### 3.2 $$\int x \cosh (a+b x+c x^2) \, dx$$

Optimal. Leaf size=111 $-\frac{\sqrt{\pi } b e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } b e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\sinh \left (a+b x+c x^2\right )}{2 c}$

[Out]

-(b*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - (b*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi
[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + Sinh[a + b*x + c*x^2]/(2*c)

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Rubi [A]  time = 0.0481427, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.385, Rules used = {5383, 5375, 2234, 2204, 2205} $-\frac{\sqrt{\pi } b e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } b e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\sinh \left (a+b x+c x^2\right )}{2 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Cosh[a + b*x + c*x^2],x]

[Out]

-(b*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - (b*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi
[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + Sinh[a + b*x + c*x^2]/(2*c)

Rule 5383

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 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 x \cosh \left (a+b x+c x^2\right ) \, dx &=\frac{\sinh \left (a+b x+c x^2\right )}{2 c}-\frac{b \int \cosh \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=\frac{\sinh \left (a+b x+c x^2\right )}{2 c}-\frac{b \int e^{-a-b x-c x^2} \, dx}{4 c}-\frac{b \int e^{a+b x+c x^2} \, dx}{4 c}\\ &=\frac{\sinh \left (a+b x+c x^2\right )}{2 c}-\frac{\left (b e^{a-\frac{b^2}{4 c}}\right ) \int e^{\frac{(b+2 c x)^2}{4 c}} \, dx}{4 c}-\frac{\left (b e^{-a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(-b-2 c x)^2}{4 c}} \, dx}{4 c}\\ &=-\frac{b 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 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{\sinh \left (a+b x+c x^2\right )}{2 c}\\ \end{align*}

Mathematica [A]  time = 0.143164, size = 130, normalized size = 1.17 $\frac{\sqrt{\pi } b \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )-\cosh \left (a-\frac{b^2}{4 c}\right )\right )-\sqrt{\pi } b \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )+\cosh \left (a-\frac{b^2}{4 c}\right )\right )+4 \sqrt{c} \sinh (a+x (b+c x))}{8 c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Cosh[a + b*x + c*x^2],x]

[Out]

(b*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])]*(-Cosh[a - b^2/(4*c)] + Sinh[a - b^2/(4*c)]) - b*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]*Sinh[a + x*(b + c*x)])/(8*c^(3/2))

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Maple [A]  time = 0.034, size = 124, normalized size = 1.1 \begin{align*} -{\frac{{{\rm e}^{-c{x}^{2}-bx-a}}}{4\,c}}-{\frac{b\sqrt{\pi }}{8}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{{\rm e}^{c{x}^{2}+bx+a}}}{4\,c}}+{\frac{b\sqrt{\pi }}{8\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-c}}}} \right ){\frac{1}{\sqrt{-c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(c*x^2+b*x+a),x)

[Out]

-1/4/c*exp(-c*x^2-b*x-a)-1/8*b/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))+1/4/c*exp
(c*x^2+b*x+a)+1/8*b/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))

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Maxima [B]  time = 1.58906, size = 825, normalized size = 7.43 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/2*x^2*cosh(c*x^2 + b*x + a) - 1/32*(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)))*b*e^(a - 1/4*b^2/c)/sqrt(c) + 1/32*(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*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))*sqrt(c)*e^(a - 1/4*b^2/c) - 1/32*(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)))*b*e^(-a + 1/4*b^2/c)/sqrt(-c) - 1/32*(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*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))*c*e^(-a + 1/4*b^2/c)/sqrt(-c)

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Fricas [B]  time = 1.84975, size = 892, normalized size = 8.04 \begin{align*} \frac{2 \, c \cosh \left (c x^{2} + b x + a\right )^{2} + \sqrt{\pi }{\left (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) + b \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left (b \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) + b \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 (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - b \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left (b \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - b \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 \, c \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + 2 \, c \sinh \left (c x^{2} + b x + a\right )^{2} - 2 \, c}{8 \,{\left (c^{2} \cosh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/8*(2*c*cosh(c*x^2 + b*x + a)^2 + sqrt(pi)*(b*cosh(c*x^2 + b*x + a)*cosh(-1/4*(b^2 - 4*a*c)/c) + b*cosh(c*x^2
+ b*x + a)*sinh(-1/4*(b^2 - 4*a*c)/c) + (b*cosh(-1/4*(b^2 - 4*a*c)/c) + b*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*cosh(c*x^2 + b*x + a)*cosh(-1/4*(b^2 -
4*a*c)/c) - b*cosh(c*x^2 + b*x + a)*sinh(-1/4*(b^2 - 4*a*c)/c) + (b*cosh(-1/4*(b^2 - 4*a*c)/c) - b*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*c*cosh(c*x^2 + b*x + a)*sinh(
c*x^2 + b*x + a) + 2*c*sinh(c*x^2 + b*x + a)^2 - 2*c)/(c^2*cosh(c*x^2 + b*x + a) + c^2*sinh(c*x^2 + b*x + a))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(c*x**2+b*x+a),x)

[Out]

Integral(x*cosh(a + b*x + c*x**2), x)

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Giac [A]  time = 1.23404, size = 163, normalized size = 1.47 \begin{align*} \frac{\frac{\sqrt{\pi } b \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 \, e^{\left (-c x^{2} - b x - a\right )}}{8 \, c} + \frac{\frac{\sqrt{\pi } b \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 \, e^{\left (c x^{2} + b x + a\right )}}{8 \, c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/8*(sqrt(pi)*b*erf(-1/2*sqrt(c)*(2*x + b/c))*e^(1/4*(b^2 - 4*a*c)/c)/sqrt(c) - 2*e^(-c*x^2 - b*x - a))/c + 1/
8*(sqrt(pi)*b*erf(-1/2*sqrt(-c)*(2*x + b/c))*e^(-1/4*(b^2 - 4*a*c)/c)/sqrt(-c) + 2*e^(c*x^2 + b*x + a))/c