Optimal. Leaf size=116 \[ \frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.0869676, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2751, 2647, 2646} \[ \frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx &=\frac{2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{5}{7} \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{7} (8 a) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{21} \left (32 a^2\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a+a \cos (c+d x)}}+\frac{16 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.225013, size = 84, normalized size = 0.72 \[ \frac{a^2 \left (315 \sin \left (\frac{1}{2} (c+d x)\right )+77 \sin \left (\frac{3}{2} (c+d x)\right )+3 \left (7 \sin \left (\frac{5}{2} (c+d x)\right )+\sin \left (\frac{7}{2} (c+d x)\right )\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{84 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.694, size = 86, normalized size = 0.7 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{21\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 6\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+3\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+4\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+8 \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 = 1.94901, size = 104, normalized size = 0.9 \begin{align*} \frac{{\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 )} \sqrt{a}}{84 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56016, size = 193, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (3 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 23 \, a^{2} \cos \left (d x + c\right ) + 46 \, a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{21 \,{\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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