Optimal. Leaf size=38 \[ -\frac {\sqrt {1+a^2 x^2}}{x}-i a \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5168, 821, 272,
65, 214} \begin {gather*} -\frac {\sqrt {a^2 x^2+1}}{x}-i a \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 5168
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac {1+i a x}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{x}+(i a) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{x}+\frac {1}{2} (i a) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{x}+\frac {i \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{a}\\ &=-\frac {\sqrt {1+a^2 x^2}}{x}-i a \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 1.24 \begin {gather*} -\frac {\sqrt {1+a^2 x^2}}{x}+i a \log (x)-i a \log \left (1+\sqrt {1+a^2 x^2}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, size = 34, normalized size = 0.89
method | result | size |
default | \(-i a \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {\sqrt {a^{2} x^{2}+1}}{x}\) | \(34\) |
risch | \(-i a \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {\sqrt {a^{2} x^{2}+1}}{x}\) | \(34\) |
meijerg | \(-\frac {\sqrt {a^{2} x^{2}+1}}{x}+\frac {i a \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }\right )}{2 \sqrt {\pi }}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 29, normalized size = 0.76 \begin {gather*} -i \, a \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 66 vs. \(2 (32) = 64\).
time = 1.20, size = 66, normalized size = 1.74 \begin {gather*} \frac {-i \, a x \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + i \, a x \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - a x - \sqrt {a^{2} x^{2} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.46, size = 26, normalized size = 0.68 \begin {gather*} - a \sqrt {1 + \frac {1}{a^{2} x^{2}}} - i a \operatorname {asinh}{\left (\frac {1}{a x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 75 vs. \(2 (32) = 64\).
time = 0.42, size = 75, normalized size = 1.97 \begin {gather*} -i \, a \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) + i \, a \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) + \frac {2 \, {\left | a \right |}}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 33, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {a^2\,x^2+1}}{x}-a\,\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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