Integrand size = 24, antiderivative size = 31 \[ \int \sqrt {a \cos (c+d x)+i a \sin (c+d x)} \, dx=-\frac {2 i \sqrt {a \cos (c+d x)+i a \sin (c+d x)}}{d} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {3150} \[ \int \sqrt {a \cos (c+d x)+i a \sin (c+d x)} \, dx=-\frac {2 i \sqrt {a \cos (c+d x)+i a \sin (c+d x)}}{d} \]
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Rule 3150
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i \sqrt {a \cos (c+d x)+i a \sin (c+d x)}}{d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \sqrt {a \cos (c+d x)+i a \sin (c+d x)} \, dx=-\frac {2 i \sqrt {a (\cos (c+d x)+i \sin (c+d x))}}{d} \]
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Time = 0.72 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {2 i \sqrt {a \,{\mathrm e}^{i \left (d x +c \right )}}}{d}\) | \(20\) |
derivativedivides | \(-\frac {2 i \sqrt {\cos \left (d x +c \right ) a +i a \sin \left (d x +c \right )}}{d}\) | \(28\) |
default | \(-\frac {2 i \sqrt {\cos \left (d x +c \right ) a +i a \sin \left (d x +c \right )}}{d}\) | \(28\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \sqrt {a \cos (c+d x)+i a \sin (c+d x)} \, dx=-\frac {2 i \, \sqrt {a} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{d} \]
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\[ \int \sqrt {a \cos (c+d x)+i a \sin (c+d x)} \, dx=\int \sqrt {i a \sin {\left (c + d x \right )} + a \cos {\left (c + d x \right )}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \sqrt {a \cos (c+d x)+i a \sin (c+d x)} \, dx=-\frac {2 i \, \sqrt {a} \sqrt {-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i}}{d \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i}} \]
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none
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \sqrt {a \cos (c+d x)+i a \sin (c+d x)} \, dx=-\frac {2 i \, \sqrt {a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )}}{d} \]
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Timed out. \[ \int \sqrt {a \cos (c+d x)+i a \sin (c+d x)} \, dx=\int \sqrt {a\,\cos \left (c+d\,x\right )+a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
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