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

Optimal. Leaf size=85 \[ \frac{\sqrt{\pi } e^{\frac{1}{4 c}-a} \text{Erf}\left (\frac{1-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } e^{a-\frac{1}{4 c}} \text{Erfi}\left (\frac{2 c x+1}{2 \sqrt{c}}\right )}{4 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0812564, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5512, 2234, 2205, 2204} \[ \frac{\sqrt{\pi } e^{\frac{1}{4 c}-a} \text{Erf}\left (\frac{1-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } e^{a-\frac{1}{4 c}} \text{Erfi}\left (\frac{2 c x+1}{2 \sqrt{c}}\right )}{4 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[E^x*Sinh[a + c*x^2],x]

[Out]

(E^(-a + 1/(4*c))*Sqrt[Pi]*Erf[(1 - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c]) + (E^(a - 1/(4*c))*Sqrt[Pi]*Erfi[(1 + 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+c x^2\right ) \, dx &=\int \left (-\frac{1}{2} e^{-a+x-c x^2}+\frac{1}{2} e^{a+x+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{2} \int e^{-a+x-c x^2} \, dx\right )+\frac{1}{2} \int e^{a+x+c x^2} \, dx\\ &=\frac{1}{2} e^{a-\frac{1}{4 c}} \int e^{\frac{(1+2 c x)^2}{4 c}} \, dx-\frac{1}{2} e^{-a+\frac{1}{4 c}} \int e^{-\frac{(1-2 c x)^2}{4 c}} \, dx\\ &=\frac{e^{-a+\frac{1}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{1-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{e^{a-\frac{1}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{1+2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.0911724, size = 80, normalized size = 0.94 \[ \frac{\sqrt{\pi } e^{\left .-\frac{1}{4}\right /c} \left ((\sinh (a)+\cosh (a)) \text{Erfi}\left (\frac{2 c x+1}{2 \sqrt{c}}\right )-e^{\left .\frac{1}{2}\right /c} (\cosh (a)-\sinh (a)) \text{Erf}\left (\frac{2 c x-1}{2 \sqrt{c}}\right )\right )}{4 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sinh[a + c*x^2],x]

[Out]

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

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Maple [A]  time = 0.109, size = 72, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{\pi }}{4}{{\rm e}^{-{\frac{4\,ac-1}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x-{\frac{1}{2}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+{\frac{\sqrt{\pi }}{4}{{\rm e}^{{\frac{4\,ac-1}{4\,c}}}}{\it Erf} \left ( \sqrt{-c}x-{\frac{1}{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+a),x)

[Out]

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

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Maxima [A]  time = 1.12056, size = 88, normalized size = 1.04 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-c} x - \frac{1}{2 \, \sqrt{-c}}\right ) e^{\left (a - \frac{1}{4 \, c}\right )}}{4 \, \sqrt{-c}} - \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{c} x - \frac{1}{2 \, \sqrt{c}}\right ) e^{\left (-a + \frac{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+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.78437, size = 286, normalized size = 3.36 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-c}{\left (\cosh \left (\frac{4 \, a c - 1}{4 \, c}\right ) + \sinh \left (\frac{4 \, a c - 1}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + 1\right )} \sqrt{-c}}{2 \, c}\right ) + \sqrt{\pi } \sqrt{c}{\left (\cosh \left (\frac{4 \, a c - 1}{4 \, c}\right ) - \sinh \left (\frac{4 \, a c - 1}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{2 \, c x - 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+a),x, algorithm="fricas")

[Out]

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

[Out]

Integral(exp(x)*sinh(a + c*x**2), x)

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Giac [A]  time = 1.15885, size = 99, normalized size = 1.16 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c}{\left (2 \, x + \frac{1}{c}\right )}\right ) e^{\left (\frac{4 \, a c - 1}{4 \, c}\right )}}{4 \, \sqrt{-c}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c}{\left (2 \, x - \frac{1}{c}\right )}\right ) e^{\left (-\frac{4 \, a c - 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+a),x, algorithm="giac")

[Out]

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