Optimal. Leaf size=275 \[ \frac {2 a^2 (12 A+11 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (168 A+187 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {4 a^2 (168 A+187 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{1155 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (168 A+187 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{3465 d \sqrt {a \sec (c+d x)+a}}+\frac {32 a^2 (168 A+187 B) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a A \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{11 d} \]
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Rubi [A] time = 0.71, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2955, 4017, 4015, 3805, 3804} \[ \frac {2 a^2 (12 A+11 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (168 A+187 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {4 a^2 (168 A+187 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{1155 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (168 A+187 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{3465 d \sqrt {a \sec (c+d x)+a}}+\frac {32 a^2 (168 A+187 B) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a A \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{11 d} \]
Antiderivative was successfully verified.
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Rule 2955
Rule 3804
Rule 3805
Rule 4015
Rule 4017
Rubi steps
\begin {align*} \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{11} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{2} a (12 A+11 B)+\frac {1}{2} a (8 A+11 B) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{99} \left (a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{231} \left (2 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {4 a^2 (168 A+187 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {\left (8 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{1155}\\ &=\frac {16 a^2 (168 A+187 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {4 a^2 (168 A+187 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {\left (16 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{3465}\\ &=\frac {32 a^2 (168 A+187 B) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (168 A+187 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {4 a^2 (168 A+187 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 131, normalized size = 0.48 \[ \frac {2 a \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a (\sec (c+d x)+1)} \left (35 (21 A+11 B) \cos ^4(c+d x)+(840 A+935 B) \cos ^3(c+d x)+6 (168 A+187 B) \cos ^2(c+d x)+8 (168 A+187 B) \cos (c+d x)+315 A \cos ^5(c+d x)+2688 A+2992 B\right )}{3465 d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 142, normalized size = 0.52 \[ \frac {2 \, {\left (315 \, A a \cos \left (d x + c\right )^{5} + 35 \, {\left (21 \, A + 11 \, B\right )} a \cos \left (d x + c\right )^{4} + 5 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right )^{3} + 6 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right )^{2} + 8 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right ) + 16 \, {\left (168 \, A + 187 \, B\right )} a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.13, size = 153, normalized size = 0.56 \[ -\frac {2 a \left (-1+\cos \left (d x +c \right )\right ) \left (315 A \left (\cos ^{5}\left (d x +c \right )\right )+735 A \left (\cos ^{4}\left (d x +c \right )\right )+385 B \left (\cos ^{4}\left (d x +c \right )\right )+840 A \left (\cos ^{3}\left (d x +c \right )\right )+935 B \left (\cos ^{3}\left (d x +c \right )\right )+1008 A \left (\cos ^{2}\left (d x +c \right )\right )+1122 B \left (\cos ^{2}\left (d x +c \right )\right )+1344 A \cos \left (d x +c \right )+1496 B \cos \left (d x +c \right )+2688 A +2992 B \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{3465 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 703, normalized size = 2.56 \[ \text {result too large to display} \]
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 \[ \int {\cos \left (c+d\,x\right )}^{11/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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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