Optimal. Leaf size=48 \[ \frac{4 \sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\sin ^{-1}(a x) \]
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Rubi [A] time = 0.786747, antiderivative size = 48, 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} \[ \frac{4 \sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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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\\ &=-\left (a \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\right )-(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 = 0.0402435, size = 51, normalized size = 1.06 \[ -\frac{4 \sqrt{1-a^2 x^2}}{a x-1}-\log \left (\sqrt{1-a^2 x^2}+1\right )-\sin ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.037, size = 75, normalized size = 1.6 \begin{align*} 4\,{\frac{ax}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{a\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+4\,{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43973, size = 105, normalized size = 2.19 \begin{align*} \frac{4 \, a x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{a \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{4}{\sqrt{-a^{2} x^{2} + 1}} - \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85225, size = 193, normalized size = 4.02 \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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a x + 1\right )^{3}}{x \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.
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Giac [B] time = 1.1919, size = 117, normalized size = 2.44 \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.
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