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 = {5183, 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 5183
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 = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {(2-i a x) \sqrt {1+i a x} \sqrt {1+a^2 x^2}}{3 a c \sqrt {1-i a x} (i+a x) \sqrt {c+a^2 c x^2}} \]
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85
method | result | size |
trager | \(\frac {\left (a^{3} x^{3}+3 a x -2 i\right ) \sqrt {a^{2} c \,x^{2}+c}}{3 c^{2} \left (a^{2} x^{2}+1\right )^{2} a}\) | \(46\) |
gosper | \(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (a x +2 i\right ) \left (i a x +1\right )^{2}}{3 a \left (a^{2} x^{2}+1\right ) \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(57\) |
default | \(-\frac {x}{c \sqrt {a^{2} c \,x^{2}+c}}+\frac {\left (i \sqrt {-a^{2}}+a \right ) \left (-\frac {1}{3 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {2 \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c +2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}}+\frac {\left (i \sqrt {-a^{2}}-a \right ) \left (\frac {1}{3 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {2 \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c -2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}}\) | \(398\) |
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Time = 0.27 (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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\[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=- \int \frac {a^{2} x^{2}}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {2 i a x}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\right )\, dx - \int \left (- \frac {1}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\right )\, dx \]
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\[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}} \,d x } \]
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Time = 0.30 (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 = 0.75 (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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