Integrand size = 16, antiderivative size = 170 \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^4} \, dx=-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {17}{8} i a^3 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {17}{8} i a^3 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5170, 101, 156, 12, 95, 304, 209, 212} \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^4} \, dx=-\frac {17}{8} i a^3 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {17}{8} i a^3 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2} \]
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 209
Rule 212
Rule 304
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a x)^{3/4}}{x^4 (1-i a x)^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac {1}{3} \int \frac {\frac {7 i a}{2}-2 a^2 x}{x^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}-\frac {1}{6} \int \frac {\frac {23 a^2}{4}+\frac {7}{2} i a^3 x}{x^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac {1}{6} \int -\frac {51 i a^3}{8 x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {1}{16} \left (17 i a^3\right ) \int \frac {1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {1}{4} \left (17 i a^3\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac {1}{8} \left (17 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 (17 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 {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {17}{8} i a^3 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {17}{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 = 93, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^4} \, dx=\frac {\sqrt [4]{1-i a x} \left (-8-22 i a x+37 a^2 x^2+23 i a^3 x^3+102 i a^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {i+a x}{i-a x}\right )\right )}{24 x^3 \sqrt [4]{1+i a x}} \]
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\[\int \frac {{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {3}{2}}}{x^{4}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^4} \, dx=\frac {51 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 51 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 51 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 51 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) + 2 \, {\left (23 \, a^{2} x^{2} - 14 i \, a x - 8\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{48 \, x^{3}} \]
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\[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^4} \, dx=\int \frac {\left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
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\[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^4} \, dx=\int { \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^4} \, dx=\int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{3/2}}{x^4} \,d x \]
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