Integrand size = 23, antiderivative size = 24 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]
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Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 8} \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]
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Rule 8
Rule 17
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b \cos (c+d x)} \int 1 \, dx}{\sqrt {\cos (c+d x)}} \\ & = \frac {x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]
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Time = 2.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {x \sqrt {\cos \left (d x +c \right ) b}}{\sqrt {\cos \left (d x +c \right )}}\) | \(21\) |
default | \(\frac {\sqrt {\cos \left (d x +c \right ) b}\, \left (d x +c \right )}{d \sqrt {\cos \left (d x +c \right )}}\) | \(28\) |
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none
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.92 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\left [\frac {\sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{2 \, d}, \frac {\sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{d}\right ] \]
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Time = 0.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x \sqrt {b \cos {\left (c + d x \right )}}}{\sqrt {\cos {\left (c + d x \right )}}} \]
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none
Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \, \sqrt {b} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \]
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\[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right )}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
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Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x\,\sqrt {b\,\cos \left (c+d\,x\right )}}{\sqrt {\cos \left (c+d\,x\right )}} \]
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