Optimal. Leaf size=333 \[ \frac{a^3 (680 A+628 B+545 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a^{5/2} (1304 A+1132 B+1015 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a (12 B+5 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d} \]
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Rubi [A] time = 0.989966, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3045, 2976, 2981, 2770, 2774, 216} \[ \frac{a^3 (680 A+628 B+545 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a^{5/2} (1304 A+1132 B+1015 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a (12 B+5 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d} \]
Antiderivative was successfully verified.
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Rule 3045
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+B \cos (c+d x)+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{1}{2} a (12 B+5 C) \cos (c+d x)\right ) \, dx}{6 a}\\ &=\frac{a (12 B+5 C) \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{60 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+4 B+5 C)+\frac{1}{4} a^2 (120 A+156 B+115 C) \cos (c+d x)\right ) \, dx}{30 a}\\ &=\frac{a^2 (120 A+156 B+115 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{60 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+252 B+235 C)+\frac{3}{8} a^3 (680 A+628 B+545 C) \cos (c+d x)\right ) \, dx}{120 a}\\ &=\frac{a^3 (680 A+628 B+545 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{60 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+1132 B+1015 C)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^3 (1304 A+1132 B+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 (680 A+628 B+545 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{60 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+1132 B+1015 C)\right ) \int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^3 (1304 A+1132 B+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+1132 B+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 (680 A+628 B+545 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{60 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+1132 B+1015 C)\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx}{1024}\\ &=\frac{a^3 (1304 A+1132 B+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+1132 B+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 (680 A+628 B+545 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{60 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+1132 B+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+1132 B+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+1132 B+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+1132 B+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 (680 A+628 B+545 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{60 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 = 2.16488, size = 205, normalized size = 0.62 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (15 \sqrt{2} (1304 A+1132 B+1015 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} (2 (7240 A+7748 B+8085 C) \cos (c+d x)+4 (920 A+1324 B+1575 C) \cos (2 (c+d x))+480 A \cos (3 (c+d x))+23240 A+1392 B \cos (3 (c+d x))+192 B \cos (4 (c+d x))+22084 B+2140 C \cos (3 (c+d x))+560 C \cos (4 (c+d x))+80 C \cos (5 (c+d x))+20965 C)\right )}{15360 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.146, size = 841, normalized size = 2.5 \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 = 6.93179, size = 679, normalized size = 2.04 \begin{align*} \frac{{\left (1280 \, C a^{2} \cos \left (d x + c\right )^{5} + 128 \,{\left (12 \, B + 35 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 48 \,{\left (40 \, A + 116 \, B + 145 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \,{\left (920 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 10 \,{\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right ) + 15 \,{\left (1304 \, A + 1132 \, B + 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 ) - 15 \,{\left ({\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (1304 \, A + 1132 \, B + 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 )}{7680 \,{\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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