Optimal. Leaf size=84 \[ -\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x)}{8 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 e^{\text{sech}^{-1}(a x)} \]
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Rubi [A] time = 0.0319382, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6335, 30, 90, 41, 216} \[ -\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x)}{8 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 e^{\text{sech}^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 90
Rule 41
Rule 216
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}(a x)} x^3 \, dx &=\frac{1}{4} e^{\text{sech}^{-1}(a x)} x^4+\frac{\int x^2 \, dx}{4 a}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x^2}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{4 a}\\ &=\frac{x^3}{12 a}+\frac{1}{4} e^{\text{sech}^{-1}(a x)} x^4-\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{8 a^3}\\ &=\frac{x^3}{12 a}+\frac{1}{4} e^{\text{sech}^{-1}(a x)} x^4-\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a^3}\\ &=\frac{x^3}{12 a}+\frac{1}{4} e^{\text{sech}^{-1}(a x)} x^4-\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{1+a x}}}+\frac{\sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \sin ^{-1}(a x)}{8 a^4}\\ \end{align*}
Mathematica [C] time = 0.126645, size = 97, normalized size = 1.15 \[ \frac{8 a^3 x^3-3 a \sqrt{\frac{1-a x}{a x+1}} \left (-2 a^3 x^4-2 a^2 x^3+a x^2+x\right )+3 i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right )}{24 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.179, size = 118, normalized size = 1.4 \begin{align*}{\frac{x{\it csgn} \left ( a \right ) }{8\,{a}^{3}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( 2\,{\it csgn} \left ( a \right ){x}^{3}{a}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-x\sqrt{-{a}^{2}{x}^{2}+1}{\it csgn} \left ( a \right ) a+\arctan \left ({x{\it csgn} \left ( a \right ) a{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{{x}^{3}}{3\,a}} \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*} \frac{x^{3}}{3 \, a} + \frac{-\frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{4 \, a^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} x}{8 \, a^{2}} + \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}} a^{2}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77774, size = 203, normalized size = 2.42 \begin{align*} \frac{8 \, a^{3} x^{3} + 3 \,{\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 3 \, \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right )}{24 \, a^{4}} \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*} \frac{\int x^{2}\, dx + \int a x^{3} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a} \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^{3}{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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