\(\int (c+d x) \text {erfi}(a+b x) \, dx\) [223]

Optimal result

Integrand size = 12, antiderivative size = 115 \[ \int (c+d x) \text {erfi}(a+b x) \, dx=-\frac {(b c-a d) e^{(a+b x)^2}}{b^2 \sqrt {\pi }}-\frac {d e^{(a+b x)^2} (a+b x)}{2 b^2 \sqrt {\pi }}+\frac {d \text {erfi}(a+b x)}{4 b^2}-\frac {(b c-a d)^2 \text {erfi}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d} \]

[Out]

1/4*d*erfi(b*x+a)/b^2-1/2*(-a*d+b*c)^2*erfi(b*x+a)/b^2/d+1/2*(d*x+c)^2*erfi(b*x+a)/d-(-a*d+b*c)*exp((b*x+a)^2)
/b^2/Pi^(1/2)-1/2*d*exp((b*x+a)^2)*(b*x+a)/b^2/Pi^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6498, 2258, 2235, 2240, 2243} \[ \int (c+d x) \text {erfi}(a+b x) \, dx=-\frac {(b c-a d)^2 \text {erfi}(a+b x)}{2 b^2 d}-\frac {e^{(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^2}+\frac {d \text {erfi}(a+b x)}{4 b^2}-\frac {d e^{(a+b x)^2} (a+b x)}{2 \sqrt {\pi } b^2}+\frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d} \]

[In]

Int[(c + d*x)*Erfi[a + b*x],x]

[Out]

-(((b*c - a*d)*E^(a + b*x)^2)/(b^2*Sqrt[Pi])) - (d*E^(a + b*x)^2*(a + b*x))/(2*b^2*Sqrt[Pi]) + (d*Erfi[a + b*x
])/(4*b^2) - ((b*c - a*d)^2*Erfi[a + b*x])/(2*b^2*d) + ((c + d*x)^2*Erfi[a + b*x])/(2*d)

Rule 2235

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]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 6498

Int[Erfi[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Erfi[a + b*x]/(d
*(m + 1))), x] - Dist[2*(b/(Sqrt[Pi]*d*(m + 1))), Int[(c + d*x)^(m + 1)*E^(a + b*x)^2, x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d}-\frac {b \int e^{(a+b x)^2} (c+d x)^2 \, dx}{d \sqrt {\pi }} \\ & = \frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d}-\frac {b \int \left (\frac {(b c-a d)^2 e^{(a+b x)^2}}{b^2}+\frac {2 d (b c-a d) e^{(a+b x)^2} (a+b x)}{b^2}+\frac {d^2 e^{(a+b x)^2} (a+b x)^2}{b^2}\right ) \, dx}{d \sqrt {\pi }} \\ & = \frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d}-\frac {d \int e^{(a+b x)^2} (a+b x)^2 \, dx}{b \sqrt {\pi }}-\frac {(2 (b c-a d)) \int e^{(a+b x)^2} (a+b x) \, dx}{b \sqrt {\pi }}-\frac {(b c-a d)^2 \int e^{(a+b x)^2} \, dx}{b d \sqrt {\pi }} \\ & = -\frac {(b c-a d) e^{(a+b x)^2}}{b^2 \sqrt {\pi }}-\frac {d e^{(a+b x)^2} (a+b x)}{2 b^2 \sqrt {\pi }}-\frac {(b c-a d)^2 \text {erfi}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d}+\frac {d \int e^{(a+b x)^2} \, dx}{2 b \sqrt {\pi }} \\ & = -\frac {(b c-a d) e^{(a+b x)^2}}{b^2 \sqrt {\pi }}-\frac {d e^{(a+b x)^2} (a+b x)}{2 b^2 \sqrt {\pi }}+\frac {d \text {erfi}(a+b x)}{4 b^2}-\frac {(b c-a d)^2 \text {erfi}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.68 \[ \int (c+d x) \text {erfi}(a+b x) \, dx=\frac {-2 e^{(a+b x)^2} (2 b c-a d+b d x)+\sqrt {\pi } \left (4 a b c+d-2 a^2 d+4 b^2 c x+2 b^2 d x^2\right ) \text {erfi}(a+b x)}{4 b^2 \sqrt {\pi }} \]

