Integrand size = 16, antiderivative size = 233 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\frac {2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}-\frac {475}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5170, 100, 156, 160, 12, 95, 218, 212, 209} \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=-\frac {475}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}} \]
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 160
Rule 209
Rule 212
Rule 218
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a x)^{5/4}}{x^5 (1-i a x)^{5/4}} \, dx \\ & = -\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {1}{4} \int \frac {-\frac {17 i a}{2}+8 a^2 x}{x^4 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {1}{12} \int \frac {-\frac {113 a^2}{4}-\frac {51}{2} i a^3 x}{x^3 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}-\frac {1}{24} \int \frac {\frac {521 i a^3}{8}-\frac {113 a^4 x}{2}}{x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac {1}{24} \int \frac {\frac {1425 a^4}{16}+\frac {521}{8} i a^5 x}{x (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx \\ & = \frac {2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac {i \int -\frac {1425 i a^5}{32 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{12 a} \\ & = \frac {2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac {1}{128} \left (475 a^4\right ) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = \frac {2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac {1}{32} \left (475 a^4\right ) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = \frac {2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}-\frac {1}{64} \left (475 a^4\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {1}{64} \left (475 a^4\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = \frac {2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac {17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac {113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}-\frac {475}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\frac {-48-184 i a x+362 a^2 x^2+747 i a^3 x^3+1946 a^4 x^4+2467 i a^5 x^5+950 i a^4 x^4 (i+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {i+a x}{i-a x}\right )}{192 x^4 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \]
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\[\int \frac {{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}}}{x^{5}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=-\frac {1425 \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 1425 i \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 1425 i \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 1425 \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (2467 \, a^{4} x^{4} + 521 i \, a^{3} x^{3} + 226 \, a^{2} x^{2} - 136 i \, a x - 48\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{384 \, x^{4}} \]
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Timed out. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\int { \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x^{5}} \,d x } \]
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Exception generated. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}}{x^5} \,d x \]
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