Integrand size = 37, antiderivative size = 266 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {16 a^2 (112 A+143 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (28 A+33 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{33 d}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]
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Time = 0.98 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {4306, 3123, 3054, 3059, 2851, 2850} \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 a^2 (28 A+33 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{231 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{385 d \sqrt {a \cos (c+d x)+a}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{1155 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (112 A+143 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{1155 d \sqrt {a \cos (c+d x)+a}}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{33 d} \]
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Rule 2850
Rule 2851
Rule 3054
Rule 3059
Rule 3123
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+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 {3 a A}{2}+\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 A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{33 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 {3}{4} a^2 (28 A+33 C)+\frac {9}{4} a^2 (8 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{99 a} \\ & = \frac {2 a^2 (28 A+33 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{33 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}{77} \left (a (112 A+143 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 (112 A+143 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (28 A+33 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{33 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}{385} \left (4 a (112 A+143 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 \\ & = \frac {8 a^2 (112 A+143 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (28 A+33 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{33 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 (112 A+143 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}{1155} \\ & = \frac {16 a^2 (112 A+143 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (28 A+33 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{33 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 = 0.79 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.55 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (1652 A+1188 C+(4228 A+4147 C) \cos (c+d x)+2 (728 A+737 C) \cos (2 (c+d x))+1456 A \cos (3 (c+d x))+1859 C \cos (3 (c+d x))+224 A \cos (4 (c+d x))+286 C \cos (4 (c+d x))+224 A \cos (5 (c+d x))+286 C \cos (5 (c+d x))) \sec ^{\frac {11}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{2310 d} \]
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Time = 1.46 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.50
method | result | size |
default | \(-\frac {2 a \left (\cos \left (d x +c \right )-1\right ) \left (\left (896 \left (\cos ^{5}\left (d x +c \right )\right )+448 \left (\cos ^{4}\left (d x +c \right )\right )+336 \left (\cos ^{3}\left (d x +c \right )\right )+280 \left (\cos ^{2}\left (d x +c \right )\right )+245 \cos \left (d x +c \right )+105\right ) A +\left (\cos ^{2}\left (d x +c \right )\right ) \left (1144 \left (\cos ^{3}\left (d x +c \right )\right )+572 \left (\cos ^{2}\left (d x +c \right )\right )+429 \cos \left (d x +c \right )+165\right ) C \right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {13}{2}}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{1155 d}\) | \(134\) |
parts | \(-\frac {2 A a \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {13}{2}}\left (d x +c \right )\right ) \left (128 \left (\cos ^{6}\left (d x +c \right )\right )-64 \left (\cos ^{5}\left (d x +c \right )\right )-16 \left (\cos ^{4}\left (d x +c \right )\right )-8 \left (\cos ^{3}\left (d x +c \right )\right )-5 \left (\cos ^{2}\left (d x +c \right )\right )-20 \cos \left (d x +c \right )-15\right ) \cot \left (d x +c \right )}{165 d}+\frac {2 C a \sin \left (d x +c \right ) \left (104 \left (\cos ^{3}\left (d x +c \right )\right )+52 \left (\cos ^{2}\left (d x +c \right )\right )+39 \cos \left (d x +c \right )+15\right ) \left (\sec ^{\frac {13}{2}}\left (d x +c \right )\right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\cos ^{3}\left (d x +c \right )\right )}{105 d \left (1+\cos \left (d x +c \right )\right )}\) | \(176\) |
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Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.52 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 \, {\left (8 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (56 \, A + 33 \, C\right )} a \cos \left (d x + c\right )^{2} + 245 \, A a \cos \left (d x + c\right ) + 105 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{1155 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )} \sqrt {\cos \left (d x + c\right )}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (230) = 460\).
Time = 0.50 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.68 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
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Time = 7.42 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.45 \[ \int (a+a \cos (c+d x))^{3/2} \left (A+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 )\,\left (12\,A+23\,C\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{15\,d}+\frac {48\,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 )\,\left (28\,A+27\,C\right )\,\sqrt {a+a\,\cos \left (c+d\,x\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 )\,\left (112\,A+143\,C\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{105\,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 )\,\left (112\,A+143\,C\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{1155\,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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