Optimal. Leaf size=238 \[ \frac{a^{5/2} (8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}+\frac{10 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d} \]
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Rubi [A] time = 0.852188, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {4221, 3044, 2975, 2976, 2981, 2774, 216} \[ \frac{a^{5/2} (8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}+\frac{10 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3044
Rule 2975
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{5}{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{5}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 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 (4 A-3 C) \cos (c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{3 a}\\ &=\frac{10 a A (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{\left (4 \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 (16 A+3 C)-\frac{3}{4} a^2 (8 A-C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{a^2 (8 A-C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{10 a A (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 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{5}{8} a^3 (8 A+3 C)-\frac{1}{8} a^3 (56 A-27 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{a^3 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{10 a A (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac{1}{8} \left (a^2 (8 A+19 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 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{10 a A (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}-\frac{\left (a^2 (8 A+19 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 )}{4 d}\\ &=\frac{a^{5/2} (8 A+19 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)}}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{10 a A (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.85915, size = 141, normalized size = 0.59 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left (6 \sqrt{2} (8 A+19 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left (\frac{1}{2} (c+d x)\right ) ((128 A+9 C) \cos (c+d x)+16 A+33 C \cos (2 (c+d x))+3 C \cos (3 (c+d x))+33 C)\right )}{48 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.214, size = 494, normalized size = 2.1 \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 = 2.15052, size = 459, normalized size = 1.93 \begin{align*} -\frac{3 \,{\left ({\left (8 \, A + 19 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (8 \, A + 19 \, C\right )} a^{2} \cos \left (d x + c\right )\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 (6 \, C a^{2} \cos \left (d x + c\right )^{3} + 33 \, C a^{2} \cos \left (d x + c\right )^{2} + 64 \, A a^{2} \cos \left (d x + c\right ) + 8 \, 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 )}}}{12 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\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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