Integrand size = 45, antiderivative size = 234 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 a^3 (8 A+10 B+11 C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (584 A+690 B+903 C) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (64 A+90 B+63 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+9 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Time = 0.92 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3122, 3054, 3059, 2850} \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 a^3 (8 A+10 B+11 C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (584 A+690 B+903 C) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (64 A+90 B+63 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+9 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Rule 2850
Rule 3054
Rule 3059
Rule 3122
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^{5/2} \left (\frac {1}{2} a (5 A+9 B)+\frac {1}{2} a (2 A+9 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{9 a} \\ & = \frac {2 a (5 A+9 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {1}{4} a^2 (64 A+90 B+63 C)+\frac {3}{4} a^2 (8 A+6 B+21 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a} \\ & = \frac {2 a^2 (64 A+90 B+63 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+9 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8 \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {63}{8} a^3 (8 A+10 B+11 C)+\frac {1}{8} a^3 (248 A+270 B+441 C) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a} \\ & = \frac {2 a^3 (8 A+10 B+11 C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (64 A+90 B+63 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+9 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{315} \left (a^2 (584 A+690 B+903 C)\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 a^3 (8 A+10 B+11 C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (584 A+690 B+903 C) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (64 A+90 B+63 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+9 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.68 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (2908 A+2790 B+2961 C+2 (1396 A+1215 B+882 C) \cos (c+d x)+4 (803 A+870 B+966 C) \cos (2 (c+d x))+584 A \cos (3 (c+d x))+690 B \cos (3 (c+d x))+588 C \cos (3 (c+d x))+584 A \cos (4 (c+d x))+690 B \cos (4 (c+d x))+903 C \cos (4 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{1260 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Time = 14.55 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {2 a^{2} \sin \left (d x +c \right ) \left (584 A \left (\cos ^{4}\left (d x +c \right )\right )+690 B \left (\cos ^{4}\left (d x +c \right )\right )+903 C \left (\cos ^{4}\left (d x +c \right )\right )+292 A \left (\cos ^{3}\left (d x +c \right )\right )+345 B \left (\cos ^{3}\left (d x +c \right )\right )+294 C \left (\cos ^{3}\left (d x +c \right )\right )+219 A \left (\cos ^{2}\left (d x +c \right )\right )+180 B \left (\cos ^{2}\left (d x +c \right )\right )+63 C \left (\cos ^{2}\left (d x +c \right )\right )+130 A \cos \left (d x +c \right )+45 B \cos \left (d x +c \right )+35 A \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{315 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {9}{2}}}\) | \(166\) |
parts | \(\frac {2 A \sin \left (d x +c \right ) \left (584 \left (\cos ^{4}\left (d x +c \right )\right )+292 \left (\cos ^{3}\left (d x +c \right )\right )+219 \left (\cos ^{2}\left (d x +c \right )\right )+130 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{315 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 B \sin \left (d x +c \right ) \left (46 \left (\cos ^{3}\left (d x +c \right )\right )+23 \left (\cos ^{2}\left (d x +c \right )\right )+12 \cos \left (d x +c \right )+3\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{21 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 C \sin \left (d x +c \right ) \left (43 \left (\cos ^{2}\left (d x +c \right )\right )+14 \cos \left (d x +c \right )+3\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{15 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {5}{2}}}\) | \(227\) |
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Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.61 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \, {\left ({\left (584 \, A + 690 \, B + 903 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + {\left (292 \, A + 345 \, B + 294 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (73 \, A + 60 \, B + 21 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 5 \, {\left (26 \, A + 9 \, B\right )} a^{2} \cos \left (d x + c\right ) + 35 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \]
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Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (204) = 408\).
Time = 0.41 (sec) , antiderivative size = 682, normalized size of antiderivative = 2.91 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Time = 9.60 (sec) , antiderivative size = 713, normalized size of antiderivative = 3.05 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Too large to display} \]
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