\(\int e^{x^2} \sinh (a+b x) \, dx\) [339]

Optimal result

Integrand size = 12, antiderivative size = 65 \[ \int e^{x^2} \sinh (a+b x) \, dx=-\frac {1}{4} e^{-a-\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-b+2 x)\right )+\frac {1}{4} e^{a-\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (b+2 x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5623, 2266, 2235} \[ \int e^{x^2} \sinh (a+b x) \, dx=\frac {1}{4} \sqrt {\pi } e^{a-\frac {b^2}{4}} \text {erfi}\left (\frac {1}{2} (b+2 x)\right )-\frac {1}{4} \sqrt {\pi } e^{-a-\frac {b^2}{4}} \text {erfi}\left (\frac {1}{2} (2 x-b)\right ) \]

[In]

Int[E^x^2*Sinh[a + b*x],x]

[Out]

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

Rule 2235

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 2266

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 5623

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]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int e^{x^2} \sinh (a+b x) \, dx=\frac {1}{4} e^{-\frac {b^2}{4}} \sqrt {\pi } \left (\text {erfi}\left (\frac {b}{2}-x\right ) (\cosh (a)-\sinh (a))+\text {erfi}\left (\frac {b}{2}+x\right ) (\cosh (a)+\sinh (a))\right ) \]

[In]

Integrate[E^x^2*Sinh[a + b*x],x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80

method	result	size
risch	\(-\frac {i \sqrt {\pi }\, {\mathrm e}^{-a -\frac {b^{2}}{4}} \operatorname {erf}\left (-i x +\frac {1}{2} i b \right )}{4}-\frac {i \sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4}} \operatorname {erf}\left (i x +\frac {1}{2} i b \right )}{4}\)	\(52\)

[In]

int(exp(x^2)*sinh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int e^{x^2} \sinh (a+b x) \, dx=\frac {1}{4} \, \sqrt {\pi } {\left (\cosh \left (\frac {1}{4} \, b^{2} - a\right ) \operatorname {erfi}\left (\frac {1}{2} \, b + x\right ) - \cosh \left (\frac {1}{4} \, b^{2} + a\right ) \operatorname {erfi}\left (-\frac {1}{2} \, b + x\right ) + \operatorname {erfi}\left (-\frac {1}{2} \, b + x\right ) \sinh \left (\frac {1}{4} \, b^{2} + a\right ) - \operatorname {erfi}\left (\frac {1}{2} \, b + x\right ) \sinh \left (\frac {1}{4} \, b^{2} - a\right )\right )} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int e^{x^2} \sinh (a+b x) \, dx=\int e^{x^{2}} \sinh {\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral(exp(x**2)*sinh(a + b*x), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int e^{x^2} \sinh (a+b x) \, dx=-\frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, b + i \, x\right ) e^{\left (-\frac {1}{4} \, b^{2} + a\right )} + \frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, b + i \, x\right ) e^{\left (-\frac {1}{4} \, b^{2} - a\right )} \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int e^{x^2} \sinh (a+b x) \, dx=\frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, b - i \, x\right ) e^{\left (-\frac {1}{4} \, b^{2} + a\right )} - \frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, b - i \, x\right ) e^{\left (-\frac {1}{4} \, b^{2} - a\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int e^{x^2} \sinh (a+b x) \, dx=\int {\mathrm {e}}^{x^2}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]

[In]

int(exp(x^2)*sinh(a + b*x),x)

[Out]

int(exp(x^2)*sinh(a + b*x), x)