Optimal. Leaf size=175 \[ \frac{16 a^2 (15 A+13 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (15 A+13 B) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (9 A-2 B) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d} \]
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Rubi [A] time = 0.279669, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {2968, 3023, 2751, 2647, 2646} \[ \frac{16 a^2 (15 A+13 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (15 A+13 B) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (9 A-2 B) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3023
Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx &=\int (a+a \cos (c+d x))^{5/2} \left (A \cos (c+d x)+B \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 B (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{2 \int (a+a \cos (c+d x))^{5/2} \left (\frac{7 a B}{2}+\frac{1}{2} a (9 A-2 B) \cos (c+d x)\right ) \, dx}{9 a}\\ &=\frac{2 (9 A-2 B) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 B (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{1}{21} (15 A+13 B) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac{2 a (15 A+13 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac{2 (9 A-2 B) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 B (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{1}{105} (8 a (15 A+13 B)) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 (15 A+13 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (15 A+13 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac{2 (9 A-2 B) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 B (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{1}{315} \left (32 a^2 (15 A+13 B)\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{64 a^3 (15 A+13 B) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{16 a^2 (15 A+13 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (15 A+13 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac{2 (9 A-2 B) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 B (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}\\ \end{align*}
Mathematica [A] time = 0.670721, size = 105, normalized size = 0.6 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((3030 A+3116 B) \cos (c+d x)+8 (90 A+127 B) \cos (2 (c+d x))+90 A \cos (3 (c+d x))+6240 A+260 B \cos (3 (c+d x))+35 B \cos (4 (c+d x))+5653 B)}{1260 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.134, size = 123, normalized size = 0.7 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{315\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 140\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -90\,A-540\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 315\,A+819\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -420\,A-630\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+315\,A+315\,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 = 2.79148, size = 232, normalized size = 1.33 \begin{align*} \frac{30 \,{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 21 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 77 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 315 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} +{\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 )} B \sqrt{a}}{2520 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62873, size = 302, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (35 \, B a^{2} \cos \left (d x + c\right )^{4} + 5 \,{\left (9 \, A + 26 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (60 \, A + 73 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (345 \, A + 292 \, B\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (345 \, A + 292 \, B\right )} a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \,{\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{5}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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