Optimal. Leaf size=213 \[ \frac{8 a (16 A+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.463261, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {3044, 2980, 2772, 2771} \[ \frac{8 a (16 A+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3044
Rule 2980
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{a A}{2}+\frac{3}{2} a (2 A+3 C) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{1}{21} (16 A+21 C) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{1}{105} (4 (16 A+21 C)) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{8 a (16 A+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{1}{315} (8 (16 A+21 C)) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{8 a (16 A+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{16 a (16 A+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.642179, size = 124, normalized size = 0.58 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (2 (88 A+63 C) \cos (c+d x)+11 (16 A+21 C) \cos (2 (c+d x))+32 A \cos (3 (c+d x))+32 A \cos (4 (c+d x))+214 A+42 C \cos (3 (c+d x))+42 C \cos (4 (c+d x))+189 C)}{315 d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.127, size = 121, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 128\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+168\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+64\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+84\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+48\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+63\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+40\,A\cos \left ( dx+c \right ) +35\,A \right ) }{315\,d\sin \left ( dx+c \right ) }\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.73003, size = 765, normalized size = 3.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49266, size = 308, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (8 \,{\left (16 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (16 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (16 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{2} + 40 \, A \cos \left (d x + c\right ) + 35 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{a \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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