Integrand size = 17, antiderivative size = 115 \[ \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx=\frac {e^{-a-\frac {b^2}{4 (1-c)}} \sqrt {\pi } \text {erfi}\left (\frac {b-2 (1-c) x}{2 \sqrt {1-c}}\right )}{4 \sqrt {1-c}}+\frac {e^{a-\frac {b^2}{4 (1+c)}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 (1+c) x}{2 \sqrt {1+c}}\right )}{4 \sqrt {1+c}} \] Output:
1/4*exp(-a-b^2/(4-4*c))*Pi^(1/2)*erfi(1/2*(b-2*(1-c)*x)/(1-c)^(1/2))/(1-c) ^(1/2)+1/4*exp(a-b^2/(4+4*c))*Pi^(1/2)*erfi(1/2*(b+2*(1+c)*x)/(1+c)^(1/2)) /(1+c)^(1/2)
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07 \[ \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx=\frac {e^{-\frac {b^2}{4+4 c}} \sqrt {\pi } \left (-\sqrt {-1+c} (1+c) e^{\frac {b^2 c}{2 \left (-1+c^2\right )}} \text {erf}\left (\frac {b+2 (-1+c) x}{2 \sqrt {-1+c}}\right ) (\cosh (a)-\sinh (a))+(-1+c) \sqrt {1+c} \text {erfi}\left (\frac {b+2 (1+c) x}{2 \sqrt {1+c}}\right ) (\cosh (a)+\sinh (a))\right )}{4 \left (-1+c^2\right )} \] Input:
Integrate[E^x^2*Sinh[a + b*x + c*x^2],x]
Output:
(Sqrt[Pi]*(-(Sqrt[-1 + c]*(1 + c)*E^((b^2*c)/(2*(-1 + c^2)))*Erf[(b + 2*(- 1 + c)*x)/(2*Sqrt[-1 + c])]*(Cosh[a] - Sinh[a])) + (-1 + c)*Sqrt[1 + c]*Er fi[(b + 2*(1 + c)*x)/(2*Sqrt[1 + c])]*(Cosh[a] + Sinh[a])))/(4*(-1 + c^2)* E^(b^2/(4 + 4*c)))
Time = 0.39 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6038, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 6038 |
\(\displaystyle \int \left (\frac {1}{2} e^{a+b x+(c+1) x^2}-\frac {1}{2} e^{-a-b x+(1-c) x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 (1-c)}} \text {erfi}\left (\frac {b-2 (1-c) x}{2 \sqrt {1-c}}\right )}{4 \sqrt {1-c}}+\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 (c+1)}} \text {erfi}\left (\frac {b+2 (c+1) x}{2 \sqrt {c+1}}\right )}{4 \sqrt {c+1}}\) |
Input:
Int[E^x^2*Sinh[a + b*x + c*x^2],x]
Output:
(E^(-a - b^2/(4*(1 - c)))*Sqrt[Pi]*Erfi[(b - 2*(1 - c)*x)/(2*Sqrt[1 - c])] )/(4*Sqrt[1 - c]) + (E^(a - b^2/(4*(1 + c)))*Sqrt[Pi]*Erfi[(b + 2*(1 + c)* x)/(2*Sqrt[1 + c])])/(4*Sqrt[1 + c])
Int[(F_)^(u_)*Sinh[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sinh[v] ^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[ v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
Time = 0.44 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}-4 a}{4 \left (c -1\right )}} \operatorname {erf}\left (\sqrt {c -1}\, x +\frac {b}{2 \sqrt {c -1}}\right )}{4 \sqrt {c -1}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}+4 a}{4+4 c}} \operatorname {erf}\left (-\sqrt {-1-c}\, x +\frac {b}{2 \sqrt {-1-c}}\right )}{4 \sqrt {-1-c}}\) | \(105\) |
Input:
int(exp(x^2)*sinh(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/4*Pi^(1/2)*exp(-1/4*(4*a*c-b^2-4*a)/(c-1))/(c-1)^(1/2)*erf((c-1)^(1/2)* x+1/2*b/(c-1)^(1/2))-1/4*Pi^(1/2)*exp(1/4*(4*a*c-b^2+4*a)/(1+c))/(-1-c)^(1 /2)*erf(-(-1-c)^(1/2)*x+1/2*b/(-1-c)^(1/2))
