Optimal. Leaf size=63 \[ -\frac{i a \sqrt{a^2 x^2+1}}{x}-\frac{\sqrt{a^2 x^2+1}}{2 x^2}+\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.0502899, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5060, 835, 807, 266, 63, 208} \[ -\frac{i a \sqrt{a^2 x^2+1}}{x}-\frac{\sqrt{a^2 x^2+1}}{2 x^2}+\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 5060
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{i \tan ^{-1}(a x)}}{x^3} \, dx &=\int \frac{1+i a x}{x^3 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{2 x^2}-\frac{1}{2} \int \frac{-2 i a+a^2 x}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{2 x^2}-\frac{i a \sqrt{1+a^2 x^2}}{x}-\frac{1}{2} a^2 \int \frac{1}{x \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{2 x^2}-\frac{i a \sqrt{1+a^2 x^2}}{x}-\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{i a \sqrt{1+a^2 x^2}}{x}-\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{i a \sqrt{1+a^2 x^2}}{x}+\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0396952, size = 57, normalized size = 0.9 \[ \frac{1}{2} \left (\frac{(-1-2 i a x) \sqrt{a^2 x^2+1}}{x^2}+a^2 \log \left (\sqrt{a^2 x^2+1}+1\right )+a^2 (-\log (x))\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.071, size = 53, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{ia}{x}\sqrt{{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976611, size = 68, normalized size = 1.08 \begin{align*} \frac{1}{2} \, a^{2} \operatorname{arsinh}\left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) - \frac{i \, \sqrt{a^{2} x^{2} + 1} a}{x} - \frac{\sqrt{a^{2} x^{2} + 1}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68001, size = 197, normalized size = 3.13 \begin{align*} \frac{a^{2} x^{2} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - a^{2} x^{2} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) - 2 i \, a^{2} x^{2} + \sqrt{a^{2} x^{2} + 1}{\left (-2 i \, a x - 1\right )}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.53996, size = 48, normalized size = 0.76 \begin{align*} - i a^{2} \sqrt{1 + \frac{1}{a^{2} x^{2}}} + \frac{a^{2} \operatorname{asinh}{\left (\frac{1}{a x} \right )}}{2} - \frac{a \sqrt{1 + \frac{1}{a^{2} x^{2}}}}{2 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13557, size = 209, normalized size = 3.32 \begin{align*} \frac{1}{2} \, a^{2} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1} + 1 \right |}\right ) - \frac{1}{2} \, a^{2} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1} - 1 \right |}\right ) + \frac{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{3} a^{2} + 2 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} a i{\left | a \right |} +{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )} a^{2} - 2 \, a i{\left | a \right |}}{{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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