Optimal. Leaf size=177 \[ -\frac {4}{45} b^6 \text {erf}(b x)^2-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac {28 b^6 \text {Ei}\left (-2 b^2 x^2\right )}{45 \pi }-\frac {8 b^5 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x}+\frac {2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}+\frac {4 b^3 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)^2}{6 x^6} \]
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Rubi [A] time = 0.29, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6364, 6391, 6373, 30, 2210, 2214} \[ -\frac {8 b^5 e^{-b^2 x^2} \text {Erf}(b x)}{45 \sqrt {\pi } x}+\frac {4 b^3 e^{-b^2 x^2} \text {Erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {2 b e^{-b^2 x^2} \text {Erf}(b x)}{15 \sqrt {\pi } x^5}-\frac {4}{45} b^6 \text {Erf}(b x)^2+\frac {28 b^6 \text {Ei}\left (-2 b^2 x^2\right )}{45 \pi }+\frac {2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}-\frac {\text {Erf}(b x)^2}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2210
Rule 2214
Rule 6364
Rule 6373
Rule 6391
Rubi steps
\begin {align*} \int \frac {\text {erf}(b x)^2}{x^7} \, dx &=-\frac {\text {erf}(b x)^2}{6 x^6}+\frac {(2 b) \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^6} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}-\frac {\text {erf}(b x)^2}{6 x^6}+\frac {\left (4 b^2\right ) \int \frac {e^{-2 b^2 x^2}}{x^5} \, dx}{15 \pi }-\frac {\left (4 b^3\right ) \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx}{15 \sqrt {\pi }}\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}+\frac {4 b^3 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)^2}{6 x^6}-\frac {\left (8 b^4\right ) \int \frac {e^{-2 b^2 x^2}}{x^3} \, dx}{45 \pi }-\frac {\left (4 b^4\right ) \int \frac {e^{-2 b^2 x^2}}{x^3} \, dx}{15 \pi }+\frac {\left (8 b^5\right ) \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2} \, dx}{45 \sqrt {\pi }}\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac {2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}+\frac {4 b^3 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {8 b^5 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x}-\frac {\text {erf}(b x)^2}{6 x^6}+2 \frac {\left (16 b^6\right ) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{45 \pi }+\frac {\left (8 b^6\right ) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{15 \pi }-\frac {\left (16 b^7\right ) \int e^{-b^2 x^2} \text {erf}(b x) \, dx}{45 \sqrt {\pi }}\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac {2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}+\frac {4 b^3 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {8 b^5 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x}-\frac {\text {erf}(b x)^2}{6 x^6}+\frac {28 b^6 \text {Ei}\left (-2 b^2 x^2\right )}{45 \pi }-\frac {1}{45} \left (8 b^6\right ) \operatorname {Subst}(\int x \, dx,x,\text {erf}(b x))\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac {2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}+\frac {4 b^3 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {8 b^5 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x}-\frac {4}{45} b^6 \text {erf}(b x)^2-\frac {\text {erf}(b x)^2}{6 x^6}+\frac {28 b^6 \text {Ei}\left (-2 b^2 x^2\right )}{45 \pi }\\ \end {align*}
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Mathematica [A] time = 0.05, size = 133, normalized size = 0.75 \[ \frac {e^{-2 b^2 x^2} \left (20 b^4 x^4-6 b^2 x^2-\pi e^{2 b^2 x^2} \left (8 b^6 x^6+15\right ) \text {erf}(b x)^2+56 b^6 x^6 e^{2 b^2 x^2} \text {Ei}\left (-2 b^2 x^2\right )-4 \sqrt {\pi } b x e^{b^2 x^2} \left (4 b^4 x^4-2 b^2 x^2+3\right ) \text {erf}(b x)\right )}{90 \pi x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 114, normalized size = 0.64 \[ \frac {56 \, b^{6} x^{6} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) - 4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )^{2} + 2 \, {\left (10 \, b^{4} x^{4} - 3 \, b^{2} x^{2}\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{90 \, \pi x^{6}} \]
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 \[ \int \frac {\operatorname {erf}\left (b x\right )^{2}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\erf \left (b x \right )^{2}}{x^{7}}\, dx \]
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 \[ \frac {2 \, b \int \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{6}}\,{d x}}{3 \, \sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right )^{2}}{6 \, x^{6}} \]
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 \[ \int \frac {{\mathrm {erf}\left (b\,x\right )}^2}{x^7} \,d x \]
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 \[ \int \frac {\operatorname {erf}^{2}{\left (b x \right )}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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