3.13 $$\int \cosh (\frac{1}{4}+x+x^2) \, dx$$

Optimal. Leaf size=39 $\frac{1}{4} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} (2 x+1)\right )-\frac{1}{4} \sqrt{\pi } \text{Erf}\left (\frac{1}{2} (-2 x-1)\right )$

[Out]

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

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Rubi [A]  time = 0.015898, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.444, Rules used = {5375, 2234, 2204, 2205} $\frac{1}{4} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} (2 x+1)\right )-\frac{1}{4} \sqrt{\pi } \text{Erf}\left (\frac{1}{2} (-2 x-1)\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[1/4 + x + x^2],x]

[Out]

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

Rule 5375

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 2234

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 2204

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 2205

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]

Rubi steps

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

Mathematica [A]  time = 0.0230148, size = 22, normalized size = 0.56 $\frac{1}{4} \sqrt{\pi } \left (\text{Erf}\left (x+\frac{1}{2}\right )+\text{Erfi}\left (x+\frac{1}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[1/4 + x + x^2],x]

[Out]

(Sqrt[Pi]*(Erf[1/2 + x] + Erfi[1/2 + x]))/4

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Maple [C]  time = 0.027, size = 25, normalized size = 0.6 \begin{align*}{\frac{\sqrt{\pi }}{4}{\it Erf} \left ({\frac{1}{2}}+x \right ) }-{\frac{i}{4}}\sqrt{\pi }{\it Erf} \left ( ix+{\frac{i}{2}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*erf(1/2+x)*Pi^(1/2)-1/4*I*Pi^(1/2)*erf(I*x+1/2*I)

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Maxima [B]  time = 1.39304, size = 127, normalized size = 3.26 \begin{align*} -\frac{{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )}{2 \,{\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 )}{2 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac{3}{2}}} + x \cosh \left (x^{2} + x + \frac{1}{4}\right ) + \frac{1}{4} \, e^{\left (\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )} + \frac{1}{4} \, e^{\left (-\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.48161, size = 61, normalized size = 1.56 \begin{align*} \frac{1}{4} \, \sqrt{\pi }{\left (\operatorname{erf}\left (x + \frac{1}{2}\right ) + \operatorname{erfi}\left (x + \frac{1}{2}\right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(pi)*(erf(x + 1/2) + erfi(x + 1/2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x^{2} + x + \frac{1}{4} \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x**2 + x + 1/4), x)

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Giac [C]  time = 1.23348, size = 28, normalized size = 0.72 \begin{align*} \frac{1}{4} \, \sqrt{\pi } \operatorname{erf}\left (x + \frac{1}{2}\right ) + \frac{1}{4} i \, \sqrt{\pi } \operatorname{erf}\left (-i \, x - \frac{1}{2} i\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(pi)*erf(x + 1/2) + 1/4*I*sqrt(pi)*erf(-I*x - 1/2*I)