Integrand size = 26, antiderivative size = 118 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {8 a \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {16 a \sin (c+d x)}{15 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]
[Out]
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2851, 2850} \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {8 a \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {16 a \sin (c+d x)}{15 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]
[In]
[Out]
Rule 2850
Rule 2851
Rubi steps \begin{align*} \text {integral}& = \frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {4}{5} \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {8 a \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {8}{15} \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}-\frac {8 a \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a-a \cos (c+d x)}}+\frac {16 a \sin (c+d x)}{15 d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \sqrt {a-a \cos (c+d x)} (7-4 \cos (c+d x)+4 \cos (2 (c+d x))) \cot \left (\frac {1}{2} (c+d x)\right )}{15 d \cos ^{\frac {5}{2}}(c+d x)} \]
[In]
[Out]
Time = 5.45 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.53
method | result | size |
default | \(\frac {2 \csc \left (d x +c \right ) \sqrt {-a \left (\cos \left (d x +c \right )-1\right )}\, \left (3+8 \left (\cos ^{3}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right )-\cos \left (d x +c \right )\right )}{15 d \cos \left (d x +c \right )^{\frac {5}{2}}}\) | \(63\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \, {\left (8 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 3\right )} \sqrt {-a \cos \left (d x + c\right ) + a}}{15 \, d \cos \left (d x + c\right )^{\frac {5}{2}} \sin \left (d x + c\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (100) = 200\).
Time = 0.31 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.87 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \, {\left (7 \, \sqrt {2} \sqrt {a} - \frac {17 \, \sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {25 \, \sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {15 \, \sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} \]
[In]
[Out]
none
Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {2 \, \sqrt {2} {\left ({\left ({\left ({\left ({\left (7 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 75\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 430\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 430\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 75\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 7\right )} \sqrt {a} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{15 \, {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1\right )}^{\frac {5}{2}} d} \]
[In]
[Out]
Time = 15.97 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {8\,\sqrt {2\,a\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\,\left (7\,\sin \left (c+d\,x\right )-4\,\sin \left (2\,c+2\,d\,x\right )+9\,\sin \left (3\,c+3\,d\,x\right )-2\,\sin \left (4\,c+4\,d\,x\right )+2\,\sin \left (5\,c+5\,d\,x\right )\right )}{15\,d\,\sqrt {1-2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\,\left (-16\,{\sin \left (c+d\,x\right )}^2-4\,{\sin \left (2\,c+2\,d\,x\right )}^2+20\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+10\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2+2\,{\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}^2\right )} \]
[In]
[Out]