Optimal. Leaf size=75 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^3}{3 a^3}+\frac{\left (4 \sqrt{\frac{1-a x}{a x+1}}+3\right ) (a x+1)^2}{6 a^3}-\frac{x}{a^2} \]
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Rubi [A] time = 0.513254, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6337, 1804, 1814, 12, 261} \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^3}{3 a^3}+\frac{\left (4 \sqrt{\frac{1-a x}{a x+1}}+3\right ) (a x+1)^2}{6 a^3}-\frac{x}{a^2} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1804
Rule 1814
Rule 12
Rule 261
Rubi steps
\begin{align*} \int e^{-\text{sech}^{-1}(a x)} x^2 \, dx &=\int \frac{x^2}{\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}} \, dx\\ &=\frac{4 \operatorname{Subst}\left (\int \frac{(-1+x)^3 x (1+x)}{\left (1+x^2\right )^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^3}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^3}{3 a^3}-\frac{2 \operatorname{Subst}\left (\int \frac{-4+6 x+12 x^2-6 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{3 a^3}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^3}{3 a^3}+\frac{(1+a x)^2 \left (3+4 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^3}+\frac{\operatorname{Subst}\left (\int \frac{24 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^3}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^3}{3 a^3}+\frac{(1+a x)^2 \left (3+4 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^3}+\frac{4 \operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^3}\\ &=-\frac{x}{a^2}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^3}{3 a^3}+\frac{(1+a x)^2 \left (3+4 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^3}\\ \end{align*}
Mathematica [A] time = 0.055459, size = 48, normalized size = 0.64 \[ \frac{3 a^2 x^2-2 (a x-1) \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2}{6 a^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.496, size = 269, normalized size = 3.6 \begin{align*} -{\frac{ax-1}{6\,{x}^{5}{a}^{8}} \left ( 3\,{a}^{6}{x}^{6} \left ({\frac{ax+1}{ax}} \right ) ^{5/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{3/2}+3\,{x}^{4}\ln \left ({a}^{2}{x}^{2} \right ) \left ({\frac{ax+1}{ax}} \right ) ^{5/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{3/2}{a}^{4}+3\,\sqrt{-{\frac{ax-1}{ax}}} \left ({\frac{ax+1}{ax}} \right ) ^{5/2}\ln \left ({a}^{2}{x}^{2} \right ){x}^{4}{a}^{4}-2\,{a}^{7}{x}^{7}-3\,{x}^{3}\ln \left ({a}^{2}{x}^{2} \right ) \left ({\frac{ax+1}{ax}} \right ) ^{5/2}\sqrt{-{\frac{ax-1}{ax}}}{a}^{3}-2\,{x}^{6}{a}^{6}+6\,{x}^{5}{a}^{5}+6\,{x}^{4}{a}^{4}-6\,{x}^{3}{a}^{3}-6\,{a}^{2}{x}^{2}+2\,ax+2 \right ) \left ({\frac{ax+1}{ax}} \right ) ^{-{\frac{5}{2}}} \left ( -{\frac{ax-1}{ax}} \right ) ^{-{\frac{5}{2}}}} \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{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91037, size = 111, normalized size = 1.48 \begin{align*} \frac{3 \, a x^{2} - 2 \,{\left (a^{2} x^{3} - x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}}{6 \, a^{2}} \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*} a \int \frac{x^{3}}{a x \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + 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{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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