Integrand size = 16, antiderivative size = 132 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^3} \, dx=-\frac {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac {1}{4} a^2 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{4} a^2 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5170, 98, 96, 95, 218, 212, 209} \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^3} \, dx=\frac {1}{4} a^2 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{4} a^2 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x} \]
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Rule 95
Rule 96
Rule 98
Rule 209
Rule 212
Rule 218
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [4]{1+i a x}}{x^3 \sqrt [4]{1-i a x}} \, dx \\ & = -\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac {1}{4} (i a) \int \frac {\sqrt [4]{1+i a x}}{x^2 \sqrt [4]{1-i a x}} \, dx \\ & = -\frac {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac {1}{8} a^2 \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac {1}{2} a^2 \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 {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac {1}{4} a^2 \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}{4} a^2 \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 {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac {1}{4} a^2 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{4} a^2 \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 = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.61 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^3} \, dx=\frac {(1-i a x)^{3/4} \left (-6-15 i a x+9 a^2 x^2+2 a^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {i+a x}{i-a x}\right )\right )}{12 x^2 (1+i a x)^{3/4}} \]
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\[\int \frac {\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}{x^{3}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^3} \, dx=\frac {a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + i \, a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - i \, a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (3 \, a^{2} x^{2} + i \, a x + 2\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{8 \, x^{2}} \]
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\[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^3} \, dx=\int \frac {\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{3}}\, dx \]
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\[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^3} \, dx=\int { \frac {\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^3} \, dx=\int \frac {\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}}{x^3} \,d x \]
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