Optimal. Leaf size=255 \[ \frac{2 \left (21 a^2 b C+5 a^3 B+21 a b^2 B+21 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 \left (9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a \left (5 a^2 B+21 a b C+18 b^2 B\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (7 a C+11 b B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.59507, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3029, 2989, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 \left (21 a^2 b C+5 a^3 B+21 a b^2 B+21 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 \left (9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a \left (5 a^2 B+21 a b C+18 b^2 B\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (7 a C+11 b B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2989
Rule 3031
Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx &=\int \frac{(a+b \cos (c+d x))^3 (B+C \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{2} a (11 b B+7 a C)+\frac{1}{2} \left (5 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)+\frac{1}{2} b (a B+7 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (11 b B+7 a C) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{4}{35} \int \frac{-\frac{5}{4} a \left (5 a^2 B+18 b^2 B+21 a b C\right )-\frac{7}{4} \left (9 a^2 b B+5 b^3 B+3 a^3 C+15 a b^2 C\right ) \cos (c+d x)-\frac{5}{4} b^2 (a B+7 b C) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (11 b B+7 a C) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (5 a^2 B+18 b^2 B+21 a b C\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{8}{105} \int \frac{-\frac{21}{8} \left (9 a^2 b B+5 b^3 B+3 a^3 C+15 a b^2 C\right )-\frac{5}{8} \left (5 a^3 B+21 a b^2 B+21 a^2 b C+21 b^3 C\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (11 b B+7 a C) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (5 a^2 B+18 b^2 B+21 a b C\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{1}{5} \left (-9 a^2 b B-5 b^3 B-3 a^3 C-15 a b^2 C\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx-\frac{1}{21} \left (-5 a^3 B-21 a b^2 B-21 a^2 b C-21 b^3 C\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (5 a^3 B+21 a b^2 B+21 a^2 b C+21 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a^2 (11 b B+7 a C) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (5 a^2 B+18 b^2 B+21 a b C\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (9 a^2 b B+5 b^3 B+3 a^3 C+15 a b^2 C\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{1}{5} \left (9 a^2 b B+5 b^3 B+3 a^3 C+15 a b^2 C\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (9 a^2 b B+5 b^3 B+3 a^3 C+15 a b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 a^3 B+21 a b^2 B+21 a^2 b C+21 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a^2 (11 b B+7 a C) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (5 a^2 B+18 b^2 B+21 a b C\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (9 a^2 b B+5 b^3 B+3 a^3 C+15 a b^2 C\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 3.42871, size = 221, normalized size = 0.87 \[ \frac{2 \left (5 \left (21 a^2 b C+5 a^3 B+21 a b^2 B+21 b^3 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-21 \left (9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{5 a \left (5 a^2 B+21 a b C+21 b^2 B\right ) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{21 \left (9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right ) \sin (c+d x)}{\sqrt{\cos (c+d x)}}+\frac{21 a^2 (a C+3 b B) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{15 a^3 B \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.678, size = 944, normalized size = 3.7 \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 \frac{{\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}}{\cos \left (d x + c\right )^{\frac{11}{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 (\frac{C b^{3} \cos \left (d x + c\right )^{4} + B a^{3} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (C a^{2} b + B a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac{9}{2}}}, 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 \frac{{\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}}{\cos \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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