Optimal. Leaf size=125 \[ -\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{-b^2 x^2} \text {Erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {2 b^3 e^{-b^2 x^2} \text {Erf}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {Erf}(b x)^2-\frac {\text {Erf}(b x)^2}{4 x^4}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi } \]
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Rubi [A]
time = 0.12, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6499, 6526,
6508, 30, 2241, 2245} \begin {gather*} \frac {1}{3} b^4 \text {Erf}(b x)^2-\frac {b e^{-b^2 x^2} \text {Erf}(b x)}{3 \sqrt {\pi } x^3}-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi }+\frac {2 b^3 e^{-b^2 x^2} \text {Erf}(b x)}{3 \sqrt {\pi } x}-\frac {\text {Erf}(b x)^2}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2241
Rule 2245
Rule 6499
Rule 6508
Rule 6526
Rubi steps
\begin {align*} \int \frac {\text {erf}(b x)^2}{x^5} \, dx &=-\frac {\text {erf}(b x)^2}{4 x^4}+\frac {b \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx}{\sqrt {\pi }}\\ &=-\frac {b e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)^2}{4 x^4}+\frac {\left (2 b^2\right ) \int \frac {e^{-2 b^2 x^2}}{x^3} \, dx}{3 \pi }-\frac {\left (2 b^3\right ) \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {2 b^3 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erf}(b x)^2}{4 x^4}-2 \frac {\left (4 b^4\right ) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{3 \pi }+\frac {\left (4 b^5\right ) \int e^{-b^2 x^2} \text {erf}(b x) \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {2 b^3 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erf}(b x)^2}{4 x^4}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi }+\frac {1}{3} \left (2 b^4\right ) \text {Subst}(\int x \, dx,x,\text {erf}(b x))\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {2 b^3 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erf}(b x)^2-\frac {\text {erf}(b x)^2}{4 x^4}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi }\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 97, normalized size = 0.78 \begin {gather*} \frac {\frac {4 b e^{-b^2 x^2} x \left (-1+2 b^2 x^2\right ) \text {Erf}(b x)}{\sqrt {\pi }}+\left (-3+4 b^4 x^4\right ) \text {Erf}(b x)^2-\frac {4 b^2 x^2 \left (e^{-2 b^2 x^2}+4 b^2 x^2 \text {Ei}\left (-2 b^2 x^2\right )\right )}{\pi }}{12 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\erf \left (b x \right )^{2}}{x^{5}}\, dx\]
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 [A]
time = 0.38, size = 94, normalized size = 0.75 \begin {gather*} -\frac {16 \, b^{4} x^{4} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + 4 \, b^{2} x^{2} e^{\left (-2 \, b^{2} x^{2}\right )} - 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2}}{12 \, \pi x^{4}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {erf}^{2}{\left (b x \right )}}{x^{5}}\, dx \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 \frac {{\mathrm {erf}\left (b\,x\right )}^2}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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