Optimal. Leaf size=90 \[ -\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {2 a^2 \sqrt {1+a^2 x^2}}{3 x}+\frac {1}{2} i a^3 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {2 a^2 \sqrt {a^2 x^2+1}}{3 x}-\frac {i a \sqrt {a^2 x^2+1}}{2 x^2}-\frac {\sqrt {a^2 x^2+1}}{3 x^3}+\frac {1}{2} i a^3 \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^4} \, dx &=\int \frac {1+i a x}{x^4 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {1}{3} \int \frac {-3 i a+2 a^2 x}{x^3 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {1}{6} \int \frac {-4 a^2-3 i a^3 x}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {2 a^2 \sqrt {1+a^2 x^2}}{3 x}-\frac {1}{2} \left (i a^3\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {2 a^2 \sqrt {1+a^2 x^2}}{3 x}-\frac {1}{4} \left (i a^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {2 a^2 \sqrt {1+a^2 x^2}}{3 x}-\frac {1}{2} (i a) \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}}{3 x^3}-\frac {i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {2 a^2 \sqrt {1+a^2 x^2}}{3 x}+\frac {1}{2} i a^3 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 70, normalized size = 0.78 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {1+a^2 x^2} \left (-2-3 i a x+4 a^2 x^2\right )}{x^3}-3 i a^3 \log (x)+3 i a^3 \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 = 0.09, size = 75, normalized size = 0.83
method | result | size |
risch | \(\frac {4 a^{4} x^{4}-3 i a^{3} x^{3}+2 a^{2} x^{2}-3 i a x -2}{6 x^{3} \sqrt {a^{2} x^{2}+1}}+\frac {i a^{3} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}\) | \(68\) |
default | \(-\frac {\sqrt {a^{2} x^{2}+1}}{3 x^{3}}+\frac {2 a^{2} \sqrt {a^{2} x^{2}+1}}{3 x}+i a \left (-\frac {\sqrt {a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}\right )\) | \(75\) |
meijerg | \(-\frac {\left (-2 a^{2} x^{2}+1\right ) \sqrt {a^{2} x^{2}+1}}{3 x^{3}}+\frac {i a^{3} \left (\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}-\frac {\sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{2 \sqrt {\pi }}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 67, normalized size = 0.74 \begin {gather*} \frac {1}{2} i \, a^{3} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) + \frac {2 \, \sqrt {a^{2} x^{2} + 1} a^{2}}{3 \, x} - \frac {i \, \sqrt {a^{2} x^{2} + 1} a}{2 \, x^{2}} - \frac {\sqrt {a^{2} x^{2} + 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.20, size = 92, normalized size = 1.02 \begin {gather*} \frac {3 i \, a^{3} x^{3} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 3 i \, a^{3} x^{3} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 4 \, a^{3} x^{3} + {\left (4 \, a^{2} x^{2} - 3 i \, a x - 2\right )} \sqrt {a^{2} x^{2} + 1}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.10, size = 75, normalized size = 0.83 \begin {gather*} \frac {2 a^{3} \sqrt {1 + \frac {1}{a^{2} x^{2}}}}{3} + \frac {i a^{3} \operatorname {asinh}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a^{2} \sqrt {1 + \frac {1}{a^{2} x^{2}}}}{2 x} - \frac {a \sqrt {1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} \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. 161 vs. \(2 (70) = 140\).
time = 0.41, size = 161, normalized size = 1.79 \begin {gather*} \frac {1}{2} i \, a^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) - \frac {1}{2} i \, a^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) - \frac {-3 i \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{5} a^{3} - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} a^{2} {\left | a \right |} + 3 \, {\left (i \, x {\left | a \right |} - i \, \sqrt {a^{2} x^{2} + 1}\right )} a^{3} + 4 \, a^{2} {\left | a \right |}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 74, normalized size = 0.82 \begin {gather*} \frac {a^3\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )}{2}-\frac {\sqrt {a^2\,x^2+1}}{3\,x^3}+\frac {2\,a^2\,\sqrt {a^2\,x^2+1}}{3\,x}-\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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