Optimal. Leaf size=162 \[ -\frac{2 x^4 e^{b^2 x^2} \text{Erfi}(b x)}{5 \sqrt{\pi } b}+\frac{4 x^2 e^{b^2 x^2} \text{Erfi}(b x)}{5 \sqrt{\pi } b^3}-\frac{4 e^{b^2 x^2} \text{Erfi}(b x)}{5 \sqrt{\pi } b^5}+\frac{43 \text{Erfi}\left (\sqrt{2} b x\right )}{40 \sqrt{2 \pi } b^5}+\frac{x^3 e^{2 b^2 x^2}}{5 \pi b^2}-\frac{11 x e^{2 b^2 x^2}}{20 \pi b^4}+\frac{1}{5} x^5 \text{Erfi}(b x)^2 \]
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Rubi [A] time = 0.225014, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6366, 6387, 6384, 2204, 2212} \[ -\frac{2 x^4 e^{b^2 x^2} \text{Erfi}(b x)}{5 \sqrt{\pi } b}+\frac{4 x^2 e^{b^2 x^2} \text{Erfi}(b x)}{5 \sqrt{\pi } b^3}-\frac{4 e^{b^2 x^2} \text{Erfi}(b x)}{5 \sqrt{\pi } b^5}+\frac{43 \text{Erfi}\left (\sqrt{2} b x\right )}{40 \sqrt{2 \pi } b^5}+\frac{x^3 e^{2 b^2 x^2}}{5 \pi b^2}-\frac{11 x e^{2 b^2 x^2}}{20 \pi b^4}+\frac{1}{5} x^5 \text{Erfi}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6366
Rule 6387
Rule 6384
Rule 2204
Rule 2212
Rubi steps
\begin{align*} \int x^4 \text{erfi}(b x)^2 \, dx &=\frac{1}{5} x^5 \text{erfi}(b x)^2-\frac{(4 b) \int e^{b^2 x^2} x^5 \text{erfi}(b x) \, dx}{5 \sqrt{\pi }}\\ &=-\frac{2 e^{b^2 x^2} x^4 \text{erfi}(b x)}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erfi}(b x)^2+\frac{4 \int e^{2 b^2 x^2} x^4 \, dx}{5 \pi }+\frac{8 \int e^{b^2 x^2} x^3 \text{erfi}(b x) \, dx}{5 b \sqrt{\pi }}\\ &=\frac{e^{2 b^2 x^2} x^3}{5 b^2 \pi }+\frac{4 e^{b^2 x^2} x^2 \text{erfi}(b x)}{5 b^3 \sqrt{\pi }}-\frac{2 e^{b^2 x^2} x^4 \text{erfi}(b x)}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erfi}(b x)^2-\frac{3 \int e^{2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac{8 \int e^{2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac{8 \int e^{b^2 x^2} x \text{erfi}(b x) \, dx}{5 b^3 \sqrt{\pi }}\\ &=-\frac{11 e^{2 b^2 x^2} x}{20 b^4 \pi }+\frac{e^{2 b^2 x^2} x^3}{5 b^2 \pi }-\frac{4 e^{b^2 x^2} \text{erfi}(b x)}{5 b^5 \sqrt{\pi }}+\frac{4 e^{b^2 x^2} x^2 \text{erfi}(b x)}{5 b^3 \sqrt{\pi }}-\frac{2 e^{b^2 x^2} x^4 \text{erfi}(b x)}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erfi}(b x)^2+\frac{3 \int e^{2 b^2 x^2} \, dx}{20 b^4 \pi }+\frac{2 \int e^{2 b^2 x^2} \, dx}{5 b^4 \pi }+\frac{8 \int e^{2 b^2 x^2} \, dx}{5 b^4 \pi }\\ &=-\frac{11 e^{2 b^2 x^2} x}{20 b^4 \pi }+\frac{e^{2 b^2 x^2} x^3}{5 b^2 \pi }-\frac{4 e^{b^2 x^2} \text{erfi}(b x)}{5 b^5 \sqrt{\pi }}+\frac{4 e^{b^2 x^2} x^2 \text{erfi}(b x)}{5 b^3 \sqrt{\pi }}-\frac{2 e^{b^2 x^2} x^4 \text{erfi}(b x)}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erfi}(b x)^2+\frac{2 \sqrt{\frac{2}{\pi }} \text{erfi}\left (\sqrt{2} b x\right )}{5 b^5}+\frac{11 \text{erfi}\left (\sqrt{2} b x\right )}{40 b^5 \sqrt{2 \pi }}\\ \end{align*}
Mathematica [A] time = 0.0455509, size = 105, normalized size = 0.65 \[ \frac{16 \pi b^5 x^5 \text{Erfi}(b x)^2-32 \sqrt{\pi } e^{b^2 x^2} \left (b^4 x^4-2 b^2 x^2+2\right ) \text{Erfi}(b x)+4 b x e^{2 b^2 x^2} \left (4 b^2 x^2-11\right )+43 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} b x\right )}{80 \pi b^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ({\it erfi} \left ( bx \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 x^{4} \operatorname{erfi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.5078, size = 278, normalized size = 1.72 \begin{align*} \frac{16 \, \pi b^{6} x^{5} \operatorname{erfi}\left (b x\right )^{2} - 32 \, \sqrt{\pi }{\left (b^{5} x^{4} - 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + 43 \, \sqrt{2} \sqrt{\pi } \sqrt{b^{2}} \operatorname{erfi}\left (\sqrt{2} \sqrt{b^{2}} x\right ) + 4 \,{\left (4 \, b^{4} x^{3} - 11 \, b^{2} x\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \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 x^{4} \operatorname{erfi}^{2}{\left (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 x^{4} \operatorname{erfi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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