Optimal. Leaf size=282 \[ -\frac{x (a x+1)^{3/4} (1-a x)^{5/4}}{3 a^2}+\frac{(a x+1)^{3/4} (1-a x)^{5/4}}{12 a^3}+\frac{3 (a x+1)^{3/4} \sqrt [4]{1-a x}}{8 a^3}+\frac{3 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt{2} a^3}-\frac{3 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt{2} a^3}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{8 \sqrt{2} a^3}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^3} \]
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Rubi [A] time = 0.198589, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {6126, 90, 80, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{x (a x+1)^{3/4} (1-a x)^{5/4}}{3 a^2}+\frac{(a x+1)^{3/4} (1-a x)^{5/4}}{12 a^3}+\frac{3 (a x+1)^{3/4} \sqrt [4]{1-a x}}{8 a^3}+\frac{3 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt{2} a^3}-\frac{3 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt{2} a^3}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{8 \sqrt{2} a^3}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^3} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 90
Rule 80
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{-\frac{1}{2} \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac{x^2 \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx\\ &=-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}-\frac{\int \frac{\sqrt [4]{1-a x} \left (-1+\frac{a x}{2}\right )}{\sqrt [4]{1+a x}} \, dx}{3 a^2}\\ &=\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{12 a^3}-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}+\frac{3 \int \frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx}{8 a^2}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}+\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{12 a^3}-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}+\frac{3 \int \frac{1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{16 a^2}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}+\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{12 a^3}-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{4 a^3}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}+\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{12 a^3}-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a^3}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}+\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{12 a^3}-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a^3}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}+\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{12 a^3}-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt{2} a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt{2} a^3}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}+\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{12 a^3}-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}+\frac{3 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt{2} a^3}-\frac{3 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt{2} a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^3}\\ &=\frac{3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}+\frac{(1-a x)^{5/4} (1+a x)^{3/4}}{12 a^3}-\frac{x (1-a x)^{5/4} (1+a x)^{3/4}}{3 a^2}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^3}-\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^3}+\frac{3 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt{2} a^3}-\frac{3 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt{2} a^3}\\ \end{align*}
Mathematica [C] time = 0.0260234, size = 62, normalized size = 0.22 \[ -\frac{(1-a x)^{5/4} \left (9\ 2^{3/4} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{5}{4},\frac{9}{4},\frac{1}{2} (1-a x)\right )+5 (a x+1)^{3/4} (4 a x-1)\right )}{60 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.111, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}}}}\, 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 \frac{x^{2}}{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.18141, size = 1369, normalized size = 4.85 \begin{align*} -\frac{36 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{3} \sqrt{\frac{\sqrt{2}{\left (a^{10} x - a^{9}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} +{\left (a^{7} x - a^{6}\right )} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - 1\right ) + 36 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{3} \sqrt{-\frac{\sqrt{2}{\left (a^{10} x - a^{9}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} -{\left (a^{7} x - a^{6}\right )} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + 1\right ) + 9 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a^{10} x - a^{9}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} +{\left (a^{7} x - a^{6}\right )} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 9 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a^{10} x - a^{9}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} -{\left (a^{7} x - a^{6}\right )} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \,{\left (8 \, a^{2} x^{2} - 10 \, a x + 11\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx \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 \frac{x^{2}}{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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