Integrand size = 14, antiderivative size = 184 \[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\frac {d e^{2 (a+b x)^2}}{2 b^2 \pi }-\frac {2 (b c-a d) e^{(a+b x)^2} \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}-\frac {d e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}+\frac {d \text {erfi}(a+b x)^2}{4 b^2}+\frac {(b c-a d) (a+b x) \text {erfi}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \text {erfi}(a+b x)^2}{2 b^2}+\frac {(b c-a d) \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^2} \] Output:
1/2*d*exp(2*(b*x+a)^2)/b^2/Pi-2*(-a*d+b*c)*exp((b*x+a)^2)*erfi(b*x+a)/b^2/ Pi^(1/2)-d*exp((b*x+a)^2)*(b*x+a)*erfi(b*x+a)/b^2/Pi^(1/2)+1/4*d*erfi(b*x+ a)^2/b^2+(-a*d+b*c)*(b*x+a)*erfi(b*x+a)^2/b^2+1/2*d*(b*x+a)^2*erfi(b*x+a)^ 2/b^2+(-a*d+b*c)*2^(1/2)/Pi^(1/2)*erfi(2^(1/2)*(b*x+a))/b^2
Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.70 \[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\frac {2 d e^{2 (a+b x)^2}-4 e^{(a+b x)^2} \sqrt {\pi } (2 b c-a d+b d x) \text {erfi}(a+b x)+\pi \left (4 a b c+d-2 a^2 d+4 b^2 c x+2 b^2 d x^2\right ) \text {erfi}(a+b x)^2+4 (b c-a d) \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} (a+b x)\right )}{4 b^2 \pi } \] Input:
Integrate[(c + d*x)*Erfi[a + b*x]^2,x]
Output:
(2*d*E^(2*(a + b*x)^2) - 4*E^(a + b*x)^2*Sqrt[Pi]*(2*b*c - a*d + b*d*x)*Er fi[a + b*x] + Pi*(4*a*b*c + d - 2*a^2*d + 4*b^2*c*x + 2*b^2*d*x^2)*Erfi[a + b*x]^2 + 4*(b*c - a*d)*Sqrt[2*Pi]*Erfi[Sqrt[2]*(a + b*x)])/(4*b^2*Pi)
Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6923, 2009}
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.
\(\displaystyle \int (c+d x) \text {erfi}(a+b x)^2 \, dx\) |
\(\Big \downarrow \) 6923 |
\(\displaystyle \frac {\int \left ((b c-a d) \text {erfi}(a+b x)^2+d (a+b x) \text {erfi}(a+b x)^2\right )d(a+b x)}{b^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b x) (b c-a d) \text {erfi}(a+b x)^2-\frac {2 e^{(a+b x)^2} (b c-a d) \text {erfi}(a+b x)}{\sqrt {\pi }}+\sqrt {\frac {2}{\pi }} (b c-a d) \text {erfi}\left (\sqrt {2} (a+b x)\right )+\frac {1}{2} d (a+b x)^2 \text {erfi}(a+b x)^2+\frac {1}{4} d \text {erfi}(a+b x)^2-\frac {d e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{\sqrt {\pi }}+\frac {d e^{2 (a+b x)^2}}{2 \pi }}{b^2}\) |
Input:
Int[(c + d*x)*Erfi[a + b*x]^2,x]
Output:
((d*E^(2*(a + b*x)^2))/(2*Pi) - (2*(b*c - a*d)*E^(a + b*x)^2*Erfi[a + b*x] )/Sqrt[Pi] - (d*E^(a + b*x)^2*(a + b*x)*Erfi[a + b*x])/Sqrt[Pi] + (d*Erfi[ a + b*x]^2)/4 + (b*c - a*d)*(a + b*x)*Erfi[a + b*x]^2 + (d*(a + b*x)^2*Erf i[a + b*x]^2)/2 + (b*c - a*d)*Sqrt[2/Pi]*Erfi[Sqrt[2]*(a + b*x)])/b^2
Int[Erfi[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [1/b^(m + 1) Subst[Int[ExpandIntegrand[Erfi[x]^2, (b*c - a*d + d*x)^m, x] , x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
\[\int \left (d x +c \right ) \operatorname {erfi}\left (b x +a \right )^{2}d x\]
Input:
int((d*x+c)*erfi(b*x+a)^2,x)
Output:
int((d*x+c)*erfi(b*x+a)^2,x)
Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.93 \[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=-\frac {4 \, \sqrt {2} \sqrt {\pi } \sqrt {-b^{2}} {\left (b c - a d\right )} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {-b^{2}} {\left (b x + a\right )}}{b}\right ) + 4 \, \sqrt {\pi } {\left (b^{2} d x + 2 \, b^{2} c - a b d\right )} \operatorname {erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (2 \, \pi b^{3} d x^{2} + 4 \, \pi b^{3} c x + \pi {\left (4 \, a b^{2} c - {\left (2 \, a^{2} - 1\right )} b d\right )}\right )} \operatorname {erfi}\left (b x + a\right )^{2} - 2 \, b d e^{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2}\right )}}{4 \, \pi b^{3}} \] Input:
integrate((d*x+c)*erfi(b*x+a)^2,x, algorithm="fricas")
Output:
-1/4*(4*sqrt(2)*sqrt(pi)*sqrt(-b^2)*(b*c - a*d)*erf(sqrt(2)*sqrt(-b^2)*(b* x + a)/b) + 4*sqrt(pi)*(b^2*d*x + 2*b^2*c - a*b*d)*erfi(b*x + a)*e^(b^2*x^ 2 + 2*a*b*x + a^2) - (2*pi*b^3*d*x^2 + 4*pi*b^3*c*x + pi*(4*a*b^2*c - (2*a ^2 - 1)*b*d))*erfi(b*x + a)^2 - 2*b*d*e^(2*b^2*x^2 + 4*a*b*x + 2*a^2))/(pi *b^3)
\[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\int \left (c + d x\right ) \operatorname {erfi}^{2}{\left (a + b x \right )}\, dx \] Input:
integrate((d*x+c)*erfi(b*x+a)**2,x)
Output:
Integral((c + d*x)*erfi(a + b*x)**2, x)
\[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\int { {\left (d x + c\right )} \operatorname {erfi}\left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*x+c)*erfi(b*x+a)^2,x, algorithm="maxima")
Output:
integrate((d*x + c)*erfi(b*x + a)^2, x)
\[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\int { {\left (d x + c\right )} \operatorname {erfi}\left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*x+c)*erfi(b*x+a)^2,x, algorithm="giac")
Output:
integrate((d*x + c)*erfi(b*x + a)^2, x)
Timed out. \[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\int {\mathrm {erfi}\left (a+b\,x\right )}^2\,\left (c+d\,x\right ) \,d x \] Input:
int(erfi(a + b*x)^2*(c + d*x),x)
Output:
int(erfi(a + b*x)^2*(c + d*x), x)
\[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=-\left (\int \mathrm {erf}\left (b i x +a i \right )^{2}d x \right ) c -\left (\int \mathrm {erf}\left (b i x +a i \right )^{2} x d x \right ) d \] Input:
int((d*x+c)*erfi(b*x+a)^2,x)
Output:
- (int(erf(a*i + b*i*x)**2,x)*c + int(erf(a*i + b*i*x)**2*x,x)*d)