Time = 0.08 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.43 \[ \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {\pi } {\left ({\left (c + 1\right )} \cosh \left (-\frac {b^{2} - 4 \, a c + 4 \, a}{4 \, {\left (c - 1\right )}}\right ) - {\left (c + 1\right )} \sinh \left (-\frac {b^{2} - 4 \, a c + 4 \, a}{4 \, {\left (c - 1\right )}}\right )\right )} \sqrt {c - 1} \operatorname {erf}\left (\frac {2 \, {\left (c - 1\right )} x + b}{2 \, \sqrt {c - 1}}\right ) + \sqrt {\pi } {\left ({\left (c - 1\right )} \cosh \left (-\frac {b^{2} - 4 \, a c - 4 \, a}{4 \, {\left (c + 1\right )}}\right ) + {\left (c - 1\right )} \sinh \left (-\frac {b^{2} - 4 \, a c - 4 \, a}{4 \, {\left (c + 1\right )}}\right )\right )} \sqrt {-c - 1} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c + 1\right )} x + b\right )} \sqrt {-c - 1}}{2 \, {\left (c + 1\right )}}\right )}{4 \, {\left (c^{2} - 1\right )}} \] Input:
integrate(exp(x^2)*sinh(c*x^2+b*x+a),x, algorithm="fricas")
Output:
-1/4*(sqrt(pi)*((c + 1)*cosh(-1/4*(b^2 - 4*a*c + 4*a)/(c - 1)) - (c + 1)*s inh(-1/4*(b^2 - 4*a*c + 4*a)/(c - 1)))*sqrt(c - 1)*erf(1/2*(2*(c - 1)*x + b)/sqrt(c - 1)) + sqrt(pi)*((c - 1)*cosh(-1/4*(b^2 - 4*a*c - 4*a)/(c + 1)) + (c - 1)*sinh(-1/4*(b^2 - 4*a*c - 4*a)/(c + 1)))*sqrt(-c - 1)*erf(1/2*(2 *(c + 1)*x + b)*sqrt(-c - 1)/(c + 1)))/(c^2 - 1)
\[ \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx=\int e^{x^{2}} \sinh {\left (a + b x + c x^{2} \right )}\, dx \] Input:
integrate(exp(x**2)*sinh(c*x**2+b*x+a),x)
Output:
Integral(exp(x**2)*sinh(a + b*x + c*x**2), x)
Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c - 1} x - \frac {b}{2 \, \sqrt {-c - 1}}\right ) e^{\left (a - \frac {b^{2}}{4 \, {\left (c + 1\right )}}\right )}}{4 \, \sqrt {-c - 1}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {c - 1} x + \frac {b}{2 \, \sqrt {c - 1}}\right ) e^{\left (-a + \frac {b^{2}}{4 \, {\left (c - 1\right )}}\right )}}{4 \, \sqrt {c - 1}} \] Input:
integrate(exp(x^2)*sinh(c*x^2+b*x+a),x, algorithm="maxima")
Output:
1/4*sqrt(pi)*erf(sqrt(-c - 1)*x - 1/2*b/sqrt(-c - 1))*e^(a - 1/4*b^2/(c + 1))/sqrt(-c - 1) - 1/4*sqrt(pi)*erf(sqrt(c - 1)*x + 1/2*b/sqrt(c - 1))*e^( -a + 1/4*b^2/(c - 1))/sqrt(c - 1)
Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c - 1} {\left (2 \, x + \frac {b}{c + 1}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c - 4 \, a}{4 \, {\left (c + 1\right )}}\right )}}{4 \, \sqrt {-c - 1}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c - 1} {\left (2 \, x + \frac {b}{c - 1}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c + 4 \, a}{4 \, {\left (c - 1\right )}}\right )}}{4 \, \sqrt {c - 1}} \] Input:
integrate(exp(x^2)*sinh(c*x^2+b*x+a),x, algorithm="giac")
Output:
-1/4*sqrt(pi)*erf(-1/2*sqrt(-c - 1)*(2*x + b/(c + 1)))*e^(-1/4*(b^2 - 4*a* c - 4*a)/(c + 1))/sqrt(-c - 1) + 1/4*sqrt(pi)*erf(-1/2*sqrt(c - 1)*(2*x + b/(c - 1)))*e^(1/4*(b^2 - 4*a*c + 4*a)/(c - 1))/sqrt(c - 1)
Timed out. \[ \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx=\int \mathrm {sinh}\left (c\,x^2+b\,x+a\right )\,{\mathrm {e}}^{x^2} \,d x \] Input:
int(sinh(a + b*x + c*x^2)*exp(x^2),x)
Output:
int(sinh(a + b*x + c*x^2)*exp(x^2), x)
\[ \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx=\int e^{x^{2}} \sinh \left (c \,x^{2}+b x +a \right )d x \] Input:
int(exp(x^2)*sinh(c*x^2+b*x+a),x)
Output:
int(e**(x**2)*sinh(a + b*x + c*x**2),x)