Integrand size = 24, antiderivative size = 73 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}+\frac {\text {arcsinh}(a x)}{a} \]
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Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5181, 49, 41, 221} \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)}{a}+\frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}} \]
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Rule 41
Rule 49
Rule 221
Rule 5181
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a x)^{3/2}}{(1+i a x)^{5/2}} \, dx \\ & = \frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\int \frac {\sqrt {1-i a x}}{(1+i a x)^{3/2}} \, dx \\ & = \frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}+\int \frac {1}{\sqrt {1-i a x} \sqrt {1+i a x}} \, dx \\ & = \frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}+\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx \\ & = \frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}+\frac {\text {arcsinh}(a x)}{a} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 i \left (\frac {2 \sqrt {1+i a x} \left (1+i a x+2 a^2 x^2\right )}{\sqrt {1-i a x} (-i+a x)^2}+3 \arcsin \left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{3 a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (57 ) = 114\).
Time = 0.35 (sec) , antiderivative size = 305, normalized size of antiderivative = 4.18
method | result | size |
default | \(\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{3 a \left (x -\frac {i}{a}\right )^{4}}-\frac {i a \left (\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )}{3}}{a^{4}}\) | \(305\) |
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Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=-\frac {8 \, a^{2} x^{2} - 16 i \, a x + 3 \, {\left (a^{2} x^{2} - 2 i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + 4 \, \sqrt {a^{2} x^{2} + 1} {\left (2 \, a x - i\right )} - 8}{3 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} \]
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\[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {\left (a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\left (a x - i\right )^{4}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (51) = 102\).
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{-3 i \, a^{4} x^{3} - 9 \, a^{3} x^{2} + 9 i \, a^{2} x + 3 \, a} + \frac {\operatorname {arsinh}\left (a x\right )}{a} - \frac {2 i \, \sqrt {a^{2} x^{2} + 1}}{3 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} - \frac {7 i \, \sqrt {a^{2} x^{2} + 1}}{3 i \, a^{2} x + 3 \, a} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.33 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=-\frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} \]
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Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}+\frac {8\,\sqrt {a^2\,x^2+1}}{3\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{3\,\left (-a^4\,x^2+a^3\,x\,2{}\mathrm {i}+a^2\right )} \]
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