Optimal. Leaf size=312 \[ \frac{a^3 (136 A+109 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{96 d}+\frac{a^{5/2} (1304 A+1015 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d} \]
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Rubi [A] time = 0.912638, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {3046, 2976, 2981, 2770, 2774, 216} \[ \frac{a^3 (136 A+109 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{96 d}+\frac{a^{5/2} (1304 A+1015 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2976
Rule 2981
Rule 2770
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (12 A+5 C)+\frac{5}{2} a C \cos (c+d x)\right ) \, dx}{6 a}\\ &=\frac{a C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac{15}{4} a^2 (8 A+5 C)+\frac{5}{4} a^2 (24 A+23 C) \cos (c+d x)\right ) \, dx}{30 a}\\ &=\frac{a^2 (24 A+23 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d}+\frac{a C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left (\frac{5}{8} a^3 (312 A+235 C)+\frac{15}{8} a^3 (136 A+109 C) \cos (c+d x)\right ) \, dx}{120 a}\\ &=\frac{a^3 (136 A+109 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (24 A+23 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d}+\frac{a C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{1}{384} \left (a^2 (1304 A+1015 C)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^3 (1304 A+1015 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (136 A+109 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (24 A+23 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d}+\frac{a C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{1}{512} \left (a^2 (1304 A+1015 C)\right ) \int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^3 (1304 A+1015 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{512 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (1304 A+1015 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (136 A+109 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (24 A+23 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d}+\frac{a C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{\left (a^2 (1304 A+1015 C)\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx}{1024}\\ &=\frac{a^3 (1304 A+1015 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{512 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (1304 A+1015 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (136 A+109 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (24 A+23 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d}+\frac{a C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d}-\frac{\left (a^2 (1304 A+1015 C)\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 )}{512 d}\\ &=\frac{a^{5/2} (1304 A+1015 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{512 d}+\frac{a^3 (1304 A+1015 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{512 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (1304 A+1015 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt{a+a \cos (c+d x)}}+\frac{a^3 (136 A+109 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (24 A+23 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d}+\frac{a C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 2.46453, size = 170, normalized size = 0.54 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (3 \sqrt{2} (1304 A+1015 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} ((2896 A+3234 C) \cos (c+d x)+4 (184 A+315 C) \cos (2 (c+d x))+96 A \cos (3 (c+d x))+4648 A+428 C \cos (3 (c+d x))+112 C \cos (4 (c+d x))+16 C \cos (5 (c+d x))+4193 C)\right )}{3072 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.162, size = 581, normalized size = 1.9 \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 = 2.50217, size = 582, normalized size = 1.87 \begin{align*} \frac{{\left (256 \, C a^{2} \cos \left (d x + c\right )^{5} + 896 \, C a^{2} \cos \left (d x + c\right )^{4} + 48 \,{\left (8 \, A + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \,{\left (184 \, A + 203 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right ) + 3 \,{\left (1304 \, A + 1015 \, C\right )} a^{2}\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 (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (1304 \, A + 1015 \, 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 )}{1536 \,{\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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