Optimal. Leaf size=132 \[ \frac{1}{8} a^3 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{a x+1}}}+\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.058869, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{1}{8} a^3 \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{a x+1}}}+\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
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^4} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}-\frac{\int \frac{1}{x^5} \, dx}{3 a}-\frac{\left (\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}{3 a}\\ &=\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\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}{12 a}\\ &=\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}-\frac{1}{4} \left (a \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}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{8} \left (a \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}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{8 x^2 \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 \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{8} \left (a^4 \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}{12 a x^4}-\frac{e^{\text{sech}^{-1}(a x)}}{3 x^3}+\frac{\sqrt{1-a x}}{12 a x^4 \sqrt{\frac{1}{1+a x}}}+\frac{a \sqrt{1-a x}}{8 x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{8} a^3 \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 = 0.0733636, size = 110, normalized size = 0.83 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} \left (a^3 x^3+a^2 x^2-2 a x-2\right )-a^4 x^4 \log (x)+a^4 x^4 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-2}{8 a x^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.184, size = 110, normalized size = 0.8 \begin{align*}{\frac{1}{8\,{x}^{3}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ({\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){x}^{4}{a}^{4}+{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-2\,\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{4\,{x}^{4}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{1}{8} \, a^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1} a^{4} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}}{8 \, x^{2}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{4 \, x^{4}}}{a} - \frac{1}{4 \, a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.80857, size = 298, normalized size = 2.26 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{5}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]