Integrand size = 25, antiderivative size = 254 \[ \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{a^{7/2} d}-\frac {177 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 \sqrt {2} a^{7/2} d}-\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {17 \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {49 \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}} \]
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Time = 0.82 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4307, 2844, 3056, 3061, 2861, 211, 2853, 222} \[ \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}-\frac {177 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {49 \sin (c+d x)}{64 a^2 d \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac {17 \sin (c+d x)}{48 a d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac {\sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \]
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Rule 211
Rule 222
Rule 2844
Rule 2853
Rule 2861
Rule 3056
Rule 3061
Rule 4307
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx \\ & = -\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {5 a}{2}-6 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2} \\ & = -\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {17 \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)} \left (\frac {51 a^2}{4}-24 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4} \\ & = -\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {17 \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {49 \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {147 a^3}{8}-48 a^3 \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6} \\ & = -\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {17 \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {49 \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}}+\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^4}-\frac {\left (177 \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}{128 a^3} \\ & = -\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {17 \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {49 \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}}-\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^4 d}+\frac {\left (177 \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 )}{64 a^2 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)}}{a^{7/2} d}-\frac {177 \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)}}{64 \sqrt {2} a^{7/2} d}-\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {17 \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {49 \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.94 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {i e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (64 \text {arcsinh}\left (e^{i (c+d x)}\right )+\frac {177 \text {arctanh}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )}{\sqrt {2}}-64 \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right )}{4 \sqrt {2} d (a (1+\cos (c+d x)))^{7/2}}+\frac {\cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)} \left (-\frac {247 \cos \left (\frac {d x}{2}\right ) \sin \left (\frac {c}{2}\right )}{12 d}-\frac {247 \cos \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )}{12 d}+\frac {379 \sec \left (\frac {c}{2}\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {d x}{2}\right )}{24 d}-\frac {41 \sec \left (\frac {c}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {d x}{2}\right )}{12 d}+\frac {\sec \left (\frac {c}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {d x}{2}\right )}{3 d}+\frac {379 \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{24 d}-\frac {41 \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{12 d}+\frac {\sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{3 d}\right )}{(a (1+\cos (c+d x)))^{7/2}} \]
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Time = 4.38 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.58
method | result | size |
default | \(\frac {{\left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right )}^{4} \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1\right ) \sqrt {\frac {a}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}}\, \left (8 \left (\csc ^{5}\left (d x +c \right )\right ) \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \left (1-\cos \left (d x +c \right )\right )^{5}-50 \left (\csc ^{3}\left (d x +c \right )\right ) \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \left (1-\cos \left (d x +c \right )\right )^{3}+384 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}\right )+189 \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )-531 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {2}}{384 d {\left (-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}\right )}^{\frac {7}{2}} {\left (-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right )}^{\frac {9}{2}} a^{4}}\) | \(401\) |
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Time = 0.46 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {531 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 768 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {2 \, {\left (247 \, \cos \left (d x + c\right )^{3} + 362 \, \cos \left (d x + c\right )^{2} + 147 \, \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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