Integrand size = 25, antiderivative size = 105 \[ \int \frac {1}{\sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d} \]
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Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4307, 2856, 2853, 222, 2861, 211} \[ \int \frac {1}{\sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d} \]
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Rule 211
Rule 222
Rule 2853
Rule 2856
Rule 2861
Rule 4307
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+a \cos (c+d x)}} \, dx \\ & = -\left (\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx\right )+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{a} \\ & = -\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a d}+\frac {\left (2 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{d} \\ & = \frac {2 \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{\sqrt {a} d}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{\sqrt {a} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\frac {i \sqrt {2} e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (-\text {arcsinh}\left (e^{i (c+d x)}\right )+\sqrt {2} \text {arctanh}\left (\frac {-1+e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+\text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \cos \left (\frac {1}{2} (c+d x)\right )}{d \sqrt {a (1+\cos (c+d x))}} \]
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Time = 3.98 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\sqrt {2}\, \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )+\arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {2}}{d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\sec \left (d x +c \right )}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, a}\) | \(108\) |
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Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 2 \, \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{a d} \]
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\[ \int \frac {1}{\sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )} \sqrt {\sec {\left (c + d x \right )}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 698, normalized size of antiderivative = 6.65 \[ \int \frac {1}{\sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \, dx=-\frac {\sqrt {2} \sqrt {a} \arctan \left (\frac {{\left ({\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{4} + 16 \, \cos \left (d x + c\right )^{4} + 16 \, \sin \left (d x + c\right )^{4} + 8 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} {\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{2} - 64 \, \cos \left (d x + c\right )^{3} + 32 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{2} + 96 \, \cos \left (d x + c\right )^{2} - 64 \, \cos \left (d x + c\right ) + 16\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\frac {8 \, {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{2}}, \frac {{\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{2} + 4 \, \cos \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )^{2} - 8 \, \cos \left (d x + c\right ) + 4}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{2}}\right )\right ) + 2 \, \sin \left (d x + c\right )}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}}, \frac {{\left ({\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{4} + 16 \, \cos \left (d x + c\right )^{4} + 16 \, \sin \left (d x + c\right )^{4} + 8 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} {\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{2} - 64 \, \cos \left (d x + c\right )^{3} + 32 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{2} + 96 \, \cos \left (d x + c\right )^{2} - 64 \, \cos \left (d x + c\right ) + 16\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\frac {8 \, {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{2}}, \frac {{\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{2} + 4 \, \cos \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )^{2} - 8 \, \cos \left (d x + c\right ) + 4}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}^{2}}\right )\right ) + 2 \, \cos \left (d x + c\right ) - 2}{{\left | 2 \, e^{\left (i \, d x + i \, c\right )} + 2 \right |}}\right ) - \sqrt {a} \arctan \left ({\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + \sin \left (d x + c\right ), {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + \cos \left (d x + c\right )\right )}{a d} \]
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\[ \int \frac {1}{\sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]
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