Optimal. Leaf size=301 \[ -\frac{a^5}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{a^5}{4 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{11 a^5}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{a^5}{8 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{35 a^5}{12 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{a^5}{12 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}+\frac{4 a^5}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}-\frac{18 a^5}{5 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^5}+\frac{2 a^5}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^6}-\frac{4 a^5}{7 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^7} \]
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Rubi [A] time = 0.575837, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6337, 1612} \[ -\frac{a^5}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{a^5}{4 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{11 a^5}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{a^5}{8 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{35 a^5}{12 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{a^5}{12 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}+\frac{4 a^5}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}-\frac{18 a^5}{5 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^5}+\frac{2 a^5}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^6}-\frac{4 a^5}{7 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^7} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1612
Rubi steps
\begin{align*} \int \frac{e^{2 \text{sech}^{-1}(a x)}}{x^6} \, dx &=\int \frac{\left (\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}\right )^2}{x^6} \, dx\\ &=-\left ((4 a) \operatorname{Subst}\left (\int \frac{x \left (a+a x^2\right )^4}{(-1+x)^8 (1+x)^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\left ((4 a) \operatorname{Subst}\left (\int \left (\frac{a^4}{(-1+x)^8}+\frac{3 a^4}{(-1+x)^7}+\frac{9 a^4}{2 (-1+x)^6}+\frac{4 a^4}{(-1+x)^5}+\frac{35 a^4}{16 (-1+x)^4}+\frac{11 a^4}{16 (-1+x)^3}+\frac{a^4}{16 (-1+x)^2}-\frac{a^4}{16 (1+x)^4}+\frac{a^4}{16 (1+x)^3}-\frac{a^4}{16 (1+x)^2}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\frac{4 a^5}{7 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^7}+\frac{2 a^5}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^6}-\frac{18 a^5}{5 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^5}+\frac{4 a^5}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^4}-\frac{35 a^5}{12 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^3}+\frac{11 a^5}{8 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}-\frac{a^5}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{a^5}{12 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}+\frac{a^5}{8 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}-\frac{a^5}{4 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}\\ \end{align*}
Mathematica [A] time = 0.0991938, size = 85, normalized size = 0.28 \[ \frac{21 a^2 x^2+2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2 \left (8 a^5 x^5-8 a^4 x^4+12 a^3 x^3-12 a^2 x^2+15 a x-15\right )-30}{105 a^2 x^7} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.24, size = 92, normalized size = 0.3 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{1}{7\,{x}^{7}}}+{\frac{{a}^{2}}{5\,{x}^{5}}} \right ) }+{\frac{ \left ( 2\,{a}^{2}{x}^{2}-2 \right ) \left ( 8\,{x}^{4}{a}^{4}+12\,{a}^{2}{x}^{2}+15 \right ) }{105\,{x}^{6}a}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{1}{7\,{a}^{2}{x}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07831, size = 88, normalized size = 0.29 \begin{align*} \frac{1}{5 \, x^{5}} + \frac{2 \,{\left (8 \, a^{6} x^{7} + 4 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - 15 \, x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{105 \, a^{2} x^{8}} - \frac{2}{7 \, a^{2} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08195, size = 174, normalized size = 0.58 \begin{align*} \frac{21 \, a^{2} x^{2} + 2 \,{\left (8 \, a^{7} x^{7} + 4 \, a^{5} x^{5} + 3 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 30}{105 \, a^{2} x^{7}} \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{2}{x^{8}}\, dx + \int - \frac{a^{2}}{x^{6}}\, dx + \int \frac{2 a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{7}}\, dx}{a^{2}} \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{{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}^{2}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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