Optimal. Leaf size=115 \[ \frac{2 a^3 \sqrt{1-a x}}{15 x \sqrt{\frac{1}{a x+1}}}+\frac{a \sqrt{1-a x}}{15 x^3 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{20 a x^5 \sqrt{\frac{1}{a x+1}}}+\frac{1}{20 a x^5}-\frac{e^{\text{sech}^{-1}(a x)}}{4 x^4} \]
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Rubi [A] time = 0.051863, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6335, 30, 103, 12, 95} \[ \frac{2 a^3 \sqrt{1-a x}}{15 x \sqrt{\frac{1}{a x+1}}}+\frac{a \sqrt{1-a x}}{15 x^3 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{20 a x^5 \sqrt{\frac{1}{a x+1}}}+\frac{1}{20 a x^5}-\frac{e^{\text{sech}^{-1}(a x)}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 103
Rule 12
Rule 95
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(a x)}}{x^5} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{4 x^4}-\frac{\int \frac{1}{x^6} \, dx}{4 a}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^6 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{4 a}\\ &=\frac{1}{20 a x^5}-\frac{e^{\text{sech}^{-1}(a x)}}{4 x^4}+\frac{\sqrt{1-a x}}{20 a x^5 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{4 a^2}{x^4 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{20 a}\\ &=\frac{1}{20 a x^5}-\frac{e^{\text{sech}^{-1}(a x)}}{4 x^4}+\frac{\sqrt{1-a x}}{20 a x^5 \sqrt{\frac{1}{1+a x}}}-\frac{1}{5} \left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^4 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{20 a x^5}-\frac{e^{\text{sech}^{-1}(a x)}}{4 x^4}+\frac{\sqrt{1-a x}}{20 a x^5 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{15 x^3 \sqrt{\frac{1}{1+a x}}}+\frac{1}{15} \left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{2 a^2}{x^2 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{20 a x^5}-\frac{e^{\text{sech}^{-1}(a x)}}{4 x^4}+\frac{\sqrt{1-a x}}{20 a x^5 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{15 x^3 \sqrt{\frac{1}{1+a x}}}-\frac{1}{15} \left (2 a^3 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^2 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{20 a x^5}-\frac{e^{\text{sech}^{-1}(a x)}}{4 x^4}+\frac{\sqrt{1-a x}}{20 a x^5 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{15 x^3 \sqrt{\frac{1}{1+a x}}}+\frac{2 a^3 \sqrt{1-a x}}{15 x \sqrt{\frac{1}{1+a x}}}\\ \end{align*}
Mathematica [A] time = 0.0669814, size = 60, normalized size = 0.52 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^2 \left (2 a^3 x^3-2 a^2 x^2+3 a x-3\right )-3}{15 a x^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.179, size = 63, normalized size = 0.6 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}-1 \right ) \left ( 2\,{a}^{2}{x}^{2}+3 \right ) }{15\,{x}^{4}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{1}{5\,a{x}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03999, size = 69, normalized size = 0.6 \begin{align*} \frac{{\left (2 \, a^{4} x^{5} + a^{2} x^{3} - 3 \, x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{15 \, a x^{6}} - \frac{1}{5 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92158, size = 128, normalized size = 1.11 \begin{align*} \frac{{\left (2 \, a^{5} x^{5} + a^{3} x^{3} - 3 \, a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 3}{15 \, a x^{5}} \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^{6}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{5}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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