Optimal. Leaf size=45 \[ \frac{4 \sqrt{1-a^2 x^2}}{a x+1}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+\sin ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.708101, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6124, 6742, 216, 266, 63, 208, 651} \[ \frac{4 \sqrt{1-a^2 x^2}}{a x+1}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+\sin ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6124
Rule 6742
Rule 216
Rule 266
Rule 63
Rule 208
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{x} \, dx &=\int \frac{(1-a x)^2}{x (1+a x) \sqrt{1-a^2 x^2}} \, dx\\ &=\int \left (\frac{a}{\sqrt{1-a^2 x^2}}+\frac{1}{x \sqrt{1-a^2 x^2}}-\frac{4 a}{(1+a x) \sqrt{1-a^2 x^2}}\right ) \, dx\\ &=a \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-(4 a) \int \frac{1}{(1+a x) \sqrt{1-a^2 x^2}} \, dx+\int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{4 \sqrt{1-a^2 x^2}}{1+a x}+\sin ^{-1}(a x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{4 \sqrt{1-a^2 x^2}}{1+a x}+\sin ^{-1}(a x)-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a^2}\\ &=\frac{4 \sqrt{1-a^2 x^2}}{1+a x}+\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0369465, size = 49, normalized size = 1.09 \[ \frac{4 \sqrt{1-a^2 x^2}}{a x+1}-\log \left (\sqrt{1-a^2 x^2}+1\right )+\sin ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.049, size = 200, normalized size = 4.4 \begin{align*}{\frac{1}{{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{2}{3} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+a\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }x+{a\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{3} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+\sqrt{-{a}^{2}{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{1}{{a}^{3} \left ( x+{a}^{-1} \right ) ^{3}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}} \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*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.06162, size = 193, normalized size = 4.29 \begin{align*} \frac{4 \, a x - 2 \,{\left (a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (a x + 1\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + 4 \, \sqrt{-a^{2} x^{2} + 1} + 4}{a x + 1} \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*} \int \frac{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{x \left (a x + 1\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.16404, size = 116, normalized size = 2.58 \begin{align*} \frac{a \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{a \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} - \frac{8 \, a}{{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]