Integrand size = 16, antiderivative size = 202 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^5} \, dx=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}-\frac {11}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {11}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5170, 101, 156, 12, 95, 218, 212, 209} \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^5} \, dx=-\frac {11}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {11}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3} \]
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 209
Rule 212
Rule 218
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [4]{1+i a x}}{x^5 \sqrt [4]{1-i a x}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}+\frac {1}{4} \int \frac {\frac {7 i a}{2}-3 a^2 x}{x^4 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}-\frac {1}{12} \int \frac {\frac {29 a^2}{4}+7 i a^3 x}{x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {1}{24} \int \frac {-\frac {83 i a^3}{8}+\frac {29 a^4 x}{4}}{x^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}-\frac {1}{24} \int -\frac {33 a^4}{16 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}+\frac {1}{128} \left (11 a^4\right ) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}+\frac {1}{32} \left (11 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 {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}-\frac {1}{64} \left (11 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 (11 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-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}-\frac {11}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {11}{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 = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.49 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^5} \, dx=-\frac {(1-i a x)^{3/4} \left (48+104 i a x-114 a^2 x^2-141 i a^3 x^3+83 a^4 x^4+22 a^4 x^4 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {i+a x}{i-a x}\right )\right )}{192 x^4 (1+i a x)^{3/4}} \]
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\[\int \frac {\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}{x^{5}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^5} \, dx=-\frac {33 \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 33 i \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 33 i \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 33 \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (83 \, a^{4} x^{4} + 25 i \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 8 i \, a x - 48\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{384 \, x^{4}} \]
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\[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^5} \, dx=\int \frac {\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{5}}\, dx \]
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\[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^5} \, dx=\int { \frac {\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}{x^{5}} \,d x } \]
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Exception generated. \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^5} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^5} \, dx=\int \frac {\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}}{x^5} \,d x \]
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