Optimal. Leaf size=67 \[ \frac{(a \sin (c+d x)+a)^{13}}{13 a^5 d}-\frac{(a \sin (c+d x)+a)^{12}}{3 a^4 d}+\frac{4 (a \sin (c+d x)+a)^{11}}{11 a^3 d} \]
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Rubi [A] time = 0.084119, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac{(a \sin (c+d x)+a)^{13}}{13 a^5 d}-\frac{(a \sin (c+d x)+a)^{12}}{3 a^4 d}+\frac{4 (a \sin (c+d x)+a)^{11}}{11 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^{10} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{10}-4 a (a+x)^{11}+(a+x)^{12}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{4 (a+a \sin (c+d x))^{11}}{11 a^3 d}-\frac{(a+a \sin (c+d x))^{12}}{3 a^4 d}+\frac{(a+a \sin (c+d x))^{13}}{13 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.421885, size = 58, normalized size = 0.87 \[ -\frac{a^8 (\sin (c+d x)+1)^8 \left (33 \sin ^2(c+d x)-77 \sin (c+d x)+46\right ) \cos ^6(c+d x)}{429 d (\sin (c+d x)-1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 513, normalized size = 7.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.947403, size = 234, normalized size = 3.49 \begin{align*} \frac{33 \, a^{8} \sin \left (d x + c\right )^{13} + 286 \, a^{8} \sin \left (d x + c\right )^{12} + 1014 \, a^{8} \sin \left (d x + c\right )^{11} + 1716 \, a^{8} \sin \left (d x + c\right )^{10} + 715 \, a^{8} \sin \left (d x + c\right )^{9} - 2574 \, a^{8} \sin \left (d x + c\right )^{8} - 5148 \, a^{8} \sin \left (d x + c\right )^{7} - 3432 \, a^{8} \sin \left (d x + c\right )^{6} + 1287 \, a^{8} \sin \left (d x + c\right )^{5} + 4290 \, a^{8} \sin \left (d x + c\right )^{4} + 3718 \, a^{8} \sin \left (d x + c\right )^{3} + 1716 \, a^{8} \sin \left (d x + c\right )^{2} + 429 \, a^{8} \sin \left (d x + c\right )}{429 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04122, size = 400, normalized size = 5.97 \begin{align*} \frac{286 \, a^{8} \cos \left (d x + c\right )^{12} - 3432 \, a^{8} \cos \left (d x + c\right )^{10} + 10296 \, a^{8} \cos \left (d x + c\right )^{8} - 9152 \, a^{8} \cos \left (d x + c\right )^{6} +{\left (33 \, a^{8} \cos \left (d x + c\right )^{12} - 1212 \, a^{8} \cos \left (d x + c\right )^{10} + 6280 \, a^{8} \cos \left (d x + c\right )^{8} - 8512 \, a^{8} \cos \left (d x + c\right )^{6} + 768 \, a^{8} \cos \left (d x + c\right )^{4} + 1024 \, a^{8} \cos \left (d x + c\right )^{2} + 2048 \, a^{8}\right )} \sin \left (d x + c\right )}{429 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 120.691, size = 558, normalized size = 8.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28251, size = 296, normalized size = 4.42 \begin{align*} \frac{a^{8} \cos \left (12 \, d x + 12 \, c\right )}{3072 \, d} - \frac{3 \, a^{8} \cos \left (10 \, d x + 10 \, c\right )}{256 \, d} + \frac{27 \, a^{8} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} + \frac{155 \, a^{8} \cos \left (6 \, d x + 6 \, c\right )}{768 \, d} - \frac{475 \, a^{8} \cos \left (4 \, d x + 4 \, c\right )}{1024 \, d} - \frac{323 \, a^{8} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac{a^{8} \sin \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac{115 \, a^{8} \sin \left (11 \, d x + 11 \, c\right )}{45056 \, d} + \frac{205 \, a^{8} \sin \left (9 \, d x + 9 \, c\right )}{6144 \, d} - \frac{7 \, a^{8} \sin \left (7 \, d x + 7 \, c\right )}{2048 \, d} - \frac{2033 \, a^{8} \sin \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac{6137 \, a^{8} \sin \left (3 \, d x + 3 \, c\right )}{12288 \, d} + \frac{4845 \, a^{8} \sin \left (d x + c\right )}{1024 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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