Integrand size = 8, antiderivative size = 68 \[ \int \text {erfi}(a+b x)^2 \, dx=-\frac {2 e^{(a+b x)^2} \text {erfi}(a+b x)}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfi}(a+b x)^2}{b}+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b} \] Output:
-2*exp((b*x+a)^2)*erfi(b*x+a)/b/Pi^(1/2)+(b*x+a)*erfi(b*x+a)^2/b+2^(1/2)/P i^(1/2)*erfi(2^(1/2)*(b*x+a))/b
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int \text {erfi}(a+b x)^2 \, dx=\frac {-2 e^{(a+b x)^2} \text {erfi}(a+b x)+\sqrt {\pi } (a+b x) \text {erfi}(a+b x)^2+\sqrt {2} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b \sqrt {\pi }} \] Input:
Integrate[Erfi[a + b*x]^2,x]
Output:
(-2*E^(a + b*x)^2*Erfi[a + b*x] + Sqrt[Pi]*(a + b*x)*Erfi[a + b*x]^2 + Sqr t[2]*Erfi[Sqrt[2]*(a + b*x)])/(b*Sqrt[Pi])
Time = 0.45 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6908, 7281, 6938, 2633}
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.
\(\displaystyle \int \text {erfi}(a+b x)^2 \, dx\) |
\(\Big \downarrow \) 6908 |
\(\displaystyle \frac {(a+b x) \text {erfi}(a+b x)^2}{b}-\frac {4 \int e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)dx}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {(a+b x) \text {erfi}(a+b x)^2}{b}-\frac {4 \int e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)d(a+b x)}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 6938 |
\(\displaystyle \frac {(a+b x) \text {erfi}(a+b x)^2}{b}-\frac {4 \left (\frac {1}{2} e^{(a+b x)^2} \text {erfi}(a+b x)-\frac {\int e^{2 (a+b x)^2}d(a+b x)}{\sqrt {\pi }}\right )}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {(a+b x) \text {erfi}(a+b x)^2}{b}-\frac {4 \left (\frac {1}{2} e^{(a+b x)^2} \text {erfi}(a+b x)-\frac {\text {erfi}\left (\sqrt {2} (a+b x)\right )}{2 \sqrt {2}}\right )}{\sqrt {\pi } b}\) |
Input:
Int[Erfi[a + b*x]^2,x]
Output:
((a + b*x)*Erfi[a + b*x]^2)/b - (4*((E^(a + b*x)^2*Erfi[a + b*x])/2 - Erfi [Sqrt[2]*(a + b*x)]/(2*Sqrt[2])))/(b*Sqrt[Pi])
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[Erfi[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(Erfi[a + b*x]^ 2/b), x] - Simp[4/Sqrt[Pi] Int[(a + b*x)*E^(a + b*x)^2*Erfi[a + b*x], x], x] /; FreeQ[{a, b}, x]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Si mp[E^(c + d*x^2)*(Erfi[a + b*x]/(2*d)), x] - Simp[b/(d*Sqrt[Pi]) Int[E^(a ^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
\[\int \operatorname {erfi}\left (b x +a \right )^{2}d x\]
Input:
int(erfi(b*x+a)^2,x)
Output:
int(erfi(b*x+a)^2,x)
Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.37 \[ \int \text {erfi}(a+b x)^2 \, dx=-\frac {2 \, \sqrt {\pi } b \operatorname {erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (\pi b^{2} x + \pi a b\right )} \operatorname {erfi}\left (b x + a\right )^{2} + \sqrt {2} \sqrt {\pi } \sqrt {-b^{2}} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {-b^{2}} {\left (b x + a\right )}}{b}\right )}{\pi b^{2}} \] Input:
integrate(erfi(b*x+a)^2,x, algorithm="fricas")
Output:
-(2*sqrt(pi)*b*erfi(b*x + a)*e^(b^2*x^2 + 2*a*b*x + a^2) - (pi*b^2*x + pi* a*b)*erfi(b*x + a)^2 + sqrt(2)*sqrt(pi)*sqrt(-b^2)*erf(sqrt(2)*sqrt(-b^2)* (b*x + a)/b))/(pi*b^2)
\[ \int \text {erfi}(a+b x)^2 \, dx=\int \operatorname {erfi}^{2}{\left (a + b x \right )}\, dx \] Input:
integrate(erfi(b*x+a)**2,x)
Output:
Integral(erfi(a + b*x)**2, x)
\[ \int \text {erfi}(a+b x)^2 \, dx=\int { \operatorname {erfi}\left (b x + a\right )^{2} \,d x } \] Input:
integrate(erfi(b*x+a)^2,x, algorithm="maxima")
Output:
integrate(erfi(b*x + a)^2, x)
\[ \int \text {erfi}(a+b x)^2 \, dx=\int { \operatorname {erfi}\left (b x + a\right )^{2} \,d x } \] Input:
integrate(erfi(b*x+a)^2,x, algorithm="giac")
Output:
integrate(erfi(b*x + a)^2, x)
Timed out. \[ \int \text {erfi}(a+b x)^2 \, dx=\int {\mathrm {erfi}\left (a+b\,x\right )}^2 \,d x \] Input:
int(erfi(a + b*x)^2,x)
Output:
int(erfi(a + b*x)^2, x)
\[ \int \text {erfi}(a+b x)^2 \, dx=-\left (\int \mathrm {erf}\left (b i x +a i \right )^{2}d x \right ) \] Input:
int(erfi(b*x+a)^2,x)
Output:
- int(erf(a*i + b*i*x)**2,x)