Integrand size = 16, antiderivative size = 203 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=-\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac {55}{8} i a^3 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {55}{8} i a^3 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\frac {55}{8} i a^3 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {55}{8} i a^3 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}} \]
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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^4 (1-i a x)^{5/4}} \, dx \\ & = -\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {1}{3} \int \frac {-\frac {13 i a}{2}+6 a^2 x}{x^3 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {1}{6} \int \frac {-\frac {61 a^2}{4}-13 i a^3 x}{x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {1}{6} \int \frac {\frac {165 i a^3}{8}-\frac {61 a^4 x}{4}}{x (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx \\ & = -\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {i \int \frac {165 a^4}{16 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{3 a} \\ & = -\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {1}{16} \left (55 i a^3\right ) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {1}{4} \left (55 i a^3\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 {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac {1}{8} \left (55 i a^3\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}{8} \left (55 i a^3\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 {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac {55}{8} i a^3 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {55}{8} i a^3 \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 = 0.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\frac {-8-34 i a x+87 a^2 x^2-226 i a^3 x^3+287 a^4 x^4+110 a^3 x^3 (i+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {i+a x}{i-a x}\right )}{24 x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \]
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\[\int \frac {{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}}}{x^{4}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\frac {165 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 165 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 165 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 165 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (287 i \, a^{3} x^{3} - 61 \, a^{2} x^{2} + 26 i \, a x + 8\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{48 \, x^{3}} \]
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Timed out. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\int { \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}}{x^4} \,d x \]
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