Optimal. Leaf size=196 \[ \frac{2 a (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{10 b (11 A+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 b (11 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 b (11 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d} \]
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Rubi [A] time = 0.237284, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3034, 3023, 2748, 2635, 2639, 2641} \[ \frac{2 a (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{10 b (11 A+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 b (11 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 b (11 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d} \]
Antiderivative was successfully verified.
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Rule 3034
Rule 3023
Rule 2748
Rule 2635
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 b C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{2}{11} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{11 a A}{2}+\frac{1}{2} b (11 A+9 C) \cos (c+d x)+\frac{11}{2} a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 b C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{4}{99} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{11}{4} a (9 A+7 C)+\frac{9}{4} b (11 A+9 C) \cos (c+d x)\right ) \, dx\\ &=\frac{2 a C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 b C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{9} (a (9 A+7 C)) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{11} (b (11 A+9 C)) \int \cos ^{\frac{7}{2}}(c+d x) \, dx\\ &=\frac{2 a (9 A+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (11 A+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 a C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 b C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{15} (a (9 A+7 C)) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{77} (5 b (11 A+9 C)) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 a (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 b (11 A+9 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 a (9 A+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (11 A+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 a C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 b C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{231} (5 b (11 A+9 C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 b (11 A+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{10 b (11 A+9 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 a (9 A+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (11 A+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 a C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 b C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 1.64802, size = 134, normalized size = 0.68 \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)} (154 a (36 A+43 C) \cos (c+d x)+770 a C \cos (3 (c+d x))+180 b (11 A+16 C) \cos (2 (c+d x))+8580 A b+315 b C \cos (4 (c+d x))+7965 b C)+1848 a (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+600 b (11 A+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{13860 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.355, size = 481, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{5} + C a \cos \left (d x + c\right )^{4} + A b \cos \left (d x + c\right )^{3} + A a \cos \left (d x + c\right )^{2}\right )} \sqrt{\cos \left (d x + c\right )}, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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