3.338 \(\int e^x \sinh (a+b x+c x^2) \, dx\)

Optimal. Leaf size=101 \[ \frac{\sqrt{\pi } e^{\frac{(1-b)^2}{4 c}-a} \text{Erf}\left (\frac{-b-2 c x+1}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } e^{a-\frac{(b+1)^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x+1}{2 \sqrt{c}}\right )}{4 \sqrt{c}} \]

[Out]

(E^(-a + (1 - b)^2/(4*c))*Sqrt[Pi]*Erf[(1 - b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c]) + (E^(a - (1 + b)^2/(4*c))*Sq
rt[Pi]*Erfi[(1 + b + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])

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Rubi [A]  time = 0.146931, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {5512, 2234, 2205, 2204} \[ \frac{\sqrt{\pi } e^{\frac{(1-b)^2}{4 c}-a} \text{Erf}\left (\frac{-b-2 c x+1}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } e^{a-\frac{(b+1)^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x+1}{2 \sqrt{c}}\right )}{4 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[E^x*Sinh[a + b*x + c*x^2],x]

[Out]

(E^(-a + (1 - b)^2/(4*c))*Sqrt[Pi]*Erf[(1 - b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c]) + (E^(a - (1 + b)^2/(4*c))*Sq
rt[Pi]*Erfi[(1 + b + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])

Rule 5512

Int[(F_)^(u_)*Sinh[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sinh[v]^n, x], x] /; FreeQ[F, x] && (Linea
rQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]

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 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]

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]

Rubi steps

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

Mathematica [A]  time = 0.166609, size = 92, normalized size = 0.91 \[ \frac{\sqrt{\pi } e^{-\frac{(b+1)^2}{4 c}} \left ((\sinh (a)+\cosh (a)) \text{Erfi}\left (\frac{b+2 c x+1}{2 \sqrt{c}}\right )-e^{\frac{b^2+1}{2 c}} (\cosh (a)-\sinh (a)) \text{Erf}\left (\frac{b+2 c x-1}{2 \sqrt{c}}\right )\right )}{4 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sinh[a + b*x + c*x^2],x]

[Out]

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

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Maple [A]  time = 0.099, size = 97, normalized size = 1. \begin{align*} -{\frac{\sqrt{\pi }}{4}{{\rm e}^{-{\frac{4\,ac-{b}^{2}+2\,b-1}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x-{\frac{1-b}{2}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{\sqrt{\pi }}{4}{{\rm e}^{{\frac{4\,ac-{b}^{2}-2\,b-1}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{1+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(exp(x)*sinh(c*x^2+b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.15465, size = 109, normalized size = 1.08 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-c} x - \frac{b + 1}{2 \, \sqrt{-c}}\right ) e^{\left (a - \frac{{\left (b + 1\right )}^{2}}{4 \, c}\right )}}{4 \, \sqrt{-c}} - \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{c} x + \frac{b - 1}{2 \, \sqrt{c}}\right ) e^{\left (-a + \frac{{\left (b - 1\right )}^{2}}{4 \, c}\right )}}{4 \, \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78112, size = 367, normalized size = 3.63 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-c}{\left (\cosh \left (-\frac{b^{2} - 4 \, a c + 2 \, b + 1}{4 \, c}\right ) + \sinh \left (-\frac{b^{2} - 4 \, a c + 2 \, b + 1}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b + 1\right )} \sqrt{-c}}{2 \, c}\right ) + \sqrt{\pi } \sqrt{c}{\left (\cosh \left (-\frac{b^{2} - 4 \, a c - 2 \, b + 1}{4 \, c}\right ) - \sinh \left (-\frac{b^{2} - 4 \, a c - 2 \, b + 1}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{2 \, c x + b - 1}{2 \, \sqrt{c}}\right )}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sinh(c*x**2+b*x+a),x)

[Out]

Integral(exp(x)*sinh(a + b*x + c*x**2), x)

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Giac [A]  time = 1.12461, size = 123, normalized size = 1.22 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c}{\left (2 \, x + \frac{b + 1}{c}\right )}\right ) e^{\left (-\frac{b^{2} - 4 \, a c + 2 \, b + 1}{4 \, c}\right )}}{4 \, \sqrt{-c}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c}{\left (2 \, x + \frac{b - 1}{c}\right )}\right ) e^{\left (\frac{b^{2} - 4 \, a c - 2 \, b + 1}{4 \, c}\right )}}{4 \, \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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