Optimal. Leaf size=137 \[ -\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d} \]
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Rubi [A] time = 0.43131, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {4221, 3044, 2981, 2774, 216} \[ -\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3044
Rule 2981
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \sqrt{a+a \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 A \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{a A}{2}-\frac{1}{2} a (2 A-C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{a}\\ &=-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 A \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{2} \left (C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 A \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}-\frac{\left (C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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 )}{d}\\ &=\frac{\sqrt{a} C \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{d}-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 A \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.267348, size = 100, normalized size = 0.73 \[ \frac{\sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left (2 \sin \left (\frac{1}{2} (c+d x)\right ) (2 A+C \cos (c+d x))+\sqrt{2} C \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.193, size = 185, normalized size = 1.4 \begin{align*}{\frac{\cos \left ( dx+c \right ) }{d \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( C\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\arctan \left ({\frac{\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \cos \left ( dx+c \right ) +C\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\arctan \left ({\frac{\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) +C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +2\,A\sin \left ( dx+c \right ) \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{3}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.2243, size = 1202, normalized size = 8.77 \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 = 1.57315, size = 285, normalized size = 2.08 \begin{align*} -\frac{{\left (C \cos \left (d x + c\right ) + 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 ) - \frac{{\left (C \cos \left (d x + c\right ) + 2 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{d \cos \left (d x + c\right ) + d} \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{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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