Optimal. Leaf size=205 \[ \frac{2 \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (4 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a b (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a b (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
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Rubi [A] time = 0.424298, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3050, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (4 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a b (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a b (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 3050
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+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} b (9 A+7 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{8 a b 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}{2} a b (7 A+5 C) \cos (c+d x)+\frac{7}{4} \left (4 a^2 C+b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (4 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{8 a b 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 (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right )+\frac{45}{4} a b (7 A+5 C) \cos (c+d x)\right ) \, dx\\ &=\frac{2 \left (4 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{8 a b 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} (2 a b (7 A+5 C)) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (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{4 a b (7 A+5 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 \left (4 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{8 a b 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} (2 a b (7 A+5 C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (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{4 a b (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a b (7 A+5 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 \left (4 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{8 a b 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 = 1.18616, size = 148, normalized size = 0.72 \[ \frac{84 \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (7 \left (36 a^2 C+36 A b^2+43 b^2 C\right ) \cos (c+d x)+5 b (168 a A+36 a C \cos (2 (c+d x))+156 a C+7 b C \cos (3 (c+d x)))\right )+120 a b (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{630 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.401, size = 587, normalized size = 2.9 \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] 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 )}^{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} + 2 \, C a b \cos \left (d x + c\right )^{3} + 2 \, A a b \cos \left (d x + c\right ) + A a^{2} +{\left (C a^{2} + A b^{2}\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] 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 )}^{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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