Optimal. Leaf size=294 \[ \frac{2 a^3 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt{a \cos (c+d x)+a}}-\frac{4 a^2 (10439 A+9230 B+8368 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+9230 B+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac{2 a (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d} \]
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Rubi [A] time = 0.959381, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {3045, 2976, 2981, 2759, 2751, 2646} \[ \frac{2 a^3 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt{a \cos (c+d x)+a}}-\frac{4 a^2 (10439 A+9230 B+8368 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+9230 B+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac{2 a (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d} \]
Antiderivative was successfully verified.
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Rule 3045
Rule 2976
Rule 2981
Rule 2759
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac{2 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (13 A+6 C)+\frac{1}{2} a (13 B+5 C) \cos (c+d x)\right ) \, dx}{13 a}\\ &=\frac{2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac{4 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac{1}{4} a^2 (143 A+78 B+96 C)+\frac{1}{4} a^2 (143 A+182 B+136 C) \cos (c+d x)\right ) \, dx}{143 a}\\ &=\frac{2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac{2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac{8 \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \left (\frac{3}{8} a^3 (715 A+598 B+560 C)+\frac{1}{8} a^3 (2717 A+2522 B+2224 C) \cos (c+d x)\right ) \, dx}{1287 a}\\ &=\frac{2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac{2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac{\left (a^2 (10439 A+9230 B+8368 C)\right ) \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \, dx}{3003}\\ &=\frac{2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac{2 a (10439 A+9230 B+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac{2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac{(2 a (10439 A+9230 B+8368 C)) \int \left (\frac{3 a}{2}-a \cos (c+d x)\right ) \sqrt{a+a \cos (c+d x)} \, dx}{15015}\\ &=\frac{2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt{a+a \cos (c+d x)}}-\frac{4 a^2 (10439 A+9230 B+8368 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{45045 d}+\frac{2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac{2 a (10439 A+9230 B+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac{2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac{\left (a^2 (10439 A+9230 B+8368 C)\right ) \int \sqrt{a+a \cos (c+d x)} \, dx}{6435}\\ &=\frac{2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt{a+a \cos (c+d x)}}-\frac{4 a^2 (10439 A+9230 B+8368 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{45045 d}+\frac{2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac{2 a (10439 A+9230 B+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac{2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}\\ \end{align*}
Mathematica [A] time = 1.76446, size = 180, normalized size = 0.61 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (4 (445588 A+454285 B+453146 C) \cos (c+d x)+(581152 A+676000 B+746519 C) \cos (2 (c+d x))+148720 A \cos (3 (c+d x))+20020 A \cos (4 (c+d x))+3233516 A+225550 B \cos (3 (c+d x))+58240 B \cos (4 (c+d x))+8190 B \cos (5 (c+d x))+2980640 B+287060 C \cos (3 (c+d x))+94010 C \cos (4 (c+d x))+23940 C \cos (5 (c+d x))+3465 C \cos (6 (c+d x))+2798182 C)}{720720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 176, normalized size = 0.6 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{45045\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 55440\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{12}+ \left ( -32760\,B-262080\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{10}+ \left ( 20020\,A+140140\,B+520520\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}+ \left ( -77220\,A-244530\,B-566280\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 117117\,A+225225\,B+369369\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -90090\,A-120120\,B-150150\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+45045\,A+45045\,B+45045\,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 = 3.26115, size = 448, normalized size = 1.52 \begin{align*} \frac{572 \,{\left (35 \, \sqrt{2} a^{2} \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 225 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 756 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 2100 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 8190 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} + 130 \,{\left (63 \, \sqrt{2} a^{2} \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 385 \, \sqrt{2} a^{2} \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 1287 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 3465 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 8778 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 31878 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a} +{\left (3465 \, \sqrt{2} a^{2} \sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ) + 20475 \, \sqrt{2} a^{2} \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 70070 \, \sqrt{2} a^{2} \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 193050 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 459459 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 1066065 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3783780 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{1441440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00845, size = 498, normalized size = 1.69 \begin{align*} \frac{2 \,{\left (3465 \, C a^{2} \cos \left (d x + c\right )^{6} + 315 \,{\left (13 \, B + 38 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 35 \,{\left (143 \, A + 416 \, B + 523 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \,{\left (3718 \, A + 4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 4 \,{\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \,{\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45045 \,{\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(-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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