Optimal. Leaf size=223 \[ \frac{2 \left (7 a^2 B+10 a b C+5 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (7 a^2 B+10 a b C+5 b^2 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left (9 a (a C+2 b B)+7 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (9 a (a C+2 b B)+7 b^2 C\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b (11 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d} \]
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Rubi [A] time = 0.423099, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3029, 2990, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 \left (7 a^2 B+10 a b C+5 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (7 a^2 B+10 a b C+5 b^2 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left (9 a (a C+2 b B)+7 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (9 a (a C+2 b B)+7 b^2 C\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b (11 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2990
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))^2 \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))^2 (B+C \cos (c+d x)) \, dx\\ &=\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac{2}{9} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{1}{2} a (9 a B+5 b C)+\frac{1}{2} \left (7 b^2 C+9 a (2 b B+a C)\right ) \cos (c+d x)+\frac{1}{2} b (9 b B+11 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (9 b B+11 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac{4}{63} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{4} \left (7 a^2 B+5 b^2 B+10 a b C\right )+\frac{7}{4} \left (7 b^2 C+9 a (2 b B+a C)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b (9 b B+11 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac{1}{7} \left (7 a^2 B+5 b^2 B+10 a b C\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{9} \left (7 b^2 C+9 a (2 b B+a C)\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 \left (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 (7 b^2 C+9 a (2 b B+a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (9 b B+11 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac{1}{21} \left (7 a^2 B+5 b^2 B+10 a b C\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left (7 b^2 C+9 a (2 b B+a C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (7 b^2 C+9 a (2 b B+a C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (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 (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 (7 b^2 C+9 a (2 b B+a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b (9 b B+11 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 1.38698, size = 167, normalized size = 0.75 \[ \frac{60 \left (7 a^2 B+10 a b C+5 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+84 \left (9 a^2 C+18 a b B+7 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (7 \left (36 a^2 C+72 a b B+43 b^2 C\right ) \cos (c+d x)+5 \left (84 a^2 B+18 b (2 a C+b B) \cos (2 (c+d x))+156 a b C+78 b^2 B+7 b^2 C \cos (3 (c+d x))\right )\right )}{630 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.641, size = 610, 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 )}^{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} + B a^{2} \cos \left (d x + c\right ) +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{2} + 2 \, B a 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] 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 )}^{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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