Optimal. Leaf size=107 \[ \frac {2 x}{b^5 \sqrt {\pi }}+\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {Erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {Erfi}(b x)}{2 b^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6522, 6519, 8,
30} \begin {gather*} \frac {2 x}{\sqrt {\pi } b^5}+\frac {2 x^3}{3 \sqrt {\pi } b^3}-\frac {x^4 e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{b^6}-\frac {x^2 e^{-b^2 x^2} \text {Erfi}(b x)}{b^4}+\frac {x^5}{5 \sqrt {\pi } b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 6519
Rule 6522
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx}{b^2}+\frac {\int x^4 \, dx}{b \sqrt {\pi }}\\ &=\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx}{b^4}+\frac {2 \int x^2 \, dx}{b^3 \sqrt {\pi }}\\ &=\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}+\frac {2 \int 1 \, dx}{b^5 \sqrt {\pi }}\\ &=\frac {2 x}{b^5 \sqrt {\pi }}+\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 68, normalized size = 0.64 \begin {gather*} \frac {\frac {60 b x+20 b^3 x^3+6 b^5 x^5}{\sqrt {\pi }}-15 e^{-b^2 x^2} \left (2+2 b^2 x^2+b^4 x^4\right ) \text {Erfi}(b x)}{30 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 103, normalized size = 0.96
method | result | size |
default | \(\frac {\left (6 x^{5} {\mathrm e}^{b^{2} x^{2}} b^{5}-15 \erfi \left (b x \right ) x^{4} b^{4} \sqrt {\pi }+20 b^{3} x^{3} {\mathrm e}^{b^{2} x^{2}}-30 \erfi \left (b x \right ) \sqrt {\pi }\, b^{2} x^{2}+60 \,{\mathrm e}^{b^{2} x^{2}} b x -30 \erfi \left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{30 b^{6} \sqrt {\pi }}\) | \(103\) |
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 [A]
time = 0.39, size = 79, normalized size = 0.74 \begin {gather*} \frac {{\left (2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 15 \, {\left (2 \, \pi + \pi b^{4} x^{4} + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{30 \, \pi b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 138.69, size = 99, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {x^{5}}{5 \sqrt {\pi } b} - \frac {x^{4} e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} + \frac {2 x^{3}}{3 \sqrt {\pi } b^{3}} - \frac {x^{2} e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{b^{4}} + \frac {2 x}{\sqrt {\pi } b^{5}} - \frac {e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \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 [B]
time = 0.25, size = 82, normalized size = 0.77 \begin {gather*} \frac {3\,b^4\,x^5+10\,b^2\,x^3+30\,x}{15\,b^5\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{b^6}+\frac {x^4\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{b^4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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