Integrand size = 21, antiderivative size = 40 \[ \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d} \]
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Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2746, 65, 212} \[ \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{d} \]
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Rule 65
Rule 212
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {a \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {(2 a) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{d} \\ & = \frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d} \]
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Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {2}\, \sqrt {a}}{d}\) | \(32\) |
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Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.30 \[ \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\left [\frac {\sqrt {2} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right )}{2 \, d}, -\frac {\sqrt {2} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a}}{\sqrt {a \sin \left (d x + c\right ) + a}}\right )}{d}\right ] \]
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\[ \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sec {\left (c + d x \right )}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.45 \[ \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} \sqrt {a} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right )}{2 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.48 \[ \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} \sqrt {a} {\left (\log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{2 \, d} \]
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Timed out. \[ \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\cos \left (c+d\,x\right )} \,d x \]
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