### 3.32 $$\int (d+e x) \cosh ^2(a+b x+c x^2) \, dx$$

Optimal. Leaf size=160 $\frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^2}{4 e}$

[Out]

(d + e*x)^2/(4*e) + ((2*c*d - b*e)*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(
3/2)) + ((2*c*d - b*e)*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + (e*S
inh[2*a + 2*b*x + 2*c*x^2])/(8*c)

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Rubi [A]  time = 0.145921, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.316, Rules used = {5395, 5383, 5375, 2234, 2204, 2205} $\frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^2}{4 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*Cosh[a + b*x + c*x^2]^2,x]

[Out]

(d + e*x)^2/(4*e) + ((2*c*d - b*e)*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(
3/2)) + ((2*c*d - b*e)*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + (e*S
inh[2*a + 2*b*x + 2*c*x^2])/(8*c)

Rule 5395

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce
[(d + e*x)^m, Cosh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]

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 (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{1}{2} (d+e x)+\frac{1}{2} (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=\frac{(d+e x)^2}{4 e}+\frac{1}{2} \int (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=\frac{(d+e x)^2}{4 e}+\frac{e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(2 c d-b e) \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c}\\ &=\frac{(d+e x)^2}{4 e}+\frac{e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(2 c d-b e) \int e^{-2 a-2 b x-2 c x^2} \, dx}{8 c}+\frac{(2 c d-b e) \int e^{2 a+2 b x+2 c x^2} \, dx}{8 c}\\ &=\frac{(d+e x)^2}{4 e}+\frac{e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{\left ((2 c d-b e) e^{2 a-\frac{b^2}{2 c}}\right ) \int e^{\frac{(2 b+4 c x)^2}{8 c}} \, dx}{8 c}+\frac{\left ((2 c d-b e) e^{-2 a+\frac{b^2}{2 c}}\right ) \int e^{-\frac{(-2 b-4 c x)^2}{8 c}} \, dx}{8 c}\\ &=\frac{(d+e x)^2}{4 e}+\frac{(2 c d-b e) e^{-2 a+\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{(2 c d-b e) e^{2 a-\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end{align*}

Mathematica [A]  time = 0.639251, size = 177, normalized size = 1.11 $\frac{\sqrt{2 \pi } (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\cosh \left (2 a-\frac{b^2}{2 c}\right )-\sinh \left (2 a-\frac{b^2}{2 c}\right )\right )+\sqrt{2 \pi } (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\sinh \left (2 a-\frac{b^2}{2 c}\right )+\cosh \left (2 a-\frac{b^2}{2 c}\right )\right )+4 \sqrt{c} (e \sinh (2 (a+x (b+c x)))+2 c x (2 d+e x))}{32 c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*Cosh[a + b*x + c*x^2]^2,x]

[Out]

((2*c*d - b*e)*Sqrt[2*Pi]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a - b^2/(2*c)] - Sinh[2*a - b^2/(2*c)]) +
(2*c*d - b*e)*Sqrt[2*Pi]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a - b^2/(2*c)] + Sinh[2*a - b^2/(2*c)])
+ 4*Sqrt[c]*(2*c*x*(2*d + e*x) + e*Sinh[2*(a + x*(b + c*x))]))/(32*c^(3/2))

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Maple [A]  time = 0.057, size = 241, normalized size = 1.5 \begin{align*}{\frac{e{x}^{2}}{4}}+{\frac{dx}{2}}+{\frac{d\sqrt{\pi }\sqrt{2}}{16}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{e{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{16\,c}}-{\frac{be\sqrt{\pi }\sqrt{2}}{32}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}-{\frac{d\sqrt{\pi }}{8}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}}+{\frac{e{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{16\,c}}+{\frac{be\sqrt{\pi }}{16\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*e*x^2+1/2*d*x+1/16*d*Pi^(1/2)*exp(-1/2*(4*a*c-b^2)/c)*2^(1/2)/c^(1/2)*erf(2^(1/2)*c^(1/2)*x+1/2*b*2^(1/2)/
c^(1/2))-1/16*e/c*exp(-2*c*x^2-2*b*x-2*a)-1/32*e*b/c^(3/2)*Pi^(1/2)*exp(-1/2*(4*a*c-b^2)/c)*2^(1/2)*erf(2^(1/2
)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))-1/8*d*Pi^(1/2)*exp(1/2*(4*a*c-b^2)/c)/(-2*c)^(1/2)*erf(-(-2*c)^(1/2)*x+b/(-
2*c)^(1/2))+1/16*e/c*exp(2*c*x^2+2*b*x+2*a)+1/16*e*b/c*Pi^(1/2)*exp(1/2*(4*a*c-b^2)/c)/(-2*c)^(1/2)*erf(-(-2*c
)^(1/2)*x+b/(-2*c)^(1/2))

