Integrand size = 10, antiderivative size = 177 \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac {2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}+\frac {4 b^3 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {8 b^5 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x}-\frac {4}{45} b^6 \text {erf}(b x)^2-\frac {\text {erf}(b x)^2}{6 x^6}+\frac {28 b^6 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{45 \pi } \]
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Time = 0.19 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6499, 6526, 6508, 30, 2241, 2245} \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=-\frac {4}{45} b^6 \text {erf}(b x)^2-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac {28 b^6 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{45 \pi }-\frac {8 b^5 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x}+\frac {2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}+\frac {4 b^3 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)^2}{6 x^6} \]
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Rule 30
Rule 2241
Rule 2245
Rule 6499
Rule 6508
Rule 6526
Rubi steps \begin{align*} \text {integral}& = -\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 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{45 \pi }-\frac {1}{45} \left (8 b^6\right ) \text {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 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{45 \pi } \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.75 \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\frac {e^{-2 b^2 x^2} \left (-6 b^2 x^2+20 b^4 x^4-4 b e^{b^2 x^2} \sqrt {\pi } x \left (3-2 b^2 x^2+4 b^4 x^4\right ) \text {erf}(b x)-e^{2 b^2 x^2} \pi \left (15+8 b^6 x^6\right ) \text {erf}(b x)^2+56 b^6 e^{2 b^2 x^2} x^6 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )}{90 \pi x^6} \]
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\[\int \frac {\operatorname {erf}\left (b x \right )^{2}}{x^{7}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.64 \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\frac {56 \, b^{6} x^{6} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) - 4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )^{2} + 2 \, {\left (10 \, b^{4} x^{4} - 3 \, b^{2} x^{2}\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{90 \, \pi x^{6}} \]
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\[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\int \frac {\operatorname {erf}^{2}{\left (b x \right )}}{x^{7}}\, dx \]
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\[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )^{2}}{x^{7}} \,d x } \]
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\[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )^{2}}{x^{7}} \,d x } \]
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Timed out. \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\int \frac {{\mathrm {erf}\left (b\,x\right )}^2}{x^7} \,d x \]
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