Optimal. Leaf size=220 \[ \frac {2 a (8 A+9 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d \sqrt {a \sec (c+d x)+a}}+\frac {4 a (8 A+9 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{105 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a (8 A+9 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {32 a (8 A+9 B) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a A \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.48, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2955, 4015, 3805, 3804} \[ \frac {2 a (8 A+9 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d \sqrt {a \sec (c+d x)+a}}+\frac {4 a (8 A+9 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{105 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a (8 A+9 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {32 a (8 A+9 B) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a A \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d \sqrt {a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2955
Rule 3804
Rule 3805
Rule 4015
Rubi steps
\begin {align*} \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{9} \left ((8 A+9 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 (8 A+9 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{21} \left (2 (8 A+9 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 (8 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (8 A+9 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{105} \left (8 (8 A+9 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\\ &=\frac {16 a (8 A+9 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {4 a (8 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (8 A+9 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{315} \left (16 (8 A+9 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\\ &=\frac {32 a (8 A+9 B) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a (8 A+9 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {4 a (8 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (8 A+9 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 119, normalized size = 0.54 \[ \frac {\sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a (\sec (c+d x)+1)} (94 (8 A+9 B) \cos (c+d x)+4 (83 A+54 B) \cos (2 (c+d x))+80 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+1321 A+90 B \cos (3 (c+d x))+1368 B)}{1260 d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 116, normalized size = 0.53 \[ \frac {2 \, {\left (35 \, A \cos \left (d x + c\right )^{4} + 5 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right ) + 128 \, A + 144 \, B\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {a \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.96, size = 130, normalized size = 0.59 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (35 A \left (\cos ^{4}\left (d x +c \right )\right )+40 A \left (\cos ^{3}\left (d x +c \right )\right )+45 B \left (\cos ^{3}\left (d x +c \right )\right )+48 A \left (\cos ^{2}\left (d x +c \right )\right )+54 B \left (\cos ^{2}\left (d x +c \right )\right )+64 A \cos \left (d x +c \right )+72 B \cos \left (d x +c \right )+128 A +144 B \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\sqrt {\cos }\left (d x +c \right )\right )}{315 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 547, normalized size = 2.49 \[ \text {result too large to display} \]
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 \[ \int {\cos \left (c+d\,x\right )}^{9/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \]
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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