Optimal. Leaf size=266 \[ \frac {2 a^2 (28 A+33 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{231 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{385 d \sqrt {a \cos (c+d x)+a}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{1155 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (112 A+143 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{1155 d \sqrt {a \cos (c+d x)+a}}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{33 d} \]
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Rubi [A] time = 0.84, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {4221, 3044, 2975, 2980, 2772, 2771} \[ \frac {2 a^2 (28 A+33 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{231 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{385 d \sqrt {a \cos (c+d x)+a}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{1155 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (112 A+143 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{1155 d \sqrt {a \cos (c+d x)+a}}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{33 d} \]
Antiderivative was successfully verified.
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Rule 2771
Rule 2772
Rule 2975
Rule 2980
Rule 3044
Rule 4221
Rubi steps
\begin {align*} \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 &=\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*}
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Mathematica [A] time = 0.95, size = 146, normalized size = 0.55 \[ \frac {a \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \sqrt {a (\cos (c+d x)+1)} ((4228 A+4147 C) \cos (c+d x)+2 (728 A+737 C) \cos (2 (c+d x))+1456 A \cos (3 (c+d x))+224 A \cos (4 (c+d x))+224 A \cos (5 (c+d x))+1652 A+1859 C \cos (3 (c+d x))+286 C \cos (4 (c+d x))+286 C \cos (5 (c+d x))+1188 C)}{2310 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 138, normalized size = 0.52 \[ \frac {2 \, {\left (8 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (56 \, A + 33 \, C\right )} a \cos \left (d x + c\right )^{2} + 245 \, A a \cos \left (d x + c\right ) + 105 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{1155 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )} \sqrt {\cos \left (d x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 152, normalized size = 0.57 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (896 A \left (\cos ^{5}\left (d x +c \right )\right )+1144 C \left (\cos ^{5}\left (d x +c \right )\right )+448 A \left (\cos ^{4}\left (d x +c \right )\right )+572 C \left (\cos ^{4}\left (d x +c \right )\right )+336 A \left (\cos ^{3}\left (d x +c \right )\right )+429 C \left (\cos ^{3}\left (d x +c \right )\right )+280 A \left (\cos ^{2}\left (d x +c \right )\right )+165 C \left (\cos ^{2}\left (d x +c \right )\right )+245 A \cos \left (d x +c \right )+105 A \right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {13}{2}} a}{1155 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.97, size = 712, normalized size = 2.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.89, size = 387, normalized size = 1.45 \[ \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 )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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