Optimal. Leaf size=305 \[ \frac{2 \left (165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (26 a^2 C+33 a b B+9 b^2 C\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left (27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left (165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]
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Rubi [A] time = 0.638124, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3029, 2990, 3033, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 \left (165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (26 a^2 C+33 a b B+9 b^2 C\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left (27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left (165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2990
Rule 3033
Rule 3023
Rule 2748
Rule 2635
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 (B+C \cos (c+d x)) \, dx\\ &=\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{2}{11} \int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\frac{1}{2} a (11 a B+5 b C)+\frac{1}{2} \left (9 b^2 C+11 a (2 b B+a C)\right ) \cos (c+d x)+\frac{1}{2} b (11 b B+15 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b^2 (11 b B+15 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{4}{99} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{4} a^2 (11 a B+5 b C)+\frac{11}{4} \left (27 a^2 b B+7 b^3 B+9 a^3 C+21 a b^2 C\right ) \cos (c+d x)+\frac{9}{4} b \left (33 a b B+26 a^2 C+9 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (33 a b B+26 a^2 C+9 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b^2 (11 b B+15 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{8}{693} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{8} \left (77 a^3 B+165 a b^2 B+165 a^2 b C+45 b^3 C\right )+\frac{77}{8} \left (27 a^2 b B+7 b^3 B+9 a^3 C+21 a b^2 C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b \left (33 a b B+26 a^2 C+9 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b^2 (11 b B+15 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{1}{9} \left (27 a^2 b B+7 b^3 B+9 a^3 C+21 a b^2 C\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{77} \left (77 a^3 B+165 a b^2 B+165 a^2 b C+45 b^3 C\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (77 a^3 B+165 a b^2 B+165 a^2 b C+45 b^3 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (27 a^2 b B+7 b^3 B+9 a^3 C+21 a b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b \left (33 a b B+26 a^2 C+9 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b^2 (11 b B+15 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{1}{15} \left (27 a^2 b B+7 b^3 B+9 a^3 C+21 a b^2 C\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (77 a^3 B+165 a b^2 B+165 a^2 b C+45 b^3 C\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (27 a^2 b B+7 b^3 B+9 a^3 C+21 a b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (77 a^3 B+165 a b^2 B+165 a^2 b C+45 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (77 a^3 B+165 a b^2 B+165 a^2 b C+45 b^3 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (27 a^2 b B+7 b^3 B+9 a^3 C+21 a b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b \left (33 a b B+26 a^2 C+9 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b^2 (11 b B+15 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 1.91069, size = 235, normalized size = 0.77 \[ \frac{240 \left (165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3696 \left (27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (180 b \left (33 a^2 C+33 a b B+16 b^2 C\right ) \cos (2 (c+d x))+154 \left (108 a^2 b B+36 a^3 C+129 a b^2 C+43 b^3 B\right ) \cos (c+d x)+15 \left (1716 a^2 b C+616 a^3 B+1716 a b^2 B+21 b^3 C \cos (4 (c+d x))+531 b^3 C\right )+770 b^2 (3 a C+b B) \cos (3 (c+d x))\right )}{27720 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.684, size = 825, normalized size = 2.7 \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} + B \cos \left (d x + c\right )\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt{\cos \left (d x + c\right )}\,{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^{3} \cos \left (d x + c\right )^{5} + B a^{3} \cos \left (d x + c\right ) +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (C a^{2} b + B a b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, B a^{2} b\right )} \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(-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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