Optimal. Leaf size=175 \[ -\frac{x^5 e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } b}+\frac{5 x^3 e^{b^2 x^2} \text{Erfi}(b x)}{6 \sqrt{\pi } b^3}-\frac{5 x e^{b^2 x^2} \text{Erfi}(b x)}{4 \sqrt{\pi } b^5}+\frac{5 \text{Erfi}(b x)^2}{16 b^6}+\frac{x^4 e^{2 b^2 x^2}}{6 \pi b^2}-\frac{7 x^2 e^{2 b^2 x^2}}{12 \pi b^4}+\frac{11 e^{2 b^2 x^2}}{12 \pi b^6}+\frac{1}{6} x^6 \text{Erfi}(b x)^2 \]
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Rubi [A] time = 0.255586, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6366, 6387, 6375, 30, 2209, 2212} \[ -\frac{x^5 e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } b}+\frac{5 x^3 e^{b^2 x^2} \text{Erfi}(b x)}{6 \sqrt{\pi } b^3}-\frac{5 x e^{b^2 x^2} \text{Erfi}(b x)}{4 \sqrt{\pi } b^5}+\frac{5 \text{Erfi}(b x)^2}{16 b^6}+\frac{x^4 e^{2 b^2 x^2}}{6 \pi b^2}-\frac{7 x^2 e^{2 b^2 x^2}}{12 \pi b^4}+\frac{11 e^{2 b^2 x^2}}{12 \pi b^6}+\frac{1}{6} x^6 \text{Erfi}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6366
Rule 6387
Rule 6375
Rule 30
Rule 2209
Rule 2212
Rubi steps
\begin{align*} \int x^5 \text{erfi}(b x)^2 \, dx &=\frac{1}{6} x^6 \text{erfi}(b x)^2-\frac{(2 b) \int e^{b^2 x^2} x^6 \text{erfi}(b x) \, dx}{3 \sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x^5 \text{erfi}(b x)}{3 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfi}(b x)^2+\frac{2 \int e^{2 b^2 x^2} x^5 \, dx}{3 \pi }+\frac{5 \int e^{b^2 x^2} x^4 \text{erfi}(b x) \, dx}{3 b \sqrt{\pi }}\\ &=\frac{e^{2 b^2 x^2} x^4}{6 b^2 \pi }+\frac{5 e^{b^2 x^2} x^3 \text{erfi}(b x)}{6 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^5 \text{erfi}(b x)}{3 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfi}(b x)^2-\frac{2 \int e^{2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }-\frac{5 \int e^{2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }-\frac{5 \int e^{b^2 x^2} x^2 \text{erfi}(b x) \, dx}{2 b^3 \sqrt{\pi }}\\ &=-\frac{7 e^{2 b^2 x^2} x^2}{12 b^4 \pi }+\frac{e^{2 b^2 x^2} x^4}{6 b^2 \pi }-\frac{5 e^{b^2 x^2} x \text{erfi}(b x)}{4 b^5 \sqrt{\pi }}+\frac{5 e^{b^2 x^2} x^3 \text{erfi}(b x)}{6 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^5 \text{erfi}(b x)}{3 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfi}(b x)^2+\frac{\int e^{2 b^2 x^2} x \, dx}{3 b^4 \pi }+\frac{5 \int e^{2 b^2 x^2} x \, dx}{6 b^4 \pi }+\frac{5 \int e^{2 b^2 x^2} x \, dx}{2 b^4 \pi }+\frac{5 \int e^{b^2 x^2} \text{erfi}(b x) \, dx}{4 b^5 \sqrt{\pi }}\\ &=\frac{11 e^{2 b^2 x^2}}{12 b^6 \pi }-\frac{7 e^{2 b^2 x^2} x^2}{12 b^4 \pi }+\frac{e^{2 b^2 x^2} x^4}{6 b^2 \pi }-\frac{5 e^{b^2 x^2} x \text{erfi}(b x)}{4 b^5 \sqrt{\pi }}+\frac{5 e^{b^2 x^2} x^3 \text{erfi}(b x)}{6 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^5 \text{erfi}(b x)}{3 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfi}(b x)^2+\frac{5 \operatorname{Subst}(\int x \, dx,x,\text{erfi}(b x))}{8 b^6}\\ &=\frac{11 e^{2 b^2 x^2}}{12 b^6 \pi }-\frac{7 e^{2 b^2 x^2} x^2}{12 b^4 \pi }+\frac{e^{2 b^2 x^2} x^4}{6 b^2 \pi }-\frac{5 e^{b^2 x^2} x \text{erfi}(b x)}{4 b^5 \sqrt{\pi }}+\frac{5 e^{b^2 x^2} x^3 \text{erfi}(b x)}{6 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^5 \text{erfi}(b x)}{3 b \sqrt{\pi }}+\frac{5 \text{erfi}(b x)^2}{16 b^6}+\frac{1}{6} x^6 \text{erfi}(b x)^2\\ \end{align*}
Mathematica [A] time = 0.0386386, size = 99, normalized size = 0.57 \[ \frac{\pi \left (8 b^6 x^6+15\right ) \text{Erfi}(b x)^2-4 \sqrt{\pi } b x e^{b^2 x^2} \left (4 b^4 x^4-10 b^2 x^2+15\right ) \text{Erfi}(b x)+4 e^{2 b^2 x^2} \left (2 b^4 x^4-7 b^2 x^2+11\right )}{48 \pi b^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ({\it erfi} \left ( bx \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28581, size = 228, normalized size = 1.3 \begin{align*} -\frac{4 \, \sqrt{\pi }{\left (4 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} -{\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname{erfi}\left (b x\right )^{2} - 4 \,{\left (2 \, b^{4} x^{4} - 7 \, b^{2} x^{2} + 11\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{48 \, \pi b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.1129, size = 168, normalized size = 0.96 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{erfi}^{2}{\left (b x \right )}}{6} - \frac{x^{5} e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{3 \sqrt{\pi } b} + \frac{x^{4} e^{2 b^{2} x^{2}}}{6 \pi b^{2}} + \frac{5 x^{3} e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{6 \sqrt{\pi } b^{3}} - \frac{7 x^{2} e^{2 b^{2} x^{2}}}{12 \pi b^{4}} - \frac{5 x e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{4 \sqrt{\pi } b^{5}} + \frac{11 e^{2 b^{2} x^{2}}}{12 \pi b^{6}} + \frac{5 \operatorname{erfi}^{2}{\left (b x \right )}}{16 b^{6}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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