Optimal. Leaf size=255 \[ \frac{2 \left (21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left (21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
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Rubi [A] time = 0.496418, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2990, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 \left (21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left (21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 2990
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))^3 (A+B \cos (c+d x)) \, dx &=\frac{2 b B \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 A+b B)+\frac{1}{2} \left (7 b^2 B+9 a (2 A b+a B)\right ) \cos (c+d x)+\frac{1}{2} b (9 A b+13 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b^2 (9 A b+13 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b B \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 A+b B)+\frac{9}{4} \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \cos (c+d x)+\frac{7}{4} b \left (27 a A b+22 a^2 B+7 b^2 B\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (27 a A b+22 a^2 B+7 b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b^2 (9 A b+13 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b B \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 (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right )+\frac{45}{8} \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b \left (27 a A b+22 a^2 B+7 b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b^2 (9 A b+13 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{7} \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b \left (27 a A b+22 a^2 B+7 b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b^2 (9 A b+13 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{21} \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b \left (27 a A b+22 a^2 B+7 b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b^2 (9 A b+13 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b B \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 = 1.15175, size = 197, normalized size = 0.77 \[ \frac{60 \left (21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+84 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (7 b \left (108 a^2 B+108 a A b+43 b^2 B\right ) \cos (c+d x)+5 \left (252 a^2 A b+84 a^3 B+18 b^2 (3 a B+A b) \cos (2 (c+d x))+234 a b^2 B+78 A b^3+7 b^3 B \cos (3 (c+d x))\right )\right )}{630 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.348, size = 745, 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 (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \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 (B b^{3} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (B a^{2} b + 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 )\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 (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \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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