Optimal. Leaf size=357 \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (5 A+7 C)+5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)\right )}{21 d}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{15 d}+\frac{2 \sin (c+d x) \left (9 a^2 b (5 A+7 C)+15 a^3 B+54 a b^2 B+8 A b^3\right )}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)} \]
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Rubi [A] time = 0.923861, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3047, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (5 A+7 C)+5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)\right )}{21 d}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{15 d}+\frac{2 \sin (c+d x) \left (9 a^2 b (5 A+7 C)+15 a^3 B+54 a b^2 B+8 A b^3\right )}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3047
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 (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2}{9} \int \frac{(a+b \cos (c+d x))^2 \left (\frac{3}{2} (2 A b+3 a B)+\frac{1}{2} (7 a A+9 b B+9 a C) \cos (c+d x)+\frac{1}{2} b (A+9 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 (2 A b+3 a B) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4}{63} \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{4} \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right )+\frac{1}{4} \left (86 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \cos (c+d x)+\frac{1}{4} b (13 A b+9 a B+63 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (2 A b+3 a B) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{8}{315} \int \frac{-\frac{15}{8} \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right )-\frac{21}{8} \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \cos (c+d x)-\frac{5}{8} b^2 (13 A b+9 a B+63 b C) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (2 A b+3 a B) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{16}{945} \int \frac{-\frac{63}{16} \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right )-\frac{45}{16} \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (2 A b+3 a B) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{1}{21} \left (-5 a^3 B-21 a b^2 B-7 b^3 (A+3 C)-3 a^2 b (5 A+7 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{15} \left (-27 a^2 b B-15 b^3 B-9 a b^2 (3 A+5 C)-a^3 (7 A+9 C)\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (2 A b+3 a B) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{1}{15} \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (2 A b+3 a B) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 6.90144, size = 414, normalized size = 1.16 \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (75 a^2 A b+105 a^2 b C+25 a^3 B+105 a b^2 B+35 A b^3+105 b^3 C\right )+2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-49 a^3 A-189 a^2 b B-63 a^3 C-189 a A b^2-315 a b^2 C-105 b^3 B\right )}{105 d}+\frac{\sqrt{\cos (c+d x)} \left (\frac{2}{45} \sec ^3(c+d x) \left (7 a^3 A \sin (c+d x)+27 a^2 b B \sin (c+d x)+9 a^3 C \sin (c+d x)+27 a A b^2 \sin (c+d x)\right )+\frac{2}{21} \sec ^2(c+d x) \left (15 a^2 A b \sin (c+d x)+21 a^2 b C \sin (c+d x)+5 a^3 B \sin (c+d x)+21 a b^2 B \sin (c+d x)+7 A b^3 \sin (c+d x)\right )+\frac{2}{15} \sec (c+d x) \left (7 a^3 A \sin (c+d x)+27 a^2 b B \sin (c+d x)+9 a^3 C \sin (c+d x)+27 a A b^2 \sin (c+d x)+45 a b^2 C \sin (c+d x)+15 b^3 B \sin (c+d x)\right )+\frac{2}{7} \sec ^4(c+d x) \left (3 a^2 A b \sin (c+d x)+a^3 B \sin (c+d x)\right )+\frac{2}{9} a^3 A \tan (c+d x) \sec ^4(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.806, size = 1292, normalized size = 3.6 \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 ) + A\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 )^{5} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac{11}{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 ) + A\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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