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Maxima [B]  time = 1.93148, size = 406, normalized size = 2.54 \begin{align*} \frac{1}{16} \,{\left (\frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} \sqrt{-c} x - \frac{\sqrt{2} b}{2 \, \sqrt{-c}}\right ) e^{\left (2 \, a - \frac{b^{2}}{2 \, c}\right )}}{\sqrt{-c}} + \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} \sqrt{c} x + \frac{\sqrt{2} b}{2 \, \sqrt{c}}\right ) e^{\left (-2 \, a + \frac{b^{2}}{2 \, c}\right )}}{\sqrt{c}} + 8 \, x\right )} d + \frac{1}{32} \,{\left (8 \, x^{2} - \frac{\sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b{\left (\operatorname{erf}\left (\sqrt{\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{\sqrt{2} e^{\left (\frac{{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\sqrt{c}}\right )} e^{\left (2 \, a - \frac{b^{2}}{2 \, c}\right )}}{\sqrt{c}} - \frac{\sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b{\left (\operatorname{erf}\left (\sqrt{\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{\sqrt{2} c e^{\left (-\frac{{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac{3}{2}}}\right )} e^{\left (-2 \, a + \frac{b^{2}}{2 \, c}\right )}}{\sqrt{-c}}\right )} e \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(-c)*x - 1/2*sqrt(2)*b/sqrt(-c))*e^(2*a - 1/2*b^2/c)/sqrt(-c) + sqrt(2)
*sqrt(pi)*erf(sqrt(2)*sqrt(c)*x + 1/2*sqrt(2)*b/sqrt(c))*e^(-2*a + 1/2*b^2/c)/sqrt(c) + 8*x)*d + 1/32*(8*x^2 -
sqrt(2)*(sqrt(pi)*(2*c*x + b)*b*(erf(sqrt(1/2)*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(3/2))
- sqrt(2)*e^(1/2*(2*c*x + b)^2/c)/sqrt(c))*e^(2*a - 1/2*b^2/c)/sqrt(c) - sqrt(2)*(sqrt(pi)*(2*c*x + b)*b*(erf(
sqrt(1/2)*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(3/2)) + sqrt(2)*c*e^(-1/2*(2*c*x + b)^2/c)/
(-c)^(3/2))*e^(-2*a + 1/2*b^2/c)/sqrt(-c))*e

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Fricas [B]  time = 1.92095, size = 1922, normalized size = 12.01 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(2*c*e*cosh(c*x^2 + b*x + a)^4 + 8*c*e*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a)^3 + 2*c*e*sinh(c*x^2 +
b*x + a)^4 - sqrt(2)*sqrt(pi)*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c)/c) + (2*c*d - b*
e)*cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((2*c*d - b*e)*cosh(-1/2*(b^2 - 4*a*c)/c) + (2*c*d - b
*e)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)*cosh(-1/2*(b^
2 - 4*a*c)/c) + (2*c*d - b*e)*cosh(c*x^2 + b*x + a)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a))*sqrt(-c
)*erf(1/2*sqrt(2)*(2*c*x + b)*sqrt(-c)/c) + sqrt(2)*sqrt(pi)*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*
(b^2 - 4*a*c)/c) - (2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((2*c*d - b*e)*cosh(-1/2
*(b^2 - 4*a*c)/c) - (2*c*d - b*e)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((2*c*d - b*e)*cosh(
c*x^2 + b*x + a)*cosh(-1/2*(b^2 - 4*a*c)/c) - (2*c*d - b*e)*cosh(c*x^2 + b*x + a)*sinh(-1/2*(b^2 - 4*a*c)/c))*
sinh(c*x^2 + b*x + a))*sqrt(c)*erf(1/2*sqrt(2)*(2*c*x + b)/sqrt(c)) + 8*(c^2*e*x^2 + 2*c^2*d*x)*cosh(c*x^2 + b
*x + a)^2 + 4*(2*c^2*e*x^2 + 4*c^2*d*x + 3*c*e*cosh(c*x^2 + b*x + a)^2)*sinh(c*x^2 + b*x + a)^2 - 2*c*e + 8*(c
*e*cosh(c*x^2 + b*x + a)^3 + 2*(c^2*e*x^2 + 2*c^2*d*x)*cosh(c*x^2 + b*x + a))*sinh(c*x^2 + b*x + a))/(c^2*cosh
(c*x^2 + b*x + a)^2 + 2*c^2*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a) + c^2*sinh(c*x^2 + b*x + a)^2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x)*cosh(a + b*x + c*x**2)**2, x)

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Giac [A]  time = 1.33314, size = 335, normalized size = 2.09 \begin{align*} -\frac{\sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (\frac{b^{2} - 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt{c}} - \frac{\sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{-c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt{-c}} + \frac{1}{4} \, x^{2} e + \frac{1}{2} \, d x + \frac{\frac{\sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (\frac{b^{2} - 4 \, a c + 2 \, c}{2 \, c}\right )}}{\sqrt{c}} - 2 \, e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a + 1\right )}}{32 \, c} + \frac{\frac{\sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{-c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} - 4 \, a c - 2 \, c}{2 \, c}\right )}}{\sqrt{-c}} + 2 \, e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a + 1\right )}}{32 \, c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(2)*sqrt(pi)*d*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c)/c)/sqrt(c) - 1/16*sqrt(2)*
sqrt(pi)*d*erf(-1/2*sqrt(2)*sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*a*c)/c)/sqrt(-c) + 1/4*x^2*e + 1/2*d*x + 1/
32*(sqrt(2)*sqrt(pi)*b*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c + 2*c)/c)/sqrt(c) - 2*e^(-2*c
*x^2 - 2*b*x - 2*a + 1))/c + 1/32*(sqrt(2)*sqrt(pi)*b*erf(-1/2*sqrt(2)*sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*
a*c - 2*c)/c)/sqrt(-c) + 2*e^(2*c*x^2 + 2*b*x + 2*a + 1))/c