Optimal. Leaf size=251 \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^2 B+14 a A b+10 a b C+5 b^2 B\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right )}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right )}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (7 a^2 B+14 a A b+10 a b C+5 b^2 B\right )}{21 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
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Rubi [A] time = 0.537068, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^2 B+14 a A b+10 a b C+5 b^2 B\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right )}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right )}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (7 a^2 B+14 a A b+10 a b C+5 b^2 B\right )}{21 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2}{9} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac{3}{2} a (3 A+C)+\frac{1}{2} (9 A b+9 a B+7 b C) \cos (c+d x)+\frac{1}{2} (9 b B+4 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (9 b B+4 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{4}{63} \int \sqrt{\cos (c+d x)} \left (\frac{21}{4} a^2 (3 A+C)+\frac{9}{4} \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \cos (c+d x)+\frac{7}{4} \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (9 b B+4 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8}{315} \int \sqrt{\cos (c+d x)} \left (\frac{21}{8} \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right )+\frac{45}{8} \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (9 b B+4 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{7} \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (9 b B+4 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{21} \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (9 b B+4 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 1.26707, size = 195, normalized size = 0.78 \[ \frac{60 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^2 B+2 a b (7 A+5 C)+5 b^2 B\right )+84 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (7 \cos (c+d x) \left (36 a^2 C+72 a b B+36 A b^2+43 b^2 C\right )+5 \left (84 a^2 B+168 a A b+18 b (2 a C+b B) \cos (2 (c+d x))+156 a b C+78 b^2 B+7 b^2 C \cos (3 (c+d x))\right )\right )}{630 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.02, size = 784, normalized size = 3.1 \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] 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 ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \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^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\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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