Integrand size = 14, antiderivative size = 113 \[ \int \frac {e^{i \arctan (a x)}}{x^5} \, dx=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}-\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}+\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}-\frac {3}{8} a^4 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5168, 849, 821, 272, 65, 214} \[ \int \frac {e^{i \arctan (a x)}}{x^5} \, dx=\frac {3 a^2 \sqrt {a^2 x^2+1}}{8 x^2}-\frac {\sqrt {a^2 x^2+1}}{4 x^4}-\frac {i a \sqrt {a^2 x^2+1}}{3 x^3}-\frac {3}{8} a^4 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {2 i a^3 \sqrt {a^2 x^2+1}}{3 x} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+i a x}{x^5 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{4 x^4}-\frac {1}{4} \int \frac {-4 i a+3 a^2 x}{x^4 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{4 x^4}-\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {1}{12} \int \frac {-9 a^2-8 i a^3 x}{x^3 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{4 x^4}-\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {1}{24} \int \frac {16 i a^3-9 a^4 x}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{4 x^4}-\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}+\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}+\frac {1}{8} \left (3 a^4\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{4 x^4}-\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}+\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}+\frac {1}{16} \left (3 a^4\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}}{4 x^4}-\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}+\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}+\frac {1}{8} \left (3 a^2\right ) \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}}{4 x^4}-\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}+\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}-\frac {3}{8} a^4 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {e^{i \arctan (a x)}}{x^5} \, dx=\frac {1}{24} \left (\frac {\sqrt {1+a^2 x^2} \left (-6-8 i a x+9 a^2 x^2+16 i a^3 x^3\right )}{x^4}+9 a^4 \log (x)-9 a^4 \log \left (1+\sqrt {1+a^2 x^2}\right )\right ) \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {i \left (16 a^{5} x^{5}-9 i a^{4} x^{4}+8 a^{3} x^{3}-3 i a^{2} x^{2}-8 a x +6 i\right )}{24 x^{4} \sqrt {a^{2} x^{2}+1}}-\frac {3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{8}\) | \(77\) |
default | \(-\frac {\sqrt {a^{2} x^{2}+1}}{4 x^{4}}-\frac {3 a^{2} \left (-\frac {\sqrt {a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}\right )}{4}+i a \left (-\frac {\sqrt {a^{2} x^{2}+1}}{3 x^{3}}+\frac {2 a^{2} \sqrt {a^{2} x^{2}+1}}{3 x}\right )\) | \(97\) |
meijerg | \(\frac {a^{4} \left (\frac {\sqrt {\pi }\, \left (-7 a^{4} x^{4}-8 a^{2} x^{2}+8\right )}{16 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (-12 a^{2} x^{2}+8\right ) \sqrt {a^{2} x^{2}+1}}{16 a^{4} x^{4}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{4} a^{4}}+\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{2 \sqrt {\pi }}-\frac {i a \left (-2 a^{2} x^{2}+1\right ) \sqrt {a^{2} x^{2}+1}}{3 x^{3}}\) | \(162\) |
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Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \frac {e^{i \arctan (a x)}}{x^5} \, dx=-\frac {9 \, a^{4} x^{4} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 9 \, a^{4} x^{4} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - 16 i \, a^{4} x^{4} - {\left (16 i \, a^{3} x^{3} + 9 \, a^{2} x^{2} - 8 i \, a x - 6\right )} \sqrt {a^{2} x^{2} + 1}}{24 \, x^{4}} \]
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Time = 2.96 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.08 \[ \int \frac {e^{i \arctan (a x)}}{x^5} \, dx=\frac {2 i a^{4} \sqrt {1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {3 a^{4} \operatorname {asinh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} - \frac {i a^{2} \sqrt {1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} + \frac {a}{8 x^{3} \sqrt {1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \]
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Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.76 \[ \int \frac {e^{i \arctan (a x)}}{x^5} \, dx=-\frac {3}{8} \, a^{4} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) + \frac {2 i \, \sqrt {a^{2} x^{2} + 1} a^{3}}{3 \, x} + \frac {3 \, \sqrt {a^{2} x^{2} + 1} a^{2}}{8 \, x^{2}} - \frac {i \, \sqrt {a^{2} x^{2} + 1} a}{3 \, x^{3}} - \frac {\sqrt {a^{2} x^{2} + 1}}{4 \, x^{4}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (89) = 178\).
Time = 0.30 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.10 \[ \int \frac {e^{i \arctan (a x)}}{x^5} \, dx=-\frac {3}{8} \, a^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) + \frac {3}{8} \, a^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) - \frac {9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{7} a^{4} - 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{5} a^{4} - 48 i \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{4} a^{3} {\left | a \right |} - 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{3} a^{4} + 64 i \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} a^{3} {\left | a \right |} + 9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )} a^{4} - 16 i \, a^{3} {\left | a \right |}}{12 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{4}} \]
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Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.84 \[ \int \frac {e^{i \arctan (a x)}}{x^5} \, dx=\frac {a^4\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8}-\frac {\sqrt {a^2\,x^2+1}}{4\,x^4}-\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{3\,x^3}+\frac {3\,a^2\,\sqrt {a^2\,x^2+1}}{8\,x^2}+\frac {a^3\,\sqrt {a^2\,x^2+1}\,2{}\mathrm {i}}{3\,x} \]
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