Integrand size = 16, antiderivative size = 55 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^2} \, dx=-\frac {i-a}{(i+a) x}-\frac {2 i b \log (x)}{(i+a)^2}+\frac {2 i b \log (i+a+b x)}{(i+a)^2} \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5203, 78} \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^2} \, dx=-\frac {2 i b \log (x)}{(a+i)^2}+\frac {2 i b \log (a+b x+i)}{(a+i)^2}-\frac {-a+i}{(a+i) x} \]
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Rule 78
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+i a+i b x}{x^2 (1-i a-i b x)} \, dx \\ & = \int \left (\frac {i-a}{(i+a) x^2}-\frac {2 i b}{(i+a)^2 x}+\frac {2 i b^2}{(i+a)^2 (i+a+b x)}\right ) \, dx \\ & = -\frac {i-a}{(i+a) x}-\frac {2 i b \log (x)}{(i+a)^2}+\frac {2 i b \log (i+a+b x)}{(i+a)^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^2} \, dx=\frac {1+a^2-2 i b x \log (x)+2 i b x \log (i+a+b x)}{(i+a)^2 x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (45 ) = 90\).
Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.75
method | result | size |
parallelrisch | \(-\frac {2 i \ln \left (x \right ) x \,a^{2} b -2 i \ln \left (b x +a +i\right ) x \,a^{2} b +1-2 i b \ln \left (x \right ) x +4 \ln \left (x \right ) x a b +2 i b \ln \left (b x +a +i\right ) x -4 \ln \left (b x +a +i\right ) x a b +2 i a^{3}-a^{4}+2 i a}{\left (a^{2}+1\right )^{2} x}\) | \(96\) |
default | \(-\frac {-a^{2}+2 i a +1}{\left (a^{2}+1\right ) x}-\frac {2 b \left (i a^{2}+2 a -i\right ) \ln \left (x \right )}{\left (a^{2}+1\right )^{2}}+\frac {2 b^{2} \left (\frac {\left (i a^{2} b +2 a b -i b \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{2}}+\frac {\left (i a^{3}-3 i a +3 a^{2}-1-\frac {\left (i a^{2} b +2 a b -i b \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}\right )}{\left (a^{2}+1\right )^{2}}\) | \(161\) |
risch | \(-\frac {i}{\left (i+a \right ) x}+\frac {a}{\left (i+a \right ) x}-\frac {b \ln \left (4 a^{4} b^{2} x^{2}+8 a^{5} b x +4 a^{6}+8 a^{2} b^{2} x^{2}+16 a^{3} b x +12 a^{4}+4 b^{2} x^{2}+8 a b x +12 a^{2}+4\right )}{i a^{2}-2 a -i}+\frac {2 i b \arctan \left (\frac {\left (2 a^{2} b +2 b \right ) x +2 a^{3}+2 a}{2 a^{2}+2}\right )}{i a^{2}-2 a -i}+\frac {2 b \ln \left (\left (-2 a^{2} b -2 b \right ) x \right )}{i a^{2}-2 a -i}\) | \(189\) |
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Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^2} \, dx=\frac {-2 i \, b x \log \left (x\right ) + 2 i \, b x \log \left (\frac {b x + a + i}{b}\right ) + a^{2} + 1}{{\left (a^{2} + 2 i \, a - 1\right )} x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (39) = 78\).
Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.84 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^2} \, dx=- \frac {2 i b \log {\left (- \frac {2 a^{3} b}{\left (a + i\right )^{2}} - \frac {6 i a^{2} b}{\left (a + i\right )^{2}} + 2 a b + \frac {6 a b}{\left (a + i\right )^{2}} + 4 b^{2} x + 2 i b + \frac {2 i b}{\left (a + i\right )^{2}} \right )}}{\left (a + i\right )^{2}} + \frac {2 i b \log {\left (\frac {2 a^{3} b}{\left (a + i\right )^{2}} + \frac {6 i a^{2} b}{\left (a + i\right )^{2}} + 2 a b - \frac {6 a b}{\left (a + i\right )^{2}} + 4 b^{2} x + 2 i b - \frac {2 i b}{\left (a + i\right )^{2}} \right )}}{\left (a + i\right )^{2}} - \frac {- a + i}{x \left (a + i\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (38) = 76\).
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.29 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^2} \, dx=\frac {2 \, {\left (a^{2} - 2 i \, a - 1\right )} b \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{4} + 2 \, a^{2} + 1} + \frac {{\left (i \, a^{2} + 2 \, a - i\right )} b \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{4} + 2 \, a^{2} + 1} - \frac {2 \, {\left (i \, a^{2} + 2 \, a - i\right )} b \log \left (x\right )}{a^{4} + 2 \, a^{2} + 1} + \frac {a^{2} - 2 i \, a - 1}{{\left (a^{2} + 1\right )} x} \]
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Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^2} \, dx=\frac {2 \, b^{2} \log \left (b x + a + i\right )}{-i \, a^{2} b + 2 \, a b + i \, b} + \frac {2 \, b \log \left ({\left | x \right |}\right )}{i \, a^{2} - 2 \, a - i} + \frac {a^{2} + 1}{{\left (a + i\right )}^{2} x} \]
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Time = 0.70 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.78 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^2} \, dx=\frac {a-\mathrm {i}}{x\,\left (a+1{}\mathrm {i}\right )}+\frac {b\,\mathrm {atanh}\left (\frac {a^2+a\,2{}\mathrm {i}-1}{{\left (a+1{}\mathrm {i}\right )}^2}-\frac {x\,\left (2\,a^4\,b^2+4\,a^2\,b^2+2\,b^2\right )}{{\left (a+1{}\mathrm {i}\right )}^2\,\left (-b\,a^3+1{}\mathrm {i}\,b\,a^2-b\,a+b\,1{}\mathrm {i}\right )}\right )\,4{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^2} \]
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