[In]

Integrate[(c + d*x)*Erfi[a + b*x],x]

[Out]

(-2*E^(a + b*x)^2*(2*b*c - a*d + b*d*x) + Sqrt[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])/(4*b^2*Sqrt[Pi])

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {erfi}\left (b x +a \right ) d a \left (b x +a \right )}{b}+\operatorname {erfi}\left (b x +a \right ) c \left (b x +a \right )+\frac {\operatorname {erfi}\left (b x +a \right ) d \left (b x +a \right )^{2}}{2 b}+\frac {-d \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{4}\right )-{\mathrm e}^{\left (b x +a \right )^{2}} b c +d a \,{\mathrm e}^{\left (b x +a \right )^{2}}}{b \sqrt {\pi }}}{b}\)	\(117\)
default	\(\frac {-\frac {\operatorname {erfi}\left (b x +a \right ) d a \left (b x +a \right )}{b}+\operatorname {erfi}\left (b x +a \right ) c \left (b x +a \right )+\frac {\operatorname {erfi}\left (b x +a \right ) d \left (b x +a \right )^{2}}{2 b}+\frac {-d \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right )}{4}\right )-{\mathrm e}^{\left (b x +a \right )^{2}} b c +d a \,{\mathrm e}^{\left (b x +a \right )^{2}}}{b \sqrt {\pi }}}{b}\)	\(117\)
parallelrisch	\(\frac {2 d \,x^{2} \operatorname {erfi}\left (b x +a \right ) b^{2} \sqrt {\pi }+4 x \,\operatorname {erfi}\left (b x +a \right ) c \,b^{2} \sqrt {\pi }-2 \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right ) a^{2} d +4 \sqrt {\pi }\, \operatorname {erfi}\left (b x +a \right ) a b c -2 \,{\mathrm e}^{\left (b x +a \right )^{2}} b d x +d \,\operatorname {erfi}\left (b x +a \right ) \sqrt {\pi }+2 d a \,{\mathrm e}^{\left (b x +a \right )^{2}}-4 \,{\mathrm e}^{\left (b x +a \right )^{2}} b c}{4 b^{2} \sqrt {\pi }}\)	\(121\)
parts	\(\frac {\operatorname {erfi}\left (b x +a \right ) d \,x^{2}}{2}+\operatorname {erfi}\left (b x +a \right ) c x -\frac {b \left ({\mathrm e}^{a^{2}} d \left (\frac {x \,{\mathrm e}^{b^{2} x^{2}+2 a b x}}{2 b^{2}}-\frac {a \left (\frac {{\mathrm e}^{b^{2} x^{2}+2 a b x}}{2 b^{2}}+\frac {i a \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{2 b^{2}}\right )}{b}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{4 b^{3}}\right )+2 \,{\mathrm e}^{a^{2}} c \left (\frac {{\mathrm e}^{b^{2} x^{2}+2 a b x}}{2 b^{2}}+\frac {i a \sqrt {\pi }\, {\mathrm e}^{-a^{2}} \operatorname {erf}\left (i b x +i a \right )}{2 b^{2}}\right )\right )}{\sqrt {\pi }}\)	\(190\)

