Optimal. Leaf size=234 \[ \frac{2 a^2 (11 A+12 B) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (187 A+168 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (187 A+168 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{8 a (187 A+168 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a B \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d} \]
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Rubi [A] time = 0.529324, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2976, 2981, 2770, 2759, 2751, 2646} \[ \frac{2 a^2 (11 A+12 B) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (187 A+168 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (187 A+168 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{8 a (187 A+168 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a B \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d} \]
Antiderivative was successfully verified.
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Rule 2976
Rule 2981
Rule 2770
Rule 2759
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\frac{2 a B \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{2}{11} \int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \left (\frac{1}{2} a (11 A+8 B)+\frac{1}{2} a (11 A+12 B) \cos (c+d x)\right ) \, dx\\ &=\frac{2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a B \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{1}{99} (a (187 A+168 B)) \int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a B \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{1}{231} (2 a (187 A+168 B)) \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a B \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac{(4 (187 A+168 B)) \int \left (\frac{3 a}{2}-a \cos (c+d x)\right ) \sqrt{a+a \cos (c+d x)} \, dx}{1155}\\ &=\frac{2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}-\frac{8 a (187 A+168 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac{2 a B \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac{1}{495} (2 a (187 A+168 B)) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt{a+a \cos (c+d x)}}-\frac{8 a (187 A+168 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac{2 a B \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac{4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}\\ \end{align*}
Mathematica [A] time = 0.956185, size = 125, normalized size = 0.53 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((35156 A+34734 B) \cos (c+d x)+8 (1507 A+1743 B) \cos (2 (c+d x))+3740 A \cos (3 (c+d x))+770 A \cos (4 (c+d x))+59158 A+4935 B \cos (3 (c+d x))+1470 B \cos (4 (c+d x))+315 B \cos (5 (c+d x))+55482 B)}{27720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.046, size = 142, normalized size = 0.6 \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{3465\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( -5040\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+ \left ( 3080\,A+18480\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}+ \left ( -9900\,A-27720\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 12474\,A+22176\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -8085\,A-10395\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+3465\,A+3465\,B \right ){\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.00005, size = 250, normalized size = 1.07 \begin{align*} \frac{22 \,{\left (35 \, \sqrt{2} a \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 135 \, \sqrt{2} a \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 378 \, \sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 1050 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3780 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} + 21 \,{\left (15 \, \sqrt{2} a \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 55 \, \sqrt{2} a \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 165 \, \sqrt{2} a \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 429 \, \sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 990 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3630 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a}}{55440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33085, size = 351, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (315 \, B a \cos \left (d x + c\right )^{5} + 35 \,{\left (11 \, A + 21 \, B\right )} a \cos \left (d x + c\right )^{4} + 5 \,{\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right )^{3} + 6 \,{\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right )^{2} + 8 \,{\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right ) + 16 \,{\left (187 \, A + 168 \, B\right )} a\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 (B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \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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