Optimal. Leaf size=194 \[ \frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{a x+1}}}+\frac{5}{128} a^7 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{a x+1}}}+\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7} \]
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Rubi [A] time = 0.0984763, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{a x+1}}}+\frac{5}{128} a^7 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{a x+1}}}+\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7} \]
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^8} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}-\frac{\int \frac{1}{x^9} \, dx}{7 a}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^9 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{7 a}\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{7 a^2}{x^7 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{56 a}\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}-\frac{1}{8} \left (a \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\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{1}{48} \left (a \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\\ &=\frac{1}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}-\frac{1}{48} \left (5 a^3 \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}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{1}{192} \left (5 a^3 \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}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}-\frac{1}{64} \left (5 a^5 \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}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{128} \left (5 a^5 \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}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{128} \left (5 a^7 \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}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{128} \left (5 a^8 \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}{56 a x^8}-\frac{e^{\text{sech}^{-1}(a x)}}{7 x^7}+\frac{\sqrt{1-a x}}{56 a x^8 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{48 x^6 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^3 \sqrt{1-a x}}{192 x^4 \sqrt{\frac{1}{1+a x}}}+\frac{5 a^5 \sqrt{1-a x}}{128 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{5}{128} a^7 \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 = 0.132619, size = 145, normalized size = 0.75 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} \left (15 a^7 x^7+15 a^6 x^6+10 a^5 x^5+10 a^4 x^4+8 a^3 x^3+8 a^2 x^2-48 a x-48\right )-15 a^8 x^8 \log (x)+15 a^8 x^8 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-48}{384 a x^8} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.203, size = 152, normalized size = 0.8 \begin{align*}{\frac{1}{384\,{x}^{7}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( 15\,{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){x}^{8}{a}^{8}+15\,\sqrt{-{a}^{2}{x}^{2}+1}{x}^{6}{a}^{6}+10\,\sqrt{-{a}^{2}{x}^{2}+1}{x}^{4}{a}^{4}+8\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-48\,\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{8\,a{x}^{8}}} \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{5}{128} \, a^{8} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{5}{128} \, \sqrt{-a^{2} x^{2} + 1} a^{8} - \frac{5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{6}}{128 \, x^{2}} - \frac{5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{4}}{64 \, x^{4}} - \frac{5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}}{48 \, x^{6}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{8 \, x^{8}}}{a} - \frac{1}{8 \, a x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13341, size = 348, normalized size = 1.79 \begin{align*} \frac{15 \, a^{8} x^{8} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - 15 \, a^{8} x^{8} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) + 2 \,{\left (15 \, a^{7} x^{7} + 10 \, a^{5} x^{5} + 8 \, a^{3} x^{3} - 48 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 96}{768 \, a x^{8}} \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^{9}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{8}}\, 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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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