3.340 \(\int e^{x^2} \sinh (a+c x^2) \, dx\)

Optimal. Leaf size=65 \[ \frac{\sqrt{\pi } e^a \text{Erfi}\left (\sqrt{c+1} x\right )}{4 \sqrt{c+1}}-\frac{\sqrt{\pi } e^{-a} \text{Erfi}\left (\sqrt{1-c} x\right )}{4 \sqrt{1-c}} \]

[Out]

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

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Rubi [A]  time = 0.0832766, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5512, 2204} \[ \frac{\sqrt{\pi } e^a \text{Erfi}\left (\sqrt{c+1} x\right )}{4 \sqrt{c+1}}-\frac{\sqrt{\pi } e^{-a} \text{Erfi}\left (\sqrt{1-c} x\right )}{4 \sqrt{1-c}} \]

Antiderivative was successfully verified.

[In]

Int[E^x^2*Sinh[a + c*x^2],x]

[Out]

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

Mathematica [A]  time = 0.107318, size = 72, normalized size = 1.11 \[ \frac{\sqrt{\pi } \left ((c-1) \sqrt{c+1} (\sinh (a)+\cosh (a)) \text{Erfi}\left (\sqrt{c+1} x\right )-\sqrt{c-1} (c+1) (\cosh (a)-\sinh (a)) \text{Erf}\left (\sqrt{c-1} x\right )\right )}{4 \left (c^2-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^2*Sinh[a + c*x^2],x]

[Out]

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

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Maple [A]  time = 0.073, size = 48, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{\pi }{{\rm e}^{-a}}}{4}{\it Erf} \left ( \sqrt{c-1}x \right ){\frac{1}{\sqrt{c-1}}}}+{\frac{\sqrt{\pi }{{\rm e}^{a}}}{4}{\it Erf} \left ( \sqrt{-1-c}x \right ){\frac{1}{\sqrt{-1-c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2)*sinh(c*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.08772, size = 63, normalized size = 0.97 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{c - 1} x\right ) e^{\left (-a\right )}}{4 \, \sqrt{c - 1}} + \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-c - 1} x\right ) e^{a}}{4 \, \sqrt{-c - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.86845, size = 235, normalized size = 3.62 \begin{align*} -\frac{\sqrt{\pi }{\left ({\left (c + 1\right )} \cosh \left (a\right ) -{\left (c + 1\right )} \sinh \left (a\right )\right )} \sqrt{c - 1} \operatorname{erf}\left (\sqrt{c - 1} x\right ) + \sqrt{\pi }{\left ({\left (c - 1\right )} \cosh \left (a\right ) +{\left (c - 1\right )} \sinh \left (a\right )\right )} \sqrt{-c - 1} \operatorname{erf}\left (\sqrt{-c - 1} x\right )}{4 \,{\left (c^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14358, size = 66, normalized size = 1.02 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{c - 1} x\right ) e^{\left (-a\right )}}{4 \, \sqrt{c - 1}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c - 1} x\right ) e^{a}}{4 \, \sqrt{-c - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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