Optimal. Leaf size=25 \[ -\tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-i \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.036575, antiderivative size = 25, 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, 844, 215, 266, 63, 208} \[ -\tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-i \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 5060
Rule 844
Rule 215
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-i \tan ^{-1}(a x)}}{x} \, dx &=\int \frac{1-i a x}{x \sqrt{1+a^2 x^2}} \, dx\\ &=-\left ((i a) \int \frac{1}{\sqrt{1+a^2 x^2}} \, dx\right )+\int \frac{1}{x \sqrt{1+a^2 x^2}} \, dx\\ &=-i \sinh ^{-1}(a x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=-i \sinh ^{-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}\\ &=-i \sinh ^{-1}(a x)-\tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0144069, size = 29, normalized size = 1.16 \[ -\log \left (\sqrt{a^2 x^2+1}+1\right )-i \sinh ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.078, size = 121, normalized size = 4.8 \begin{align*} -\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }-{ia\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}}}}}+\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78253, size = 143, normalized size = 5.72 \begin{align*} -\log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) + i \, \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\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{\sqrt{a^{2} x^{2} + 1}}{x \left (i a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13407, size = 92, normalized size = 3.68 \begin{align*} \frac{a i \log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1} + 1 \right |}\right ) + \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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