Optimal. Leaf size=353 \[ -\frac{5 a^6}{16 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{5 a^6}{16 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{3 a^6}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{a^6}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{5 a^6}{12 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{11 a^6}{6 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}+\frac{a^6}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}+\frac{9 a^6}{4 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{a^6}{10 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^5}-\frac{19 a^6}{10 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^5}+\frac{a^6}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^6}-\frac{2 a^6}{7 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^7} \]
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Rubi [A] time = 0.604905, antiderivative size = 353, 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} \[ -\frac{5 a^6}{16 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{5 a^6}{16 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{3 a^6}{8 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}+\frac{a^6}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{5 a^6}{12 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3}-\frac{11 a^6}{6 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}+\frac{a^6}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^4}+\frac{9 a^6}{4 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{a^6}{10 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^5}-\frac{19 a^6}{10 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^5}+\frac{a^6}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^6}-\frac{2 a^6}{7 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^7} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1612
Rubi steps
\begin{align*} \int \frac{e^{-\text{sech}^{-1}(a x)}}{x^7} \, dx &=\int \frac{1}{x^7 \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 )} \, dx\\ &=-\left ((4 a) \operatorname{Subst}\left (\int \frac{x \left (a+a x^2\right )^5}{(-1+x)^6 (1+x)^8} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\left ((4 a) \operatorname{Subst}\left (\int \left (\frac{a^5}{8 (-1+x)^6}+\frac{a^5}{4 (-1+x)^5}+\frac{5 a^5}{16 (-1+x)^4}+\frac{3 a^5}{16 (-1+x)^3}+\frac{5 a^5}{64 (-1+x)^2}-\frac{a^5}{2 (1+x)^8}+\frac{3 a^5}{2 (1+x)^7}-\frac{19 a^5}{8 (1+x)^6}+\frac{9 a^5}{4 (1+x)^5}-\frac{11 a^5}{8 (1+x)^4}+\frac{a^5}{2 (1+x)^3}-\frac{5 a^5}{64 (1+x)^2}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\frac{a^6}{10 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^5}+\frac{a^6}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^4}-\frac{5 a^6}{12 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^3}+\frac{3 a^6}{8 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}-\frac{5 a^6}{16 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{2 a^6}{7 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^7}+\frac{a^6}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^6}-\frac{19 a^6}{10 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^5}+\frac{9 a^6}{4 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}-\frac{11 a^6}{6 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}+\frac{a^6}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}-\frac{5 a^6}{16 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}\\ \end{align*}
Mathematica [A] time = 0.0969453, size = 76, normalized size = 0.22 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} \left (8 a^5 x^5-8 a^4 x^4+12 a^3 x^3-12 a^2 x^2+15 a x-15\right ) (a x+1)^2+15}{105 a x^7} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7}} \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) ^{-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{1}{x^{7}{\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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Fricas [A] time = 2.09432, size = 153, normalized size = 0.43 \begin{align*} -\frac{{\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}} + 15}{105 \, a 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*} a \int \frac{1}{a x^{7} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + x^{6}}\, 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{1}{x^{7}{\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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