Integrand size = 16, antiderivative size = 56 \[ \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\frac {e^c \sqrt {\pi } \text {erf}(b x)^2}{8 b}-\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \] 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 = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09 \[ \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\frac {(\cosh (c)-\sinh (c)) \left (-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )+\pi \text {erf}(b x)^2 (\cosh (2 c)+\sinh (2 c))\right )}{8 b \sqrt {\pi }} \] Input:
Integrate[Erf[b*x]*Sinh[c - b^2*x^2],x]
Output:
((Cosh[c] - Sinh[c])*(-4*b^2*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x ^2] + Pi*Erf[b*x]^2*(Cosh[2*c] + Sinh[2*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 = {6964, 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.
\(\displaystyle \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 6964 |
\(\displaystyle \frac {1}{2} \int e^{c-b^2 x^2} \text {erf}(b x)dx-\frac {1}{2} \int e^{b^2 x^2-c} \text {erf}(b x)dx\) |
\(\Big \downarrow \) 6927 |
\(\displaystyle \frac {\sqrt {\pi } e^c \int \text {erf}(b x)d\text {erf}(b x)}{4 b}-\frac {1}{2} \int e^{b^2 x^2-c} \text {erf}(b x)dx\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\sqrt {\pi } e^c \text {erf}(b x)^2}{8 b}-\frac {1}{2} \int e^{b^2 x^2-c} \text {erf}(b x)dx\) |
\(\Big \downarrow \) 6930 |
\(\displaystyle \frac {\sqrt {\pi } e^c \text {erf}(b x)^2}{8 b}-\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}\) |
Input:
Int[Erf[b*x]*Sinh[c - b^2*x^2],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[Erf[(b_.)*(x_)]*Sinh[(c_.) + (d_.)*(x_)^2], 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]
\[\int -\operatorname {erf}\left (b x \right ) \sinh \left (b^{2} x^{2}-c \right )d x\]
Input:
int(-erf(b*x)*sinh(b^2*x^2-c),x)
Output:
int(-erf(b*x)*sinh(b^2*x^2-c),x)
\[ \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erf}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \] Input:
integrate(-erf(b*x)*sinh(b^2*x^2-c),x, algorithm="fricas")
Output:
integral(-erf(b*x)*sinh(b^2*x^2 - c), x)
\[ \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=- \int \sinh {\left (b^{2} x^{2} - c \right )} \operatorname {erf}{\left (b x \right )}\, dx \] Input:
integrate(-erf(b*x)*sinh(b**2*x**2-c),x)
Output:
-Integral(sinh(b**2*x**2 - c)*erf(b*x), x)
\[ \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erf}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \] Input:
integrate(-erf(b*x)*sinh(b^2*x^2-c),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)
\[ \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erf}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \] Input:
integrate(-erf(b*x)*sinh(b^2*x^2-c),x, algorithm="giac")
Output:
integrate(-erf(b*x)*sinh(b^2*x^2 - c), x)
Timed out. \[ \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int \mathrm {sinh}\left (c-b^2\,x^2\right )\,\mathrm {erf}\left (b\,x\right ) \,d x \] Input:
int(sinh(c - b^2*x^2)*erf(b*x),x)
Output:
int(sinh(c - b^2*x^2)*erf(b*x), x)
\[ \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=-\left (\int \mathrm {erf}\left (b x \right ) \sinh \left (b^{2} x^{2}-c \right )d x \right ) \] Input:
int(-erf(b*x)*sinh(b^2*x^2-c),x)
Output:
- int(erf(b*x)*sinh(b**2*x**2 - c),x)