Optimal. Leaf size=169 \[ \frac{16 a^2 (21 A+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (21 A+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (21 A+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d} \]
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Rubi [A] time = 0.208198, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3024, 2751, 2647, 2646} \[ \frac{16 a^2 (21 A+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (21 A+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (21 A+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C (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{1}{2} a (9 A+7 C)-a C \cos (c+d x)\right ) \, dx}{9 a}\\ &=-\frac{4 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{1}{21} (21 A+13 C) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac{2 a (21 A+13 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac{4 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{1}{105} (8 a (21 A+13 C)) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 (21 A+13 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (21 A+13 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac{4 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{1}{315} \left (32 a^2 (21 A+13 C)\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{64 a^3 (21 A+13 C) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{16 a^2 (21 A+13 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (21 A+13 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac{4 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}\\ \end{align*}
Mathematica [A] time = 0.465865, size = 95, normalized size = 0.56 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (4 (588 A+779 C) \cos (c+d x)+4 (63 A+254 C) \cos (2 (c+d x))+7476 A+260 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+5653 C)}{1260 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 118, 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\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-540\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+ \left ( 63\,A+819\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -210\,A-630\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+315\,A+315\,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 = 2.17856, size = 209, normalized size = 1.24 \begin{align*} \frac{84 \,{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 25 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 150 \, \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 )} C \sqrt{a}}{2520 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63681, size = 290, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (35 \, C a^{2} \cos \left (d x + c\right )^{4} + 130 \, C a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (21 \, A + 73 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (147 \, A + 146 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (903 \, A + 584 \, C\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 (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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