Optimal. Leaf size=111 \[ -\frac {5 \text {erfi}\left (\sqrt {2} b x\right )}{6 \sqrt {2 \pi } b^3}-\frac {2 x^2 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } b}+\frac {x e^{2 b^2 x^2}}{3 \pi b^2}+\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } b^3}+\frac {1}{3} x^3 \text {erfi}(b x)^2 \]
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Rubi [A] time = 0.12, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6366, 6387, 6384, 2204, 2212} \[ -\frac {2 x^2 e^{b^2 x^2} \text {Erfi}(b x)}{3 \sqrt {\pi } b}+\frac {2 e^{b^2 x^2} \text {Erfi}(b x)}{3 \sqrt {\pi } b^3}-\frac {5 \text {Erfi}\left (\sqrt {2} b x\right )}{6 \sqrt {2 \pi } b^3}+\frac {x e^{2 b^2 x^2}}{3 \pi b^2}+\frac {1}{3} x^3 \text {Erfi}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2212
Rule 6366
Rule 6384
Rule 6387
Rubi steps
\begin {align*} \int x^2 \text {erfi}(b x)^2 \, dx &=\frac {1}{3} x^3 \text {erfi}(b x)^2-\frac {(4 b) \int e^{b^2 x^2} x^3 \text {erfi}(b x) \, dx}{3 \sqrt {\pi }}\\ &=-\frac {2 e^{b^2 x^2} x^2 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfi}(b x)^2+\frac {4 \int e^{2 b^2 x^2} x^2 \, dx}{3 \pi }+\frac {4 \int e^{b^2 x^2} x \text {erfi}(b x) \, dx}{3 b \sqrt {\pi }}\\ &=\frac {e^{2 b^2 x^2} x}{3 b^2 \pi }+\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{3 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^2 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfi}(b x)^2-\frac {\int e^{2 b^2 x^2} \, dx}{3 b^2 \pi }-\frac {4 \int e^{2 b^2 x^2} \, dx}{3 b^2 \pi }\\ &=\frac {e^{2 b^2 x^2} x}{3 b^2 \pi }+\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{3 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^2 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfi}(b x)^2-\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{3 b^3}-\frac {\text {erfi}\left (\sqrt {2} b x\right )}{6 b^3 \sqrt {2 \pi }}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 87, normalized size = 0.78 \[ \frac {4 \pi b^3 x^3 \text {erfi}(b x)^2-8 \sqrt {\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text {erfi}(b x)+4 b x e^{2 b^2 x^2}-5 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )}{12 \pi b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 91, normalized size = 0.82 \[ \frac {4 \, \pi b^{4} x^{3} \operatorname {erfi}\left (b x\right )^{2} + 4 \, b^{2} x e^{\left (2 \, b^{2} x^{2}\right )} - 8 \, \sqrt {\pi } {\left (b^{3} x^{2} - b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - 5 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erfi}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{12 \, \pi b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {erfi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int x^{2} \erfi \left (b x \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {erfi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {erfi}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {erfi}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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