\(\int \text {erfi}(a+b x) \, dx\) [224]

Optimal result

Integrand size = 6, antiderivative size = 35 \[ \int \text {erfi}(a+b x) \, dx=-\frac {e^{(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfi}(a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6486} \[ \int \text {erfi}(a+b x) \, dx=\frac {(a+b x) \text {erfi}(a+b x)}{b}-\frac {e^{(a+b x)^2}}{\sqrt {\pi } b} \]

[In]

Int[Erfi[a + b*x],x]

[Out]

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

Rule 6486

Int[Erfi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(Erfi[a + b*x]/b), x] - Simp[E^(a + b*x)^2/(b*Sqrt[P
i]), x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfi}(a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \text {erfi}(a+b x) \, dx=\frac {-\frac {e^{(a+b x)^2}}{\sqrt {\pi }}+(a+b x) \text {erfi}(a+b x)}{b} \]

[In]

Integrate[Erfi[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\left (b x +a \right ) \operatorname {erfi}\left (b x +a \right )-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\)	\(31\)
default	\(\frac {\left (b x +a \right ) \operatorname {erfi}\left (b x +a \right )-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\)	\(31\)
parallelrisch	\(\frac {x \,\operatorname {erfi}\left (b x +a \right ) \sqrt {\pi }\, b +a \,\operatorname {erfi}\left (b x +a \right ) \sqrt {\pi }-{\mathrm e}^{\left (b x +a \right )^{2}}}{\sqrt {\pi }\, b}\)	\(42\)
parts	\(x \,\operatorname {erfi}\left (b x +a \right )-\frac {2 b \left (\frac {{\mathrm e}^{b^{2} x^{2}+2 a b x +a^{2}}}{2 b^{2}}+\frac {i a \sqrt {\pi }\, \operatorname {erf}\left (i b x +i a \right )}{2 b^{2}}\right )}{\sqrt {\pi }}\)	\(60\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \text {erfi}(a+b x) \, dx=\frac {{\left (\pi b x + \pi a\right )} \operatorname {erfi}\left (b x + a\right ) - \sqrt {\pi } e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{\pi b} \]

[In]

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

[Out]

((pi*b*x + pi*a)*erfi(b*x + a) - sqrt(pi)*e^(b^2*x^2 + 2*a*b*x + a^2))/(pi*b)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.46 \[ \int \text {erfi}(a+b x) \, dx=\begin {cases} \frac {a \operatorname {erfi}{\left (a + b x \right )}}{b} + x \operatorname {erfi}{\left (a + b x \right )} - \frac {e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erfi}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*erfi(a + b*x)/b + x*erfi(a + b*x) - exp(a**2)*exp(b**2*x**2)*exp(2*a*b*x)/(sqrt(pi)*b), Ne(b, 0))
, (x*erfi(a), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \text {erfi}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {erfi}\left (b x + a\right ) - \frac {e^{\left ({\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \]

[In]

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

[Out]

((b*x + a)*erfi(b*x + a) - e^((b*x + a)^2)/sqrt(pi))/b

Giac [F]

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

[In]

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

[Out]

integrate(erfi(b*x + a), x)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \text {erfi}(a+b x) \, dx=x\,\mathrm {erfi}\left (a+b\,x\right )+\frac {a\,\mathrm {erfi}\left (a+b\,x\right )}{b}-\frac {{\mathrm {e}}^{a^2}\,{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^{2\,a\,b\,x}}{b\,\sqrt {\pi }} \]

[In]

int(erfi(a + b*x),x)

[Out]

x*erfi(a + b*x) + (a*erfi(a + b*x))/b - (exp(a^2)*exp(b^2*x^2)*exp(2*a*b*x))/(b*pi^(1/2))