Optimal. Leaf size=68 \[ \frac{(a+b x) \text{Erfi}(a+b x)^2}{b}-\frac{2 e^{(a+b x)^2} \text{Erfi}(a+b x)}{\sqrt{\pi } b}+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} (a+b x)\right )}{b} \]
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Rubi [A] time = 0.134533, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6354, 6384, 2204} \[ \frac{(a+b x) \text{Erfi}(a+b x)^2}{b}-\frac{2 e^{(a+b x)^2} \text{Erfi}(a+b x)}{\sqrt{\pi } b}+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 6354
Rule 6384
Rule 2204
Rubi steps
\begin{align*} \int \text{erfi}(a+b x)^2 \, dx &=\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 }}\\ &=\frac{(a+b x) \text{erfi}(a+b x)^2}{b}-\frac{4 \operatorname{Subst}\left (\int e^{x^2} x \text{erfi}(x) \, dx,x,a+b x\right )}{b \sqrt{\pi }}\\ &=-\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{4 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{b \pi }\\ &=-\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}\\ \end{align*}
Mathematica [A] time = 0.0410586, size = 64, normalized size = 0.94 \[ \frac{\sqrt{\pi } (a+b x) \text{Erfi}(a+b x)^2-2 e^{(a+b x)^2} \text{Erfi}(a+b x)+\sqrt{2} \text{Erfi}\left (\sqrt{2} (a+b x)\right )}{\sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ({\it erfi} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.47335, size = 231, normalized size = 3.4 \begin{align*} -\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{erfi}\left (\frac{\sqrt{2} \sqrt{b^{2}}{\left (b x + a\right )}}{b}\right )}{\pi b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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