Optimal. Leaf size=223 \[ \frac{2 a (99 A+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (99 A+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+80 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}+\frac{2 a C \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.50594, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {3046, 2981, 2770, 2759, 2751, 2646} \[ \frac{2 a (99 A+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (99 A+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+80 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}+\frac{2 a C \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2981
Rule 2770
Rule 2759
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{2 \int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \left (\frac{1}{2} a (11 A+8 C)+\frac{1}{2} a C \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 a C \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{1}{99} (99 A+80 C) \int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a (99 A+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a C \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{1}{231} (2 (99 A+80 C)) \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a (99 A+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a C \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{4 (99 A+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d}+\frac{(4 (99 A+80 C)) \int \left (\frac{3 a}{2}-a \cos (c+d x)\right ) \sqrt{a+a \cos (c+d x)} \, dx}{1155 a}\\ &=\frac{2 a (99 A+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a C \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}-\frac{8 (99 A+80 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac{2 C \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{4 (99 A+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d}+\frac{1}{495} (2 (99 A+80 C)) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (99 A+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a C \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}-\frac{8 (99 A+80 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac{2 C \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{4 (99 A+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d}\\ \end{align*}
Mathematica [A] time = 0.936908, size = 114, normalized size = 0.51 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (2 (9306 A+9095 C) \cos (c+d x)+16 (297 A+415 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+30096 A+3175 C \cos (3 (c+d x))+700 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+26420 C)}{27720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.272, size = 135, normalized size = 0.6 \begin{align*}{\frac{2\,a\sqrt{2}}{3465\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( -10080\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+30800\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -3960\,A-39600\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 8316\,A+27720\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -6930\,A-11550\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+3465\,A+3465\,C \right ){\frac{1}{\sqrt{a \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.12468, size = 216, normalized size = 0.97 \begin{align*} \frac{396 \,{\left (5 \, \sqrt{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 7 \, \sqrt{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 35 \, \sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 105 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} + 5 \,{\left (63 \, \sqrt{2} \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 77 \, \sqrt{2} \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 495 \, \sqrt{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 693 \, \sqrt{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 2310 \, \sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 6930 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{55440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31158, size = 308, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (315 \, C \cos \left (d x + c\right )^{5} + 350 \, C \cos \left (d x + c\right )^{4} + 5 \,{\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right )^{3} + 6 \,{\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right ) + 1584 \, A + 1280 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \,{\left (d \cos \left (d x + c\right ) + d\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 )} \sqrt{a \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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