Optimal. Leaf size=63 \[ -\frac {\sqrt {1+a^2 x^2}}{2 x^2}-\frac {i a \sqrt {1+a^2 x^2}}{x}+\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 = {5168, 849, 821,
272, 65, 214} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 5168
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 \text {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} \text {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*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.90 \begin {gather*} \frac {1}{2} \left (\frac {(-1-2 i a x) \sqrt {1+a^2 x^2}}{x^2}-a^2 \log (x)+a^2 \log \left (1+\sqrt {1+a^2 x^2}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, size = 53, normalized size = 0.84
method | result | size |
default | \(-\frac {\sqrt {a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}-\frac {i a \sqrt {a^{2} x^{2}+1}}{x}\) | \(53\) |
risch | \(-\frac {i \left (2 a^{3} x^{3}-i a^{2} x^{2}+2 a x -i\right )}{2 x^{2} \sqrt {a^{2} x^{2}+1}}+\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}\) | \(60\) |
meijerg | \(\frac {a^{2} \left (\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}-\frac {\sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{2 \sqrt {\pi }}-\frac {i a \sqrt {a^{2} x^{2}+1}}{x}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 48, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, a^{2} \operatorname {arsinh}\left (\frac {1}{a {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.37, size = 83, normalized size = 1.32 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.97, size = 48, normalized size = 0.76 \begin {gather*} - 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 {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. 153 vs. \(2 (51) = 102\).
time = 0.42, size = 153, normalized size = 2.43 \begin {gather*} \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 i \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} a {\left | a \right |} + {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )} a^{2} - 2 i \, a {\left | a \right |}}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 52, normalized size = 0.83 \begin {gather*} \frac {a^2\,\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )}{2}-\frac {\sqrt {a^2\,x^2+1}}{2\,x^2}-\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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