Integrand size = 25, antiderivative size = 54 \[ \int \frac {e^{-2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {2 i (1-i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c+a^2 c x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5182, 667, 197} \[ \int \frac {e^{-2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x}{3 c \sqrt {a^2 c x^2+c}}+\frac {2 i (1-i a x)}{3 a \left (a^2 c x^2+c\right )^{3/2}} \]
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Rule 197
Rule 667
Rule 5182
Rubi steps \begin{align*} \text {integral}& = c \int \frac {(1-i a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \\ & = \frac {2 i (1-i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {1}{3} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \\ & = \frac {2 i (1-i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-i a x} (2+i a x) \sqrt {1+a^2 x^2}}{3 a c \sqrt {1+i a x} (-i+a x) \sqrt {c+a^2 c x^2}} \]
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(-\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (-a x +2 i\right ) \left (a^{2} x^{2}+1\right )}{3 a \left (i a x +1\right )^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(56\) |
default | \(-\frac {x}{c \sqrt {a^{2} c \,x^{2}+c}}-\frac {2 i \left (\frac {i}{3 a c \left (x -\frac {i}{a}\right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}+\frac {i \left (2 \left (x -\frac {i}{a}\right ) a^{2} c +2 i a c \right )}{3 a \,c^{2} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}\right )}{a}\) | \(137\) |
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Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (a x - 2 i\right )}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 i \, a^{2} c^{2} x - a c^{2}\right )}} \]
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\[ \int \frac {e^{-2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=- \int \frac {a^{2} x^{2}}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} - 2 i a^{3} c x^{3} \sqrt {a^{2} c x^{2} + c} - 2 i a c x \sqrt {a^{2} c x^{2} + c} - c \sqrt {a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} - 2 i a^{3} c x^{3} \sqrt {a^{2} c x^{2} + c} - 2 i a c x \sqrt {a^{2} c x^{2} + c} - c \sqrt {a^{2} c x^{2} + c}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x}{3 \, \sqrt {a^{2} c x^{2} + c} c} + \frac {2 i}{3 i \, \sqrt {a^{2} c x^{2} + c} a^{2} c x + 3 \, \sqrt {a^{2} c x^{2} + c} a c} \]
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Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {a^{2} c} {\left (3 \, \sqrt {a^{2} c} x - 3 \, \sqrt {a^{2} c x^{2} + c} - i \, \sqrt {c}\right )}}{3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c} - i \, \sqrt {c}\right )}^{3} a^{2} c} \]
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Time = 0.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.59 \[ \int \frac {e^{-2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {a^3\,x^3+3\,a\,x+2{}\mathrm {i}}{3\,a\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{3/2}} \]
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