[In]

int((d*x+c)*erfi(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b*(-1/b*erfi(b*x+a)*d*a*(b*x+a)+erfi(b*x+a)*c*(b*x+a)+1/2/b*erfi(b*x+a)*d*(b*x+a)^2+1/b/Pi^(1/2)*(-d*(1/2*(b
*x+a)*exp((b*x+a)^2)-1/4*Pi^(1/2)*erfi(b*x+a))-exp((b*x+a)^2)*b*c+d*a*exp((b*x+a)^2)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int (c+d x) \text {erfi}(a+b x) \, dx=-\frac {2 \, \sqrt {\pi } {\left (b d x + 2 \, b c - a d\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (2 \, \pi b^{2} d x^{2} + 4 \, \pi b^{2} c x + \pi {\left (4 \, a b c - {\left (2 \, a^{2} - 1\right )} d\right )}\right )} \operatorname {erfi}\left (b x + a\right )}{4 \, \pi b^{2}} \]

[In]

integrate((d*x+c)*erfi(b*x+a),x, algorithm="fricas")

[Out]

-1/4*(2*sqrt(pi)*(b*d*x + 2*b*c - a*d)*e^(b^2*x^2 + 2*a*b*x + a^2) - (2*pi*b^2*d*x^2 + 4*pi*b^2*c*x + pi*(4*a*
b*c - (2*a^2 - 1)*d))*erfi(b*x + a))/(pi*b^2)

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.55 \[ \int (c+d x) \text {erfi}(a+b x) \, dx=\begin {cases} - \frac {a^{2} d \operatorname {erfi}{\left (a + b x \right )}}{2 b^{2}} + \frac {a c \operatorname {erfi}{\left (a + b x \right )}}{b} + \frac {a d e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{2 \sqrt {\pi } b^{2}} + c x \operatorname {erfi}{\left (a + b x \right )} + \frac {d x^{2} \operatorname {erfi}{\left (a + b x \right )}}{2} - \frac {c e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} - \frac {d x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{2 \sqrt {\pi } b} + \frac {d \operatorname {erfi}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \operatorname {erfi}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x+c)*erfi(b*x+a),x)

[Out]

Piecewise((-a**2*d*erfi(a + b*x)/(2*b**2) + a*c*erfi(a + b*x)/b + a*d*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(2
*sqrt(pi)*b**2) + c*x*erfi(a + b*x) + d*x**2*erfi(a + b*x)/2 - c*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(p
i)*b) - d*x*exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(2*sqrt(pi)*b) + d*erfi(a + b*x)/(4*b**2), Ne(b, 0)), ((c*x
+ d*x**2/2)*erfi(a), True))

Maxima [F]

\[ \int (c+d x) \text {erfi}(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {erfi}\left (b x + a\right ) \,d x } \]

[In]

integrate((d*x+c)*erfi(b*x+a),x, algorithm="maxima")

[Out]

integrate((d*x + c)*erfi(b*x + a), x)

Giac [F]

\[ \int (c+d x) \text {erfi}(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {erfi}\left (b x + a\right ) \,d x } \]

[In]

integrate((d*x+c)*erfi(b*x+a),x, algorithm="giac")

[Out]

integrate((d*x + c)*erfi(b*x + a), x)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.92 \[ \int (c+d x) \text {erfi}(a+b x) \, dx=\frac {\frac {{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {a\,d}{2}-b\,c\right )}{b^2}-\frac {d\,x\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b}}{\sqrt {\pi }}+\mathrm {erfi}\left (a+b\,x\right )\,\left (\frac {d\,x^2}{2}+c\,x\right )+\frac {\mathrm {erfi}\left (a+b\,x\right )\,\left (-2\,d\,a^2\,b+4\,c\,a\,b^2+d\,b\right )}{4\,b^3} \]

[In]

int(erfi(a + b*x)*(c + d*x),x)

[Out]

((exp(a^2 + b^2*x^2 + 2*a*b*x)*((a*d)/2 - b*c))/b^2 - (d*x*exp(a^2 + b^2*x^2 + 2*a*b*x))/(2*b))/pi^(1/2) + erf
i(a + b*x)*(c*x + (d*x^2)/2) + (erfi(a + b*x)*(b*d + 4*a*b^2*c - 2*a^2*b*d))/(4*b^3)