Integrand size = 10, antiderivative size = 162 \[ \int x^4 \text {erfi}(b x)^2 \, dx=-\frac {11 e^{2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {4 e^{b^2 x^2} \text {erfi}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{b^2 x^2} x^2 \text {erfi}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2+\frac {43 \text {erfi}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }} \]
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Time = 0.15 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6501, 6522, 6519, 2235, 2243} \[ \int x^4 \text {erfi}(b x)^2 \, dx=\frac {43 \text {erfi}\left (\sqrt {2} b x\right )}{40 \sqrt {2 \pi } b^5}-\frac {2 x^4 e^{b^2 x^2} \text {erfi}(b x)}{5 \sqrt {\pi } b}+\frac {x^3 e^{2 b^2 x^2}}{5 \pi b^2}-\frac {4 e^{b^2 x^2} \text {erfi}(b x)}{5 \sqrt {\pi } b^5}-\frac {11 x e^{2 b^2 x^2}}{20 \pi b^4}+\frac {4 x^2 e^{b^2 x^2} \text {erfi}(b x)}{5 \sqrt {\pi } b^3}+\frac {1}{5} x^5 \text {erfi}(b x)^2 \]
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Rule 2235
Rule 2243
Rule 6501
Rule 6519
Rule 6522
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {erfi}(b x)^2-\frac {(4 b) \int e^{b^2 x^2} x^5 \text {erfi}(b x) \, dx}{5 \sqrt {\pi }} \\ & = -\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2+\frac {4 \int e^{2 b^2 x^2} x^4 \, dx}{5 \pi }+\frac {8 \int e^{b^2 x^2} x^3 \text {erfi}(b x) \, dx}{5 b \sqrt {\pi }} \\ & = \frac {e^{2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{b^2 x^2} x^2 \text {erfi}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2-\frac {3 \int e^{2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{b^2 x^2} x \text {erfi}(b x) \, dx}{5 b^3 \sqrt {\pi }} \\ & = -\frac {11 e^{2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {4 e^{b^2 x^2} \text {erfi}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{b^2 x^2} x^2 \text {erfi}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2+\frac {3 \int e^{2 b^2 x^2} \, dx}{20 b^4 \pi }+\frac {2 \int e^{2 b^2 x^2} \, dx}{5 b^4 \pi }+\frac {8 \int e^{2 b^2 x^2} \, dx}{5 b^4 \pi } \\ & = -\frac {11 e^{2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {4 e^{b^2 x^2} \text {erfi}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{b^2 x^2} x^2 \text {erfi}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2+\frac {2 \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{5 b^5}+\frac {11 \text {erfi}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.65 \[ \int x^4 \text {erfi}(b x)^2 \, dx=\frac {4 b e^{2 b^2 x^2} x \left (-11+4 b^2 x^2\right )-32 e^{b^2 x^2} \sqrt {\pi } \left (2-2 b^2 x^2+b^4 x^4\right ) \text {erfi}(b x)+16 b^5 \pi x^5 \text {erfi}(b x)^2+43 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )}{80 b^5 \pi } \]
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\[\int x^{4} \operatorname {erfi}\left (b x \right )^{2}d x\]
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Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.70 \[ \int x^4 \text {erfi}(b x)^2 \, dx=\frac {16 \, \pi b^{6} x^{5} \operatorname {erfi}\left (b x\right )^{2} - 32 \, \sqrt {\pi } {\left (b^{5} x^{4} - 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - 43 \, \sqrt {2} \sqrt {\pi } \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right ) + 4 \, {\left (4 \, b^{4} x^{3} - 11 \, b^{2} x\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \]
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\[ \int x^4 \text {erfi}(b x)^2 \, dx=\int x^{4} \operatorname {erfi}^{2}{\left (b x \right )}\, dx \]
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\[ \int x^4 \text {erfi}(b x)^2 \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right )^{2} \,d x } \]
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\[ \int x^4 \text {erfi}(b x)^2 \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right )^{2} \,d x } \]
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Timed out. \[ \int x^4 \text {erfi}(b x)^2 \, dx=\int x^4\,{\mathrm {erfi}\left (b\,x\right )}^2 \,d x \]
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