Integrand size = 15, antiderivative size = 75 \[ \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\frac {e^{-c} \sqrt {\pi } \text {erfc}(b x)^2}{8 b}+\frac {e^c \sqrt {\pi } \text {erfi}(b x)}{4 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]
-1/2*b*exp(c)*x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/Pi^(1/2)+1/8*erfc(b*x )^2*Pi^(1/2)/b/exp(c)+1/4*exp(c)*erfi(b*x)*Pi^(1/2)/b
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11 \[ \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\frac {-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right ) (\cosh (c)+\sinh (c))+\pi \left (-2 \text {erf}(b x) (\cosh (c)-\sinh (c))+\text {erf}(b x)^2 (\cosh (c)-\sinh (c))+2 \text {erfi}(b x) (\cosh (c)+\sinh (c))\right )}{8 b \sqrt {\pi }} \]
(-4*b^2*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2]*(Cosh[c] + Sinh[c ]) + Pi*(-2*Erf[b*x]*(Cosh[c] - Sinh[c]) + Erf[b*x]^2*(Cosh[c] - Sinh[c]) + 2*Erfi[b*x]*(Cosh[c] + Sinh[c])))/(8*b*Sqrt[Pi])
Time = 0.48 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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.
\(\displaystyle \int \text {erfc}(b x) \sinh \left (b^2 x^2+c\right ) \, dx\) |
\(\Big \downarrow \) 6965 |
\(\displaystyle \frac {1}{2} \int e^{b^2 x^2+c} \text {erfc}(b x)dx-\frac {1}{2} \int e^{-b^2 x^2-c} \text {erfc}(b x)dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle \frac {1}{2} \left (\int e^{b^2 x^2+c}dx-\int e^{b^2 x^2+c} \text {erf}(b x)dx\right )+\frac {\sqrt {\pi } e^{-c} \text {erfc}(b x)^2}{8 b}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\int e^{b^2 x^2+c} \text {erf}(b x)dx\right )+\frac {\sqrt {\pi } e^{-c} \text {erfc}(b x)^2}{8 b}\) |
\(\Big \downarrow \) 6930 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}\right )+\frac {\sqrt {\pi } e^{-c} \text {erfc}(b x)^2}{8 b}\) |
(Sqrt[Pi]*Erfc[b*x]^2)/(8*b*E^c) + ((E^c*Sqrt[Pi]*Erfi[b*x])/(2*b) - (b*E^ c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi])/2
3.3.3.3.1 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. \[ \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\int \mathrm {sinh}\left (b^2\,x^2+c\right )\,\mathrm {erfc}\left (b\,x\right ) \,d x \]