Integrand size = 16, antiderivative size = 77 \[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=-\frac {e^c \sqrt {\pi } \text {erfc}(b x)^2}{8 b}-\frac {e^{-c} \sqrt {\pi } \text {erfi}(b x)}{4 b}+\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]
1/2*b*x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/exp(c)/Pi^(1/2)-1/8*exp(c)*er fc(b*x)^2*Pi^(1/2)/b-1/4*erfi(b*x)*Pi^(1/2)/b/exp(c)
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09 \[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=-\frac {(\cosh (c)-\sinh (c)) \left (-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )+\pi \left (2 \text {erfi}(b x)-2 \text {erf}(b x) (\cosh (2 c)+\sinh (2 c))+\text {erf}(b x)^2 (\cosh (2 c)+\sinh (2 c))\right )\right )}{8 b \sqrt {\pi }} \]
-1/8*((Cosh[c] - Sinh[c])*(-4*b^2*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2] + Pi*(2*Erfi[b*x] - 2*Erf[b*x]*(Cosh[2*c] + Sinh[2*c]) + Erf[b*x] ^2*(Cosh[2*c] + Sinh[2*c]))))/(b*Sqrt[Pi])
Time = 0.47 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6965, 6928, 15, 6931, 2633, 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.
\(\displaystyle \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 6965 |
\(\displaystyle \frac {1}{2} \int e^{c-b^2 x^2} \text {erfc}(b x)dx-\frac {1}{2} \int e^{b^2 x^2-c} \text {erfc}(b x)dx\) |
\(\Big \downarrow \) 6928 |
\(\displaystyle -\frac {1}{2} \int e^{b^2 x^2-c} \text {erfc}(b x)dx-\frac {\sqrt {\pi } e^c \int \text {erfc}(b x)d\text {erfc}(b x)}{4 b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {1}{2} \int e^{b^2 x^2-c} \text {erfc}(b x)dx-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}\) |
\(\Big \downarrow \) 6931 |
\(\displaystyle \frac {1}{2} \left (\int e^{b^2 x^2-c} \text {erf}(b x)dx-\int e^{b^2 x^2-c}dx\right )-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {1}{2} \left (\int e^{b^2 x^2-c} \text {erf}(b x)dx-\frac {\sqrt {\pi } e^{-c} \text {erfi}(b x)}{2 b}\right )-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}\) |
\(\Big \downarrow \) 6930 |
\(\displaystyle \frac {1}{2} \left (\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}-\frac {\sqrt {\pi } e^{-c} \text {erfi}(b x)}{2 b}\right )-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}\) |
-1/8*(E^c*Sqrt[Pi]*Erfc[b*x]^2)/b + (-1/2*(Sqrt[Pi]*Erfi[b*x])/(b*E^c) + ( b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^c*Sqrt[Pi]))/2
3.3.4.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(-E^ c)*(Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erfc[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[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)], x_Symbol] :> Int[E^(c + d*x^ 2), x] - Int[E^(c + d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^ 2]
Int[Erfc[(b_.)*(x_)]*Sinh[(c_.) + (d_.)*(x_)^2], x_Symbol] :> Simp[1/2 In t[E^(c + d*x^2)*Erfc[b*x], x], x] - Simp[1/2 Int[E^(-c - d*x^2)*Erfc[b*x] , x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
\[\int -\operatorname {erfc}\left (b x \right ) \sinh \left (b^{2} x^{2}-c \right )d x\]
\[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erfc}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \]
\[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=- \int \sinh {\left (b^{2} x^{2} - c \right )} \operatorname {erfc}{\left (b x \right )}\, dx \]
\[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erfc}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \]
\[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erfc}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \]
Timed out. \[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int \mathrm {sinh}\left (c-b^2\,x^2\right )\,\mathrm {erfc}\left (b\,x\right ) \,d x \]