Optimal. Leaf size=163 \[ \frac{a^3 \sqrt{1-a x}}{16 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{1}{16} a^5 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{24 x^4 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{a x+1}}}+\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5} \]
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Rubi [A] time = 0.0769087, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6335, 30, 103, 12, 92, 208} \[ \frac{a^3 \sqrt{1-a x}}{16 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{1}{16} a^5 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{24 x^4 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{a x+1}}}+\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 103
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(a x)}}{x^6} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}-\frac{\int \frac{1}{x^7} \, dx}{5 a}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^7 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{5 a}\\ &=\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{5 a^2}{x^5 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{30 a}\\ &=\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{1+a x}}}-\frac{1}{6} \left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^5 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{24 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{1}{24} \left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{3 a^2}{x^3 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{24 x^4 \sqrt{\frac{1}{1+a x}}}-\frac{1}{8} \left (a^3 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^3 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{24 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a^3 \sqrt{1-a x}}{16 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{16} \left (a^3 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{a^2}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{24 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a^3 \sqrt{1-a x}}{16 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{16} \left (a^5 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{24 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a^3 \sqrt{1-a x}}{16 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{16} \left (a^6 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )\\ &=\frac{1}{30 a x^6}-\frac{e^{\text{sech}^{-1}(a x)}}{5 x^5}+\frac{\sqrt{1-a x}}{30 a x^6 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{24 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a^3 \sqrt{1-a x}}{16 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{16} a^5 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )\\ \end{align*}
Mathematica [A] time = 0.104171, size = 129, normalized size = 0.79 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} \left (3 a^5 x^5+3 a^4 x^4+2 a^3 x^3+2 a^2 x^2-8 a x-8\right )-3 a^6 x^6 \log (x)+3 a^6 x^6 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-8}{48 a x^6} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.184, size = 132, normalized size = 0.8 \begin{align*}{\frac{1}{48\,{x}^{5}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){x}^{6}{a}^{6}+3\,\sqrt{-{a}^{2}{x}^{2}+1}{x}^{4}{a}^{4}+2\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-8\,\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{6\,{x}^{6}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{\frac{1}{16} \, a^{6} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{1}{16} \, \sqrt{-a^{2} x^{2} + 1} a^{6} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{4}}{16 \, x^{2}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}}{8 \, x^{4}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}}}{a} - \frac{1}{6 \, a x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19449, size = 324, normalized size = 1.99 \begin{align*} \frac{3 \, a^{6} x^{6} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) + 2 \,{\left (3 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 16}{96 \, a x^{6}} \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^{7}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{6}}\, 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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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