Integrand size = 25, antiderivative size = 63 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2 i (1+i a x)}{a \sqrt {c+a^2 c x^2}}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a \sqrt {c}} \]
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Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5183, 667, 223, 212} \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a \sqrt {c}}-\frac {2 i (1+i a x)}{a \sqrt {a^2 c x^2+c}} \]
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Rule 212
Rule 223
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 )^{3/2}} \, dx \\ & = -\frac {2 i (1+i a x)}{a \sqrt {c+a^2 c x^2}}-\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {2 i (1+i a x)}{a \sqrt {c+a^2 c x^2}}-\text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right ) \\ & = -\frac {2 i (1+i a x)}{a \sqrt {c+a^2 c x^2}}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a \sqrt {c}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.44 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2 i \sqrt {1+a^2 x^2} \left (\sqrt {1+i a x}+\sqrt {1-i a x} \arcsin \left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{a \sqrt {1-i a x} \sqrt {c+a^2 c x^2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (53 ) = 106\).
Time = 0.36 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.24
method | result | size |
default | \(-\frac {\ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+c}\right )}{\sqrt {a^{2} c}}+\frac {\left (i \sqrt {-a^{2}}+a \right ) \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{a^{3} c \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}-\frac {\left (i \sqrt {-a^{2}}-a \right ) \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{a^{3} c \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}\) | \(204\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (50) = 100\).
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.41 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {{\left (a^{2} c x + i \, a c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {2 \, {\left (a^{2} c x + \sqrt {a^{2} c x^{2} + c} a^{2} c \sqrt {\frac {1}{a^{2} c}}\right )}}{x}\right ) - {\left (a^{2} c x + i \, a c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {2 \, {\left (a^{2} c x - \sqrt {a^{2} c x^{2} + c} a^{2} c \sqrt {\frac {1}{a^{2} c}}\right )}}{x}\right ) - 4 \, \sqrt {a^{2} c x^{2} + c}}{2 \, {\left (a^{2} c x + i \, a c\right )}} \]
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\[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=- \int \frac {a^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {2 i a x}{a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} c x^{2} + c}}\right )\, dx - \int \left (- \frac {1}{a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} c x^{2} + c}}\right )\, dx \]
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\[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{2}}{\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 1\right )}} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} + c} \right |}\right )}{a \sqrt {c}} - \frac {4}{{\left (i \, \sqrt {a^{2} c} x - i \, \sqrt {a^{2} c x^{2} + c} - \sqrt {c}\right )} a} \]
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Timed out. \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\left (1+a\,x\,1{}\mathrm {i}\right )}^2}{\sqrt {c\,a^2\,x^2+c}\,\left (a^2\,x^2+1\right )} \,d x \]
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