Optimal. Leaf size=242 \[ \frac{5 a^{5/2} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d} \]
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Rubi [A] time = 0.859843, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {4221, 3044, 2976, 2981, 2774, 216} \[ \frac{5 a^{5/2} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3044
Rule 2976
Rule 2981
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} \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{(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^{5/2} \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{(a+a \cos (c+d x))^{5/2} \left (\frac{5 a A}{2}-\frac{1}{2} a (6 A-C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{a}\\ &=-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \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{(a+a \cos (c+d x))^{3/2} \left (\frac{1}{4} a^2 (24 A+C)-\frac{3}{4} a^2 (8 A-3 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{a^2 (8 A-3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{1}{8} a^3 (72 A+13 C)-\frac{1}{8} a^3 (24 A-49 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{16} \left (5 a^2 (8 A+5 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^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}-\frac{\left (5 a^2 (8 A+5 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 )}{8 d}\\ &=\frac{5 a^{5/2} (8 A+5 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)}}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 1.00323, size = 142, normalized size = 0.59 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left (15 \sqrt{2} (8 A+5 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}+2 \sin \left (\frac{1}{2} (c+d x)\right ) (3 (8 A+27 C) \cos (c+d x)+48 A+17 C \cos (2 (c+d x))+2 C \cos (3 (c+d x))+17 C)\right )}{48 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.168, size = 361, normalized size = 1.5 \begin{align*}{\frac{{a}^{2}\cos \left ( dx+c \right ) }{24\,d \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( 8\,C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+120\,A\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 ) +75\,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 ) +34\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +120\,A\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 ) +24\,A\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +75\,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 ) +75\,C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +48\,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 [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 = 2.23903, size = 432, normalized size = 1.79 \begin{align*} -\frac{15 \,{\left ({\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (8 \, A + 5 \, C\right )} a^{2}\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 (8 \, C a^{2} \cos \left (d x + c\right )^{3} + 34 \, C a^{2} \cos \left (d x + c\right )^{2} + 3 \,{\left (8 \, A + 25 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, A a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{24 \,{\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(-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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