Optimal. Leaf size=55 \[ -\frac{8 a^2 \left (\frac{1-a x}{a x+1}\right )^{3/2}}{3 \left (1-\frac{1-a x}{a x+1}\right )^3}-\frac{1}{3 a x^3} \]
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Rubi [C] time = 0.0366551, antiderivative size = 84, normalized size of antiderivative = 1.53, 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, 103, 12, 95} \[ \frac{\sqrt{1-a x}}{6 a x^3 \sqrt{\frac{1}{a x+1}}}+\frac{1}{6 a x^3}-\frac{e^{\text{sech}^{-1}(a x)}}{2 x^2}+\frac{a \sqrt{1-a x}}{3 x \sqrt{\frac{1}{a x+1}}} \]
Warning: Unable to verify antiderivative.
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Rule 6335
Rule 30
Rule 103
Rule 12
Rule 95
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(a x)}}{x^3} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{2 x^2}-\frac{\int \frac{1}{x^4} \, dx}{2 a}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^4 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{2 a}\\ &=\frac{1}{6 a x^3}-\frac{e^{\text{sech}^{-1}(a x)}}{2 x^2}+\frac{\sqrt{1-a x}}{6 a x^3 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{2 a^2}{x^2 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{6 a}\\ &=\frac{1}{6 a x^3}-\frac{e^{\text{sech}^{-1}(a x)}}{2 x^2}+\frac{\sqrt{1-a x}}{6 a x^3 \sqrt{\frac{1}{1+a x}}}-\frac{1}{3} \left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^2 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{6 a x^3}-\frac{e^{\text{sech}^{-1}(a x)}}{2 x^2}+\frac{\sqrt{1-a x}}{6 a x^3 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{3 x \sqrt{\frac{1}{1+a x}}}\\ \end{align*}
Mathematica [A] time = 0.0431778, size = 43, normalized size = 0.78 \[ \frac{(a x-1) \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2-1}{3 a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 53, normalized size = 1. \begin{align*}{\frac{{a}^{2}{x}^{2}-1}{3\,{x}^{2}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{1}{3\,{x}^{3}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04497, size = 58, normalized size = 1.05 \begin{align*} \frac{{\left (a^{2} x^{3} - x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{3 \, a x^{4}} - \frac{1}{3 \, a x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83451, size = 108, normalized size = 1.96 \begin{align*} \frac{{\left (a^{3} x^{3} - a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1}{3 \, a x^{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*} \frac{\int \frac{1}{x^{4}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{3}}\, 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 \frac{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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