Optimal. Leaf size=227 \[ \frac{a (48 A+40 B+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{64 d}+\frac{a (48 A+40 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d} \]
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Rubi [A] time = 0.52814, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3045, 2981, 2770, 2774, 216} \[ \frac{a (48 A+40 B+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{64 d}+\frac{a (48 A+40 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d} \]
Antiderivative was successfully verified.
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Rule 3045
Rule 2981
Rule 2770
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d}+\frac{\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left (\frac{1}{2} a (8 A+5 C)+\frac{1}{2} a (8 B+C) \cos (c+d x)\right ) \, dx}{4 a}\\ &=\frac{a (8 B+C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d}+\frac{1}{48} (48 A+40 B+35 C) \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a (48 A+40 B+35 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \cos (c+d x)}}+\frac{a (8 B+C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d}+\frac{1}{64} (48 A+40 B+35 C) \int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a (48 A+40 B+35 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{64 d \sqrt{a+a \cos (c+d x)}}+\frac{a (48 A+40 B+35 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \cos (c+d x)}}+\frac{a (8 B+C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d}+\frac{1}{128} (48 A+40 B+35 C) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{a (48 A+40 B+35 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{64 d \sqrt{a+a \cos (c+d x)}}+\frac{a (48 A+40 B+35 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \cos (c+d x)}}+\frac{a (8 B+C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d}-\frac{(48 A+40 B+35 C) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a}}} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{64 d}\\ &=\frac{\sqrt{a} (48 A+40 B+35 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{64 d}+\frac{a (48 A+40 B+35 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{64 d \sqrt{a+a \cos (c+d x)}}+\frac{a (48 A+40 B+35 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \cos (c+d x)}}+\frac{a (8 B+C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.89565, size = 144, normalized size = 0.63 \[ \frac{\sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (3 \sqrt{2} (48 A+40 B+35 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} (2 (48 A+40 B+53 C) \cos (c+d x)+144 A+4 (8 B+7 C) \cos (2 (c+d x))+152 B+12 C \cos (3 (c+d x))+133 C)\right )}{384 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.146, size = 622, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 7.3838, size = 467, normalized size = 2.06 \begin{align*} \frac{{\left (48 \, C \cos \left (d x + c\right )^{3} + 8 \,{\left (8 \, B + 7 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (48 \, A + 40 \, B + 35 \, C\right )} \cos \left (d x + c\right ) + 144 \, A + 120 \, B + 105 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 3 \,{\left ({\left (48 \, A + 40 \, B + 35 \, C\right )} \cos \left (d x + c\right ) + 48 \, A + 40 \, B + 35 \, C\right )} \sqrt{a} \arctan \left (\frac{\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 )}{192 \,{\left (d \cos \left (d x + c\right ) + d\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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