Optimal. Leaf size=146 \[ \frac{4 a^3 \sqrt{1-a x}}{105 x^3 \sqrt{\frac{1}{a x+1}}}+\frac{8 a^5 \sqrt{1-a x}}{105 x \sqrt{\frac{1}{a x+1}}}+\frac{a \sqrt{1-a x}}{35 x^5 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{a x+1}}}+\frac{1}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0711514, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{4 a^3 \sqrt{1-a x}}{105 x^3 \sqrt{\frac{1}{a x+1}}}+\frac{8 a^5 \sqrt{1-a x}}{105 x \sqrt{\frac{1}{a x+1}}}+\frac{a \sqrt{1-a x}}{35 x^5 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{a x+1}}}+\frac{1}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6335
Rule 30
Rule 103
Rule 12
Rule 95
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(a x)}}{x^7} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6}-\frac{\int \frac{1}{x^8} \, dx}{6 a}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^8 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{6 a}\\ &=\frac{1}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{6 a^2}{x^6 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{42 a}\\ &=\frac{1}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{1+a x}}}-\frac{1}{7} \left (a \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\\ &=\frac{1}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{35 x^5 \sqrt{\frac{1}{1+a x}}}+\frac{1}{35} \left (a \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\\ &=\frac{1}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{35 x^5 \sqrt{\frac{1}{1+a x}}}-\frac{1}{35} \left (4 a^3 \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}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{35 x^5 \sqrt{\frac{1}{1+a x}}}+\frac{4 a^3 \sqrt{1-a x}}{105 x^3 \sqrt{\frac{1}{1+a x}}}+\frac{1}{105} \left (4 a^3 \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}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{35 x^5 \sqrt{\frac{1}{1+a x}}}+\frac{4 a^3 \sqrt{1-a x}}{105 x^3 \sqrt{\frac{1}{1+a x}}}-\frac{1}{105} \left (8 a^5 \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}{42 a x^7}-\frac{e^{\text{sech}^{-1}(a x)}}{6 x^6}+\frac{\sqrt{1-a x}}{42 a x^7 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{35 x^5 \sqrt{\frac{1}{1+a x}}}+\frac{4 a^3 \sqrt{1-a x}}{105 x^3 \sqrt{\frac{1}{1+a x}}}+\frac{8 a^5 \sqrt{1-a x}}{105 x \sqrt{\frac{1}{1+a x}}}\\ \end{align*}
Mathematica [A] time = 0.085222, size = 76, normalized size = 0.52 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^2 \left (8 a^5 x^5-8 a^4 x^4+12 a^3 x^3-12 a^2 x^2+15 a x-15\right )-15}{105 a x^7} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.191, size = 71, normalized size = 0.5 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}-1 \right ) \left ( 8\,{x}^{4}{a}^{4}+12\,{a}^{2}{x}^{2}+15 \right ) }{105\,{x}^{6}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{1}{7\,a{x}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03706, size = 81, normalized size = 0.55 \begin{align*} \frac{{\left (8 \, a^{6} x^{7} + 4 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - 15 \, x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{105 \, a x^{8}} - \frac{1}{7 \, a x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.12879, size = 151, normalized size = 1.03 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{8}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{7}}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]