Optimal. Leaf size=109 \[ \frac {3 x^2}{4 b^3 \sqrt {\pi }}+\frac {x^4}{4 b \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {Erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {Erfi}(b x)}{2 b^2}+\frac {3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{4 b^3 \sqrt {\pi }} \]
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Rubi [A]
time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6522, 6513, 30}
\begin {gather*} \frac {3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{4 \sqrt {\pi } b^3}+\frac {3 x^2}{4 \sqrt {\pi } b^3}-\frac {x^3 e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2}-\frac {3 x e^{-b^2 x^2} \text {Erfi}(b x)}{4 b^4}+\frac {x^4}{4 \sqrt {\pi } b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 6513
Rule 6522
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 \int e^{-b^2 x^2} x^2 \text {erfi}(b x) \, dx}{2 b^2}+\frac {\int x^3 \, dx}{b \sqrt {\pi }}\\ &=\frac {x^4}{4 b \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 \int e^{-b^2 x^2} \text {erfi}(b x) \, dx}{4 b^4}+\frac {3 \int x \, dx}{2 b^3 \sqrt {\pi }}\\ &=\frac {3 x^2}{4 b^3 \sqrt {\pi }}+\frac {x^4}{4 b \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{4 b^3 \sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 0.39 \begin {gather*} \frac {x^2 \left (1+b^2 x^2-\, _2F_2\left (1,1;-\frac {1}{2},2;-b^2 x^2\right )\right )}{4 b^3 \sqrt {\pi }} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int x^{4} \erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^4\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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