Optimal. Leaf size=69 \[ \frac{x e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{\sqrt{\pi } e^c \text{Erfi}(b x)^2}{8 b^3}-\frac{e^{2 b^2 x^2+c}}{4 \sqrt{\pi } b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0757135, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6387, 6375, 30, 2209} \[ \frac{x e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{\sqrt{\pi } e^c \text{Erfi}(b x)^2}{8 b^3}-\frac{e^{2 b^2 x^2+c}}{4 \sqrt{\pi } b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6387
Rule 6375
Rule 30
Rule 2209
Rubi steps
\begin{align*} \int e^{c+b^2 x^2} x^2 \text{erfi}(b x) \, dx &=\frac{e^{c+b^2 x^2} x \text{erfi}(b x)}{2 b^2}-\frac{\int e^{c+b^2 x^2} \text{erfi}(b x) \, dx}{2 b^2}-\frac{\int e^{c+2 b^2 x^2} x \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{c+2 b^2 x^2}}{4 b^3 \sqrt{\pi }}+\frac{e^{c+b^2 x^2} x \text{erfi}(b x)}{2 b^2}-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfi}(b x))}{4 b^3}\\ &=-\frac{e^{c+2 b^2 x^2}}{4 b^3 \sqrt{\pi }}+\frac{e^{c+b^2 x^2} x \text{erfi}(b x)}{2 b^2}-\frac{e^c \sqrt{\pi } \text{erfi}(b x)^2}{8 b^3}\\ \end{align*}
Mathematica [A] time = 0.0160547, size = 58, normalized size = 0.84 \[ -\frac{e^c \left (-4 \sqrt{\pi } b x e^{b^2 x^2} \text{Erfi}(b x)+2 e^{2 b^2 x^2}+\pi \text{Erfi}(b x)^2\right )}{8 \sqrt{\pi } b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.225, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}{x}^{2}{\it erfi} \left ( bx \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.61336, size = 131, normalized size = 1.9 \begin{align*} \frac{{\left (4 \, \pi b x \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt{\pi }{\left (\pi \operatorname{erfi}\left (b x\right )^{2} + 2 \, e^{\left (2 \, b^{2} x^{2}\right )}\right )}\right )} e^{c}}{8 \, \pi b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 5.38369, size = 68, normalized size = 0.99 \begin{align*} \begin{cases} \frac{x e^{c} e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{2 b^{2}} - \frac{e^{c} e^{2 b^{2} x^{2}}}{4 \sqrt{\pi } b^{3}} - \frac{\sqrt{\pi } e^{c} \operatorname{erfi}^{2}{\left (b x \right )}}{8 b^{3}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]