Optimal. Leaf size=90 \[ \frac{4 a^2 \sqrt{1-a^2 x^2}}{a x+1}+\frac{3 a \sqrt{1-a^2 x^2}}{x}-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{9}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.735463, antiderivative size = 90, 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} \[ \frac{4 a^2 \sqrt{1-a^2 x^2}}{a x+1}+\frac{3 a \sqrt{1-a^2 x^2}}{x}-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{9}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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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\\ &=-\left ((3 a) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx\right )+\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 = 0.0609339, size = 75, normalized size = 0.83 \[ \sqrt{1-a^2 x^2} \left (\frac{4 a^2}{a x+1}+\frac{3 a}{x}-\frac{1}{2 x^2}\right )-\frac{9}{2} a^2 \log \left (\sqrt{1-a^2 x^2}+1\right )+\frac{9}{2} a^2 \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.066, size = 319, normalized size = 3.5 \begin{align*} 3\,{\frac{a \left ( -{a}^{2}{x}^{2}+1 \right ) ^{5/2}}{x}}+3\,{a}^{3}x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3/2}+{\frac{9\,x{a}^{3}}{2}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{9\,{a}^{3}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{ \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}}}}-3\,{a}^{2} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{3/2}-{\frac{9\,x{a}^{3}}{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{9\,{a}^{3}}{2}\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{3\,{a}^{2}}{2} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{2}}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{9\,{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{1}{2\,{x}^{2}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{1}{a \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.
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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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95905, size = 196, normalized size = 2.18 \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.
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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^{3} \left (a x + 1\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21042, size = 289, normalized size = 3.21 \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.
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