Integrand size = 23, antiderivative size = 82 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=-\frac {\sqrt {2} \arcsin \left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{d}+\frac {2 \sqrt {\sec (c+d x)} \sin (c+d x)}{d \sqrt {1+\cos (c+d x)}} \]
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Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4307, 2858, 2860, 222} \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d \sqrt {\cos (c+d x)+1}}-\frac {\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arcsin \left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )}{d} \]
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Rule 222
Rule 2858
Rule 2860
Rule 4307
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {1+\cos (c+d x)}} \, dx \\ & = \frac {2 \sqrt {\sec (c+d x)} \sin (c+d x)}{d \sqrt {1+\cos (c+d x)}}-\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx \\ & = \frac {2 \sqrt {\sec (c+d x)} \sin (c+d x)}{d \sqrt {1+\cos (c+d x)}}+\frac {\left (\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,-\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )}{d} \\ & = -\frac {\sqrt {2} \arcsin \left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{d}+\frac {2 \sqrt {\sec (c+d x)} \sin (c+d x)}{d \sqrt {1+\cos (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.32 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.17 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{2} \cos (c+d x) (2+\cos (c+d x)) \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (1-\cos (c+d x)+\text {arctanh}\left (\sqrt {-\sec (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right ) \cos (c+d x) \sqrt {2-2 \sec (c+d x)}\right )-\frac {1}{10} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},-\sec (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x) \tan (c+d x)\right )}{d \sqrt {1+\cos (c+d x)}} \]
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Time = 5.88 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.68
method | result | size |
default | \(\frac {\left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \sqrt {2}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+2 \sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {2+2 \cos \left (d x +c \right )}\, \sqrt {2}}{2 d \left (1+\cos \left (d x +c \right )\right )}\) | \(138\) |
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Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\frac {{\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right ) + \frac {2 \, \sqrt {\cos \left (d x + c\right ) + 1} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{d \cos \left (d x + c\right ) + d} \]
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\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\int \frac {\sec ^{\frac {3}{2}}{\left (c + d x \right )}}{\sqrt {\cos {\left (c + d x \right )} + 1}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 648, normalized size of antiderivative = 7.90 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\frac {2 \, \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 ) \sin \left (d x + c\right ) - 2 \, {\left (\cos \left (d x + c\right ) - 1\right )} \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 ) - \sqrt {2} {\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}} \arctan \left (\frac {{\left ({\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{4} + \cos \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{4} + 2 \, {\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 | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2} - 4 \, \cos \left (d x + c\right )^{3} + 2 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2}}, \frac {{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2} + \cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2}}\right )\right ) + \sin \left (d x + c\right )}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}}, \frac {{\left ({\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{4} + \cos \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{4} + 2 \, {\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 | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2} - 4 \, \cos \left (d x + c\right )^{3} + 2 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2}}, \frac {{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2} + \cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2}}\right )\right ) + \cos \left (d x + c\right ) - 1}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}}\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}} d} \]
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\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {\cos \left (d x + c\right ) + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\cos (c+d x)}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{\sqrt {\cos \left (c+d\,x\right )+1}} \,d x \]
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