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 = 0.05137, 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 = 0.0407194, 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 [B] time = 0.079, size = 219, normalized size = 3.5 \begin{align*} -{\frac{1}{2\,{x}^{2}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{{a}^{2}}{2}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{ia}{x} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-i{a}^{3}x\sqrt{{a}^{2}{x}^{2}+1}-{i{a}^{3}\ln \left ({{a}^{2}x{\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{a}^{2}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }+{i{a}^{3}\ln \left ({ \left ( ia+{a}^{2} \left ( x-{\frac{i}{a}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) } \right ){\frac{1}{\sqrt{{a}^{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{\sqrt{a^{2} x^{2} + 1}}{{\left (i \, a x + 1\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81421, size = 196, normalized size = 3.11 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} x^{2} + 1}}{x^{3} \left (i a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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