Integrand size = 18, antiderivative size = 210 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {(i-a-b x) \sqrt [4]{1-i (a+b x)}}{(i-a) x \sqrt [4]{1+i (a+b x)}}-\frac {i b \arctan \left (\frac {\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{(i-a)^{5/4} (i+a)^{3/4}}-\frac {i b \text {arctanh}\left (\frac {\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{(i-a)^{5/4} (i+a)^{3/4}} \]
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Time = 0.08 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5202, 294, 218, 214, 211} \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {i b \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{(-a+i)^{5/4} (a+i)^{3/4}}-\frac {i b \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{(-a+i)^{5/4} (a+i)^{3/4}}-\frac {\sqrt [4]{1-i (a+b x)} (-a-b x+i)}{(-a+i) x \sqrt [4]{1+i (a+b x)}} \]
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Rule 211
Rule 214
Rule 218
Rule 294
Rule 5202
Rubi steps \begin{align*} \text {integral}& = -\left ((8 i b) \text {Subst}\left (\int \frac {x^4}{\left (1-i a-(1+i a) x^4\right )^2} \, dx,x,\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )\right ) \\ & = -\frac {(i-a-b x) \sqrt [4]{1-i (a+b x)}}{(i-a) x \sqrt [4]{1+i (a+b x)}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{1-i a+(-1-i a) x^4} \, dx,x,\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{i-a} \\ & = -\frac {(i-a-b x) \sqrt [4]{1-i (a+b x)}}{(i-a) x \sqrt [4]{1+i (a+b x)}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {i+a}-\sqrt {i-a} x^2} \, dx,x,\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{(1+i a) \sqrt {i+a}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {i+a}+\sqrt {i-a} x^2} \, dx,x,\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{(1+i a) \sqrt {i+a}} \\ & = -\frac {(i-a-b x) \sqrt [4]{1-i (a+b x)}}{(i-a) x \sqrt [4]{1+i (a+b x)}}-\frac {i b \arctan \left (\frac {\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{(i-a)^{5/4} (i+a)^{3/4}}-\frac {i b \text {arctanh}\left (\frac {\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{(i-a)^{5/4} (i+a)^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {\sqrt [4]{-i (i+a+b x)} \left (1+a^2+i b x+a b x-2 i b x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )\right )}{\left (1+a^2\right ) x \sqrt [4]{1+i a+i b x}} \]
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\[\int \frac {1}{\sqrt {\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}\, x^{2}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (141) = 282\).
Time = 0.28 (sec) , antiderivative size = 707, normalized size of antiderivative = 3.37 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x^2} \, dx=\frac {\left (-\frac {b^{4}}{a^{8} - 2 i \, a^{7} + 2 \, a^{6} - 6 i \, a^{5} - 6 i \, a^{3} - 2 \, a^{2} - 2 i \, a - 1}\right )^{\frac {1}{4}} {\left (-i \, a - 1\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + {\left (a^{6} - 2 i \, a^{5} + a^{4} - 4 i \, a^{3} - a^{2} - 2 i \, a - 1\right )} \left (-\frac {b^{4}}{a^{8} - 2 i \, a^{7} + 2 \, a^{6} - 6 i \, a^{5} - 6 i \, a^{3} - 2 \, a^{2} - 2 i \, a - 1}\right )^{\frac {3}{4}}}{b^{3}}\right ) + \left (-\frac {b^{4}}{a^{8} - 2 i \, a^{7} + 2 \, a^{6} - 6 i \, a^{5} - 6 i \, a^{3} - 2 \, a^{2} - 2 i \, a - 1}\right )^{\frac {1}{4}} {\left (i \, a + 1\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - {\left (a^{6} - 2 i \, a^{5} + a^{4} - 4 i \, a^{3} - a^{2} - 2 i \, a - 1\right )} \left (-\frac {b^{4}}{a^{8} - 2 i \, a^{7} + 2 \, a^{6} - 6 i \, a^{5} - 6 i \, a^{3} - 2 \, a^{2} - 2 i \, a - 1}\right )^{\frac {3}{4}}}{b^{3}}\right ) + \left (-\frac {b^{4}}{a^{8} - 2 i \, a^{7} + 2 \, a^{6} - 6 i \, a^{5} - 6 i \, a^{3} - 2 \, a^{2} - 2 i \, a - 1}\right )^{\frac {1}{4}} {\left (a - i\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - {\left (i \, a^{6} + 2 \, a^{5} + i \, a^{4} + 4 \, a^{3} - i \, a^{2} + 2 \, a - i\right )} \left (-\frac {b^{4}}{a^{8} - 2 i \, a^{7} + 2 \, a^{6} - 6 i \, a^{5} - 6 i \, a^{3} - 2 \, a^{2} - 2 i \, a - 1}\right )^{\frac {3}{4}}}{b^{3}}\right ) - \left (-\frac {b^{4}}{a^{8} - 2 i \, a^{7} + 2 \, a^{6} - 6 i \, a^{5} - 6 i \, a^{3} - 2 \, a^{2} - 2 i \, a - 1}\right )^{\frac {1}{4}} {\left (a - i\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - {\left (-i \, a^{6} - 2 \, a^{5} - i \, a^{4} - 4 \, a^{3} + i \, a^{2} - 2 \, a + i\right )} \left (-\frac {b^{4}}{a^{8} - 2 i \, a^{7} + 2 \, a^{6} - 6 i \, a^{5} - 6 i \, a^{3} - 2 \, a^{2} - 2 i \, a - 1}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 2 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{2 \, {\left (a - i\right )} x} \]
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\[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x^2} \, dx=\int \frac {1}{x^{2} \sqrt {\frac {i \left (a + b x - i\right )}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}}}\, dx \]
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\[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \sqrt {\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x^2} \, dx=\int \frac {1}{x^2\,\sqrt {\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}}} \,d x \]
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