Integrand size = 15, antiderivative size = 101 \[ \int e^x \cosh \left (a+b x+c x^2\right ) \, dx=-\frac {e^{-a+\frac {(1-b)^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {1-b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e^{a-\frac {(1+b)^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {1+b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \] Output:
-1/4*exp(-a+1/4*(1-b)^2/c)*Pi^(1/2)*erf(1/2*(-2*c*x-b+1)/c^(1/2))/c^(1/2)+ 1/4*exp(a-1/4*(1+b)^2/c)*Pi^(1/2)*erfi(1/2*(2*c*x+b+1)/c^(1/2))/c^(1/2)
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.90 \[ \int e^x \cosh \left (a+b x+c x^2\right ) \, dx=\frac {e^{-\frac {(1+b)^2}{4 c}} \sqrt {\pi } \left (e^{\frac {1+b^2}{2 c}} \text {erf}\left (\frac {-1+b+2 c x}{2 \sqrt {c}}\right ) (\cosh (a)-\sinh (a))+\text {erfi}\left (\frac {1+b+2 c x}{2 \sqrt {c}}\right ) (\cosh (a)+\sinh (a))\right )}{4 \sqrt {c}} \] Input:
Integrate[E^x*Cosh[a + b*x + c*x^2],x]
Output:
(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*Sq rt[c]*E^((1 + b)^2/(4*c)))
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6039, 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.
\(\displaystyle \int e^x \cosh \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 6039 |
\(\displaystyle \int \left (\frac {1}{2} e^{-a+(1-b) x-c x^2}+\frac {1}{2} e^{a+(b+1) x+c x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\pi } e^{a-\frac {(b+1)^2}{4 c}} \text {erfi}\left (\frac {b+2 c x+1}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{\frac {(1-b)^2}{4 c}-a} \text {erf}\left (\frac {-b-2 c x+1}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\) |
Input:
Int[E^x*Cosh[a + b*x + c*x^2],x]
Output:
-1/4*(E^(-a + (1 - b)^2/(4*c))*Sqrt[Pi]*Erf[(1 - b - 2*c*x)/(2*Sqrt[c])])/ Sqrt[c] + (E^(a - (1 + b)^2/(4*c))*Sqrt[Pi]*Erfi[(1 + b + 2*c*x)/(2*Sqrt[c ])])/(4*Sqrt[c])
Int[Cosh[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cosh[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 = 1.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}+2 b -1}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x -\frac {1-b}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}-2 b -1}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {1+b}{2 \sqrt {-c}}\right )}{4 \sqrt {-c}}\) | \(97\) |
Input:
int(exp(x)*cosh(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
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))
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.29 \[ \int e^x \cosh \left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {\pi } \sqrt {-c} {\left (\cosh \left (-\frac {b^{2} - 4 \, a c + 2 \, b + 1}{4 \, c}\right ) + \sinh \left (-\frac {b^{2} - 4 \, a c + 2 \, b + 1}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b + 1\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {\pi } \sqrt {c} {\left (\cosh \left (-\frac {b^{2} - 4 \, a c - 2 \, b + 1}{4 \, c}\right ) - \sinh \left (-\frac {b^{2} - 4 \, a c - 2 \, b + 1}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 \, c x + b - 1}{2 \, \sqrt {c}}\right )}{4 \, c} \] Input:
integrate(exp(x)*cosh(c*x^2+b*x+a),x, algorithm="fricas")
Output:
-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
\[ \int e^x \cosh \left (a+b x+c x^2\right ) \, dx=\int e^{x} \cosh {\left (a + b x + c x^{2} \right )}\, dx \] Input:
integrate(exp(x)*cosh(c*x**2+b*x+a),x)
Output:
Integral(exp(x)*cosh(a + b*x + c*x**2), x)
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int e^x \cosh \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c} x - \frac {b + 1}{2 \, \sqrt {-c}}\right ) e^{\left (a - \frac {{\left (b + 1\right )}^{2}}{4 \, c}\right )}}{4 \, \sqrt {-c}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {c} x + \frac {b - 1}{2 \, \sqrt {c}}\right ) e^{\left (-a + \frac {{\left (b - 1\right )}^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} \] Input:
integrate(exp(x)*cosh(c*x^2+b*x+a),x, algorithm="maxima")
Output:
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)
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.90 \[ \int e^x \cosh \left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {b + 1}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c + 2 \, b + 1}{4 \, c}\right )}}{4 \, \sqrt {-c}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b - 1}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c - 2 \, b + 1}{4 \, c}\right )}}{4 \, \sqrt {c}} \] Input:
integrate(exp(x)*cosh(c*x^2+b*x+a),x, algorithm="giac")
Output:
-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)
Timed out. \[ \int e^x \cosh \left (a+b x+c x^2\right ) \, dx=\int {\mathrm {e}}^x\,\mathrm {cosh}\left (c\,x^2+b\,x+a\right ) \,d x \] Input:
int(exp(x)*cosh(a + b*x + c*x^2),x)
Output:
int(exp(x)*cosh(a + b*x + c*x^2), x)
\[ \int e^x \cosh \left (a+b x+c x^2\right ) \, dx=\int e^{x} \cosh \left (c \,x^{2}+b x +a \right )d x \] Input:
int(exp(x)*cosh(c*x^2+b*x+a),x)
Output:
int(e**x*cosh(a + b*x + c*x**2),x)