Optimal. Leaf size=289 \[ \frac {2 (19 A+21 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}-\frac {2 (29 A+21 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (257 A+273 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{315 d \sqrt {a \cos (c+d x)+a}}-\frac {\sqrt {2} (A+C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}-\frac {2 A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{63 d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 1.02, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {4221, 3044, 2984, 12, 2782, 205} \[ \frac {2 (19 A+21 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}-\frac {2 (29 A+21 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (257 A+273 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{315 d \sqrt {a \cos (c+d x)+a}}-\frac {\sqrt {2} (A+C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}-\frac {2 A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{63 d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2782
Rule 2984
Rule 3044
Rule 4221
Rubi steps
\begin {align*} \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx\\ &=\frac {2 A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {a A}{2}+\frac {1}{2} a (8 A+9 C) \cos (c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{9 a}\\ &=-\frac {2 A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{4} a^2 (19 A+21 C)-\frac {3}{2} a^2 A \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{63 a^2}\\ &=\frac {2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{8} a^3 (29 A+21 C)+\frac {3}{2} a^3 (19 A+21 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{315 a^3}\\ &=-\frac {2 (29 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{16} a^4 (257 A+273 C)-\frac {3}{8} a^4 (29 A+21 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{945 a^4}\\ &=\frac {2 (257 A+273 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}-\frac {2 (29 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int -\frac {945 a^5 (A+C)}{32 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{945 a^5}\\ &=\frac {2 (257 A+273 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}-\frac {2 (29 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\left ((A+C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx\\ &=\frac {2 (257 A+273 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}-\frac {2 (29 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (2 a (A+C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \operatorname {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 )}{d}\\ &=-\frac {\sqrt {2} (A+C) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{\sqrt {a} d}+\frac {2 (257 A+273 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}-\frac {2 (29 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 A \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 9.03, size = 271, normalized size = 0.94 \[ \frac {2 e^{-\frac {1}{2} i (c+d x)} \cos \left (\frac {1}{2} (c+d x)\right ) \left (-315 i (A+C) \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )-\frac {1}{4} \sin \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+i \sin \left (\frac {1}{2} (c+d x)\right )\right ) (2 (107 A+63 C) \cos (c+d x)-8 (157 A+168 C) \cos (2 (c+d x))+58 A \cos (3 (c+d x))-257 A \cos (4 (c+d x))-1279 A+42 C \cos (3 (c+d x))-273 C \cos (4 (c+d x))-1071 C)\right )}{315 d \sqrt {a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 190, normalized size = 0.66 \[ \frac {\frac {315 \, \sqrt {2} {\left ({\left (A + C\right )} a \cos \left (d x + c\right )^{5} + {\left (A + C\right )} a \cos \left (d x + c\right )^{4}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{\sqrt {a}} + \frac {2 \, {\left ({\left (257 \, A + 273 \, C\right )} \cos \left (d x + c\right )^{4} - {\left (29 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (19 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{2} - 5 \, A \cos \left (d x + c\right ) + 35 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}} \]
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 [B] time = 0.56, size = 775, normalized size = 2.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}}{\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]
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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