Integrand size = 18, antiderivative size = 211 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{(1+i a) x}-\frac {3 i b \arctan \left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{7/4} \sqrt [4]{i+a}}-\frac {3 i b \text {arctanh}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{7/4} \sqrt [4]{i+a}} \]
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Time = 0.09 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5203, 96, 95, 218, 214, 211} \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {3 i b \arctan \left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(-a+i)^{7/4} \sqrt [4]{a+i}}-\frac {3 i b \text {arctanh}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(-a+i)^{7/4} \sqrt [4]{a+i}}-\frac {(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{(1+i a) x} \]
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Rule 95
Rule 96
Rule 211
Rule 214
Rule 218
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a-i b x)^{3/4}}{x^2 (1+i a+i b x)^{3/4}} \, dx \\ & = -\frac {(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{(1+i a) x}+\frac {(3 b) \int \frac {1}{x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}} \, dx}{2 (i-a)} \\ & = -\frac {(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{(1+i a) x}+\frac {(6 b) \text {Subst}\left (\int \frac {1}{-1-i a-(-1+i a) x^4} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{i-a} \\ & = -\frac {(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{(1+i a) x}-\frac {(3 i b) \text {Subst}\left (\int \frac {1}{\sqrt {i-a}-\sqrt {i+a} x^2} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/2}}-\frac {(3 i b) \text {Subst}\left (\int \frac {1}{\sqrt {i-a}+\sqrt {i+a} x^2} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/2}} \\ & = -\frac {(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{(1+i a) x}-\frac {3 i b \arctan \left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{7/4} \sqrt [4]{i+a}}-\frac {3 i b \text {arctanh}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{7/4} \sqrt [4]{i+a}} \\ \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 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {(-i (i+a+b x))^{3/4} \left (1+a^2+i b x+a b x-2 i b x \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )\right )}{\left (1+a^2\right ) x (1+i a+i b x)^{3/4}} \]
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\[\int \frac {1}{{\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}^{\frac {3}{2}} x^{2}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (137) = 274\).
Time = 0.28 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.91 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\frac {3 \, \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (-i \, a - 1\right )} x \log \left (\frac {b \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a^{2} - 2 i \, a - 1\right )}}{b}\right ) + 3 \, \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (i \, a + 1\right )} x \log \left (\frac {b \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a^{2} - 2 i \, a - 1\right )}}{b}\right ) - 3 \, \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a - i\right )} x \log \left (\frac {b \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (i \, a^{2} + 2 \, a - i\right )}}{b}\right ) + 3 \, \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a - i\right )} x \log \left (\frac {b \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (-i \, a^{2} - 2 \, a + i\right )}}{b}\right ) + 2 \, {\left (b x + a + i\right )} \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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Timed out. \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \left (\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\int \frac {1}{x^2\,{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2}} \,d x \]
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