Optimal. Leaf size=200 \[ \frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}+\frac{a^3}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}-\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{a^3}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}+\frac{a^3}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}-\frac{a^3}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{1}{4} a^3 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]
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Rubi [A] time = 0.500469, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6337, 1612, 207} \[ \frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}+\frac{a^3}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}-\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{a^3}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}+\frac{a^3}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}-\frac{a^3}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}-\frac{1}{4} a^3 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1612
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-\text{sech}^{-1}(a x)}}{x^4} \, dx &=\int \frac{1}{x^4 \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\\ &=(4 a) \operatorname{Subst}\left (\int \frac{x \left (a+a x^2\right )^2}{(-1+x)^3 (1+x)^5} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=(4 a) \operatorname{Subst}\left (\int \left (\frac{a^2}{8 (-1+x)^3}+\frac{a^2}{16 (-1+x)^2}+\frac{a^2}{2 (1+x)^5}-\frac{3 a^2}{4 (1+x)^4}+\frac{a^2}{2 (1+x)^3}-\frac{a^2}{8 (1+x)^2}+\frac{a^2}{16 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{a^3}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}+\frac{a^3}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}-\frac{a^3}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^3}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}+\frac{1}{4} a^3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^3}{4 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{a^3}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}+\frac{a^3}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}-\frac{a^3}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a^3}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{1}{4} a^3 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0996863, size = 110, normalized size = 0.55 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} \left (a^3 x^3+a^2 x^2-2 a x-2\right )-a^4 x^4 \log (x)+a^4 x^4 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )+2}{8 a x^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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^{4}{\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 = 2.07651, size = 300, normalized size = 1.5 \begin{align*} -\frac{a^{4} x^{4} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) + 2 \,{\left (a^{3} x^{3} - 2 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 4}{16 \, a x^{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*} a \int \frac{1}{a x^{4} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + x^{3}}\, 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^{4}{\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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