Integrand size = 24, antiderivative size = 19 \[ \int \frac {e^{3 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=-\frac {i}{2 a (1-i a x)^2} \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5181, 32} \[ \int \frac {e^{3 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=-\frac {i}{2 a (1-i a x)^2} \]
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Rule 32
Rule 5181
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(1-i a x)^3} \, dx \\ & = -\frac {i}{2 a (1-i a x)^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {e^{3 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {i}{2 a (i+a x)^2} \]
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {i}{2 a \left (a x +i\right )^{2}}\) | \(15\) |
risch | \(\frac {i}{2 a \left (a x +i\right )^{2}}\) | \(15\) |
norman | \(\frac {x +\frac {3}{2} i a \,x^{2}+\frac {1}{2} i a^{3} x^{4}}{\left (a^{2} x^{2}+1\right )^{2}}\) | \(31\) |
gosper | \(-\frac {\left (a x +i\right ) \left (i a x +1\right )^{3}}{2 a \left (a^{2} x^{2}+1\right )^{3}}\) | \(32\) |
parallelrisch | \(\frac {i a^{3} x^{4}+3 i a \,x^{2}+2 x}{2 \left (a^{2} x^{2}+1\right )^{2}}\) | \(34\) |
meijerg | \(\frac {\frac {x \sqrt {a^{2}}\, \left (3 a^{2} x^{2}+5\right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \sqrt {a^{2}}\, \arctan \left (a x \right )}{2 a}}{4 \sqrt {a^{2}}}+\frac {3 i a \,x^{2} \left (a^{2} x^{2}+2\right )}{4 \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \left (-\frac {x \left (a^{2}\right )^{\frac {3}{2}} \left (-3 a^{2} x^{2}+3\right )}{6 a^{2} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\left (a^{2}\right )^{\frac {3}{2}} \arctan \left (a x \right )}{2 a^{3}}\right )}{4 \sqrt {a^{2}}}-\frac {i a^{3} x^{4}}{4 \left (a^{2} x^{2}+1\right )^{2}}\) | \(154\) |
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none
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{3 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {i}{2 \, {\left (a^{3} x^{2} + 2 i \, a^{2} x - a\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{3 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {i}{2 a^{3} x^{2} + 4 i a^{2} x - 2 a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (13) = 26\).
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {e^{3 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=-\frac {-i \, a^{2} x^{2} - 2 \, a x + i}{2 \, {\left (a^{5} x^{4} + 2 \, a^{3} x^{2} + a\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {e^{3 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {i}{2 \, {\left (a x + i\right )}^{2} a} \]
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Time = 0.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {e^{3 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {1{}\mathrm {i}}{2\,\left (a^3\,x^2+a^2\,x\,2{}\mathrm {i}-a\right )} \]
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