Optimal. Leaf size=91 \[ \frac{4 a^2 \sqrt{1-a^2 x^2}}{1-a x}-\frac{3 a \sqrt{1-a^2 x^2}}{x}-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{9}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.741374, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6124, 6742, 266, 51, 63, 208, 264, 651} \[ \frac{4 a^2 \sqrt{1-a^2 x^2}}{1-a x}-\frac{3 a \sqrt{1-a^2 x^2}}{x}-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{9}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6124
Rule 6742
Rule 266
Rule 51
Rule 63
Rule 208
Rule 264
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac{(1+a x)^2}{x^3 (1-a x) \sqrt{1-a^2 x^2}} \, dx\\ &=\int \left (\frac{1}{x^3 \sqrt{1-a^2 x^2}}+\frac{3 a}{x^2 \sqrt{1-a^2 x^2}}+\frac{4 a^2}{x \sqrt{1-a^2 x^2}}-\frac{4 a^3}{(-1+a x) \sqrt{1-a^2 x^2}}\right ) \, dx\\ &=(3 a) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx+\left (4 a^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx-\left (4 a^3\right ) \int \frac{1}{(-1+a x) \sqrt{1-a^2 x^2}} \, dx+\int \frac{1}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 a \sqrt{1-a^2 x^2}}{x}+\frac{4 a^2 \sqrt{1-a^2 x^2}}{1-a x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-a^2 x}} \, dx,x,x^2\right )+\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{3 a \sqrt{1-a^2 x^2}}{x}+\frac{4 a^2 \sqrt{1-a^2 x^2}}{1-a x}-4 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )+\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{3 a \sqrt{1-a^2 x^2}}{x}+\frac{4 a^2 \sqrt{1-a^2 x^2}}{1-a x}-4 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{3 a \sqrt{1-a^2 x^2}}{x}+\frac{4 a^2 \sqrt{1-a^2 x^2}}{1-a x}-\frac{9}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0703125, size = 75, normalized size = 0.82 \[ \sqrt{1-a^2 x^2} \left (-\frac{4 a^2}{a x-1}-\frac{3 a}{x}-\frac{1}{2 x^2}\right )-\frac{9}{2} a^2 \log \left (\sqrt{1-a^2 x^2}+1\right )+\frac{9}{2} a^2 \log (x) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 108, normalized size = 1.2 \begin{align*}{x{a}^{3}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+3\,a \left ( -{\frac{1}{x\sqrt{-{a}^{2}{x}^{2}+1}}}+2\,{\frac{{a}^{2}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{9\,{a}^{2}}{2} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) }-{\frac{1}{2\,{x}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.952971, size = 138, normalized size = 1.52 \begin{align*} \frac{7 \, a^{3} x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{9}{2} \, a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{9 \, a^{2}}{2 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{3 \, a}{\sqrt{-a^{2} x^{2} + 1} x} - \frac{1}{2 \, \sqrt{-a^{2} x^{2} + 1} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.01856, size = 196, normalized size = 2.15 \begin{align*} \frac{8 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 9 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (14 \, a^{2} x^{2} - 5 \, a x - 1\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \,{\left (a x^{3} - x^{2}\right )}} \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 (a x + 1\right )^{3}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.21473, size = 288, normalized size = 3.16 \begin{align*} -\frac{{\left (a^{3} + \frac{11 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a}{x} - \frac{76 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{9 \, a^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} - \frac{\frac{12 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a{\left | a \right |}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]