Integrand size = 25, antiderivative size = 254 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=-\frac {7 \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}+\frac {637 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}} \]
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Time = 0.95 (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 = {2844, 3056, 3062, 3061, 2861, 211, 2853, 222} \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=-\frac {7 \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac {637 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {189 \sin (c+d x) \sqrt {\cos (c+d x)}}{64 a^3 d \sqrt {a \cos (c+d x)+a}}-\frac {259 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac {7 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}} \]
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Rule 211
Rule 222
Rule 2844
Rule 2853
Rule 2861
Rule 3056
Rule 3061
Rule 3062
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7 a}{2}-7 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2} \\ & = -\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {105 a^2}{4}-\frac {77}{2} a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4} \\ & = -\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (\frac {777 a^3}{8}-\frac {567}{4} a^3 \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6} \\ & = -\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\int \frac {-\frac {567 a^4}{8}+168 a^4 \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^7} \\ & = -\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {7 \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{2 a^4}+\frac {637 \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3} \\ & = -\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {7 \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 {637 \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 {7 \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}+\frac {637 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}} \\ \end{align*}
Time = 1.10 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {a (1+\cos (c+d x))} \left (4536 \arcsin \left (\sqrt {1-\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+15288 \arcsin \left (\sqrt {\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )-7644 \sqrt {2} \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+1442 \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)+1099 \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)+192 \sqrt {1-\cos (c+d x)} \cos ^{\frac {7}{2}}(c+d x)+567 \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}\right ) \sin (c+d x)}{192 a^4 d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(447\) vs. \(2(211)=422\).
Time = 12.47 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.76
method | result | size |
default | \(\frac {\left (192 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+1099 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1344 \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 ) \left (\cos ^{3}\left (d x +c \right )\right )+1442 \sqrt {2}\, \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-1911 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )-4032 \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 ) \left (\cos ^{2}\left (d x +c \right )\right )+567 \sin \left (d x +c \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-5733 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{2}\left (d x +c \right )\right )-4032 \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 ) \cos \left (d x +c \right )-5733 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \cos \left (d x +c \right )-1344 \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 )-1911 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \sqrt {2}}{384 d \left (1+\cos \left (d x +c \right )\right )^{4} \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, a^{4}}\) | \(448\) |
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Time = 0.54 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=-\frac {1911 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 2 \, {\left (192 \, \cos \left (d x + c\right )^{3} + 1099 \, \cos \left (d x + c\right )^{2} + 1442 \, \cos \left (d x + c\right ) + 567\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2688 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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Timed out. \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{9/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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