Integrand size = 16, antiderivative size = 56 \[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\frac {e^c \sqrt {\pi } \text {erf}(b x)^2}{8 b}+\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \] Output:
1/8*exp(c)*Pi^(1/2)*erf(b*x)^2/b+1/2*b*x^2*hypergeom([1, 1],[3/2, 2],b^2*x ^2)/exp(c)/Pi^(1/2)
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.34 \[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\frac {4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right ) (-\cosh (c)+\sinh (c))+\pi \text {erf}(b x) (2 \text {erfi}(b x) (\cosh (c)-\sinh (c))+\text {erf}(b x) (\cosh (c)+\sinh (c)))}{8 b \sqrt {\pi }} \] Input:
Integrate[Cosh[c - b^2*x^2]*Erf[b*x],x]
Output:
(4*b^2*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)]*(-Cosh[c] + Sin h[c]) + Pi*Erf[b*x]*(2*Erfi[b*x]*(Cosh[c] - Sinh[c]) + Erf[b*x]*(Cosh[c] + Sinh[c])))/(8*b*Sqrt[Pi])
Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6967, 6927, 15, 6930}
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.
\(\displaystyle \int \text {erf}(b x) \cosh \left (c-b^2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 6967 |
\(\displaystyle \frac {1}{2} \int e^{c-b^2 x^2} \text {erf}(b x)dx+\frac {1}{2} \int e^{b^2 x^2-c} \text {erf}(b x)dx\) |
\(\Big \downarrow \) 6927 |
\(\displaystyle \frac {1}{2} \int e^{b^2 x^2-c} \text {erf}(b x)dx+\frac {\sqrt {\pi } e^c \int \text {erf}(b x)d\text {erf}(b x)}{4 b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} \int e^{b^2 x^2-c} \text {erf}(b x)dx+\frac {\sqrt {\pi } e^c \text {erf}(b x)^2}{8 b}\) |
\(\Big \downarrow \) 6930 |
\(\displaystyle \frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\sqrt {\pi } e^c \text {erf}(b x)^2}{8 b}\) |
Input:
Int[Cosh[c - b^2*x^2]*Erf[b*x],x]
Output:
(E^c*Sqrt[Pi]*Erf[b*x]^2)/(8*b) + (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2 }, b^2*x^2])/(2*E^c*Sqrt[Pi])
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[E^c*( Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erf[b*x]], x] /; FreeQ[{b, c, d, n} , x] && EqQ[d, -b^2]
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/ Sqrt[Pi])*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
Int[Cosh[(c_.) + (d_.)*(x_)^2]*Erf[(b_.)*(x_)], x_Symbol] :> Simp[1/2 Int [E^(c + d*x^2)*Erf[b*x], x], x] + Simp[1/2 Int[E^(-c - d*x^2)*Erf[b*x], x ], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
\[\int \cosh \left (b^{2} x^{2}-c \right ) \operatorname {erf}\left (b x \right )d x\]
Input:
int(cosh(b^2*x^2-c)*erf(b*x),x)
Output:
int(cosh(b^2*x^2-c)*erf(b*x),x)
\[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\int { \cosh \left (b^{2} x^{2} - c\right ) \operatorname {erf}\left (b x\right ) \,d x } \] Input:
integrate(cosh(b^2*x^2-c)*erf(b*x),x, algorithm="fricas")
Output:
integral(cosh(b^2*x^2 - c)*erf(b*x), x)
\[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\int \cosh {\left (b^{2} x^{2} - c \right )} \operatorname {erf}{\left (b x \right )}\, dx \] Input:
integrate(cosh(b**2*x**2-c)*erf(b*x),x)
Output:
Integral(cosh(b**2*x**2 - c)*erf(b*x), x)
\[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\int { \cosh \left (b^{2} x^{2} - c\right ) \operatorname {erf}\left (b x\right ) \,d x } \] Input:
integrate(cosh(b^2*x^2-c)*erf(b*x),x, algorithm="maxima")
Output:
1/8*sqrt(pi)*erf(b*x)^2*e^c/b + 1/2*integrate(erf(b*x)*e^(b^2*x^2 - c), x)
\[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\int { \cosh \left (b^{2} x^{2} - c\right ) \operatorname {erf}\left (b x\right ) \,d x } \] Input:
integrate(cosh(b^2*x^2-c)*erf(b*x),x, algorithm="giac")
Output:
integrate(cosh(b^2*x^2 - c)*erf(b*x), x)
Timed out. \[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\int \mathrm {cosh}\left (c-b^2\,x^2\right )\,\mathrm {erf}\left (b\,x\right ) \,d x \] Input:
int(cosh(c - b^2*x^2)*erf(b*x),x)
Output:
int(cosh(c - b^2*x^2)*erf(b*x), x)
\[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\int \cosh \left (b^{2} x^{2}-c \right ) \mathrm {erf}\left (b x \right )d x \] Input:
int(cosh(b^2*x^2-c)*erf(b*x),x)
Output:
int(cosh(b**2*x**2 - c)*erf(b*x),x)