {1}{4}-x-x^2} \, dx-\frac {1}{4} \int e^{\frac {1}{4}+x+x^2} \, dx+\frac {1}{4} \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx \\ & = -\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 )+\frac {1}{8} \int e^{-\frac {1}{4}-x-x^2} \, dx+\frac {1}{8} \int e^{\frac {1}{4}+x+x^2} \, dx+\frac {1}{4} \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx-\frac {1}{4} \int e^{\frac {1}{4} (1+2 x)^2} \, dx \\ & = -\frac {1}{8} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right )-\frac {1}{8} \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 )+\frac {1}{8} \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx+\frac {1}{8} \int e^{\frac {1}{4} (1+2 x)^2} \, 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 ) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[x^2*Cosh[1/4 + x + x^2],x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14

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

[In]

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

[Out]

-1/4*x*exp(-1/4*(1+2*x)^2)+1/8*exp(-1/4*(1+2*x)^2)+3/16*erf(1/2+x)*Pi^(1/2)+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 = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.95 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {2 \, {\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac {1}{4}\right )^{2} + 4 \, {\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac {1}{4}\right ) \sinh \left (x^{2} + x + \frac {1}{4}\right ) + 2 \, {\left (2 \, x - 1\right )} \sinh \left (x^{2} + x + \frac {1}{4}\right )^{2} + \sqrt {\pi } {\left (3 \, \cosh \left (x^{2} + x + \frac {1}{4}\right ) \operatorname {erf}\left (x + \frac {1}{2}\right ) - \cosh \left (x^{2} + x + \frac {1}{4}\right ) \operatorname {erfi}\left (x + \frac {1}{2}\right ) + {\left (3 \, \operatorname {erf}\left (x + \frac {1}{2}\right ) - \operatorname {erfi}\left (x + \frac {1}{2}\right )\right )} \sinh \left (x^{2} + x + \frac {1}{4}\right )\right )} - 4 \, x + 2}{16 \, {\left (\cosh \left (x^{2} + x + \frac {1}{4}\right ) + \sinh \left (x^{2} + x + \frac {1}{4}\right )\right )}} \]

[In]

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

[Out]

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)*si
nh(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*er
f(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]

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

[In]

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

[Out]

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 = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.77 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{3} \, x^{3} \cosh \left (x^{2} + x + \frac {1}{4}\right ) - \frac {{\left (2 \, x + 1\right )}^{5} \Gamma \left (\frac {5}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{6 \, {\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (2 \, x + 1\right )}^{5} \Gamma \left (\frac {5}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{6 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {5}{2}}} - \frac {{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{8 \, {\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{8 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {3}{2}}} + \frac {1}{48} \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} + \frac {1}{48} \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} + \frac {1}{4} \, \Gamma \left (2, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right ) - \frac {1}{4} \, \Gamma \left (2, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right ) \]

[In]

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

[Out]

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 = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int x^2 \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{8} \, {\left (2 \, x - 1\right )} e^{\left (x^{2} + x + \frac {1}{4}\right )} - \frac {1}{8} \, {\left (2 \, x - 1\right )} e^{\left (-x^{2} - x - \frac {1}{4}\right )} + \frac {3}{16} \, \sqrt {\pi } \operatorname {erf}\left (x + \frac {1}{2}\right ) - \frac {1}{16} i \, \sqrt {\pi } \operatorname {erf}\left (-i \, x - \frac {1}{2} i\right ) \]

[In]

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

[Out]

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

[In]

int(x^2*cosh(x + x^2 + 1/4),x)

[Out]

int(x^2*cosh(x + x^2 + 1/4), x)