Integrand size = 45, antiderivative size = 284 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {16 a^2 (336 A+374 B+429 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {8 a^2 (336 A+374 B+429 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (336 A+374 B+429 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (84 A+110 B+99 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (3 A+11 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]
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Time = 1.03 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4306, 3122, 3054, 3059, 2851, 2850} \[ \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 a^2 (84 A+110 B+99 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{1155 d \sqrt {a \cos (c+d x)+a}}+\frac {8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3465 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{3465 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a (3 A+11 B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{99 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d} \]
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Rule 2850
Rule 2851
Rule 3054
Rule 3059
Rule 3122
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx \\ & = \frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (3 A+11 B)+\frac {1}{2} a (6 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{11 a} \\ & = \frac {2 a (3 A+11 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {1}{4} a^2 (84 A+110 B+99 C)+\frac {3}{4} a^2 (24 A+22 B+33 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{99 a} \\ & = \frac {2 a^2 (84 A+110 B+99 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (3 A+11 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{231} \left (a (336 A+374 B+429 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (336 A+374 B+429 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (84 A+110 B+99 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (3 A+11 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (4 a (336 A+374 B+429 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{1155} \\ & = \frac {8 a^2 (336 A+374 B+429 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (336 A+374 B+429 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (84 A+110 B+99 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (3 A+11 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (8 a (336 A+374 B+429 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{3465} \\ & = \frac {16 a^2 (336 A+374 B+429 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {8 a^2 (336 A+374 B+429 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (336 A+374 B+429 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (84 A+110 B+99 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (3 A+11 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.66 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (4956 A+4114 B+3564 C+(12684 A+12386 B+12441 C) \cos (c+d x)+(4368 A+4862 B+4422 C) \cos (2 (c+d x))+4368 A \cos (3 (c+d x))+4862 B \cos (3 (c+d x))+5577 C \cos (3 (c+d x))+672 A \cos (4 (c+d x))+748 B \cos (4 (c+d x))+858 C \cos (4 (c+d x))+672 A \cos (5 (c+d x))+748 B \cos (5 (c+d x))+858 C \cos (5 (c+d x))) \sec ^{\frac {11}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{6930 d} \]
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Time = 2.22 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.64
method | result | size |
default | \(-\frac {2 a \left (\cos \left (d x +c \right )-1\right ) \left (\left (2688 \left (\cos ^{5}\left (d x +c \right )\right )+1344 \left (\cos ^{4}\left (d x +c \right )\right )+1008 \left (\cos ^{3}\left (d x +c \right )\right )+840 \left (\cos ^{2}\left (d x +c \right )\right )+735 \cos \left (d x +c \right )+315\right ) A +\cos \left (d x +c \right ) \left (2992 \left (\cos ^{4}\left (d x +c \right )\right )+1496 \left (\cos ^{3}\left (d x +c \right )\right )+1122 \left (\cos ^{2}\left (d x +c \right )\right )+935 \cos \left (d x +c \right )+385\right ) B +\left (\cos ^{2}\left (d x +c \right )\right ) \left (3432 \left (\cos ^{3}\left (d x +c \right )\right )+1716 \left (\cos ^{2}\left (d x +c \right )\right )+1287 \cos \left (d x +c \right )+495\right ) C \right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {13}{2}}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{3465 d}\) | \(182\) |
parts | \(-\frac {2 A a \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {13}{2}}\left (d x +c \right )\right ) \left (128 \left (\cos ^{6}\left (d x +c \right )\right )-64 \left (\cos ^{5}\left (d x +c \right )\right )-16 \left (\cos ^{4}\left (d x +c \right )\right )-8 \left (\cos ^{3}\left (d x +c \right )\right )-5 \left (\cos ^{2}\left (d x +c \right )\right )-20 \cos \left (d x +c \right )-15\right ) \cot \left (d x +c \right )}{165 d}+\frac {2 B a \sin \left (d x +c \right ) \left (272 \left (\cos ^{4}\left (d x +c \right )\right )+136 \left (\cos ^{3}\left (d x +c \right )\right )+102 \left (\cos ^{2}\left (d x +c \right )\right )+85 \cos \left (d x +c \right )+35\right ) \left (\sec ^{\frac {13}{2}}\left (d x +c \right )\right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\cos ^{2}\left (d x +c \right )\right )}{315 d \left (1+\cos \left (d x +c \right )\right )}+\frac {2 C a \sin \left (d x +c \right ) \left (104 \left (\cos ^{3}\left (d x +c \right )\right )+52 \left (\cos ^{2}\left (d x +c \right )\right )+39 \cos \left (d x +c \right )+15\right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {13}{2}}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{105 d \left (1+\cos \left (d x +c \right )\right )}\) | \(267\) |
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Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.55 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 \, {\left (8 \, {\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \, {\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (168 \, A + 187 \, B + 99 \, C\right )} a \cos \left (d x + c\right )^{2} + 35 \, {\left (21 \, A + 11 \, B\right )} a \cos \left (d x + c\right ) + 315 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )} \sqrt {\cos \left (d x + c\right )}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1065 vs. \(2 (248) = 496\).
Time = 0.39 (sec) , antiderivative size = 1065, normalized size of antiderivative = 3.75 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
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Time = 8.00 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.40 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {\sqrt {\frac {1}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (-\frac {16\,C\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{3\,d}-\frac {16\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (12\,A+18\,B+23\,C\right )}{15\,d}+\frac {16\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (84\,A+76\,B+81\,C\right )}{35\,d}+\frac {16\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (336\,A+374\,B+429\,C\right )}{315\,d}+\frac {32\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (336\,A+374\,B+429\,C\right )}{3465\,d}\right )}{20\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+20\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )+10\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+10\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )+2\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )+2\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )} \]
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