Optimal. Leaf size=181 \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac{a^3 \sin ^{n+8}(c+d x)}{d (n+8)} \]
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Rubi [A] time = 0.180944, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2836, 88} \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac{a^3 \sin ^{n+8}(c+d x)}{d (n+8)} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 88
Rubi steps
\begin{align*} \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 \left (\frac{x}{a}\right )^n (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^7 \left (\frac{x}{a}\right )^n+3 a^7 \left (\frac{x}{a}\right )^{1+n}+a^7 \left (\frac{x}{a}\right )^{2+n}-5 a^7 \left (\frac{x}{a}\right )^{3+n}-5 a^7 \left (\frac{x}{a}\right )^{4+n}+a^7 \left (\frac{x}{a}\right )^{5+n}+3 a^7 \left (\frac{x}{a}\right )^{6+n}+a^7 \left (\frac{x}{a}\right )^{7+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac{3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac{a^3 \sin ^{3+n}(c+d x)}{d (3+n)}-\frac{5 a^3 \sin ^{4+n}(c+d x)}{d (4+n)}-\frac{5 a^3 \sin ^{5+n}(c+d x)}{d (5+n)}+\frac{a^3 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac{3 a^3 \sin ^{7+n}(c+d x)}{d (7+n)}+\frac{a^3 \sin ^{8+n}(c+d x)}{d (8+n)}\\ \end{align*}
Mathematica [A] time = 0.564413, size = 123, normalized size = 0.68 \[ \frac{a^3 \sin ^{n+1}(c+d x) \left (\frac{\sin ^7(c+d x)}{n+8}+\frac{3 \sin ^6(c+d x)}{n+7}+\frac{\sin ^5(c+d x)}{n+6}-\frac{5 \sin ^4(c+d x)}{n+5}-\frac{5 \sin ^3(c+d x)}{n+4}+\frac{\sin ^2(c+d x)}{n+3}+\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 11.083, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{n} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.48964, size = 1554, normalized size = 8.59 \begin{align*} \frac{{\left ({\left (a^{3} n^{7} + 28 \, a^{3} n^{6} + 322 \, a^{3} n^{5} + 1960 \, a^{3} n^{4} + 6769 \, a^{3} n^{3} + 13132 \, a^{3} n^{2} + 13068 \, a^{3} n + 5040 \, a^{3}\right )} \cos \left (d x + c\right )^{8} + 32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} -{\left (5 \, a^{3} n^{7} + 142 \, a^{3} n^{6} + 1654 \, a^{3} n^{5} + 10180 \, a^{3} n^{4} + 35485 \, a^{3} n^{3} + 69358 \, a^{3} n^{2} + 69416 \, a^{3} n + 26880 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 23520 \, a^{3} n^{2} + 2 \,{\left (2 \, a^{3} n^{7} + 49 \, a^{3} n^{6} + 470 \, a^{3} n^{5} + 2230 \, a^{3} n^{4} + 5438 \, a^{3} n^{3} + 6361 \, a^{3} n^{2} + 2730 \, a^{3} n\right )} \cos \left (d x + c\right )^{4} + 39968 \, a^{3} n + 21840 \, a^{3} + 8 \,{\left (2 \, a^{3} n^{6} + 45 \, a^{3} n^{5} + 380 \, a^{3} n^{4} + 1470 \, a^{3} n^{3} + 2498 \, a^{3} n^{2} + 1365 \, a^{3} n\right )} \cos \left (d x + c\right )^{2} +{\left (32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} - 3 \,{\left (a^{3} n^{7} + 29 \, a^{3} n^{6} + 343 \, a^{3} n^{5} + 2135 \, a^{3} n^{4} + 7504 \, a^{3} n^{3} + 14756 \, a^{3} n^{2} + 14832 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 24000 \, a^{3} n^{2} + 2 \,{\left (2 \, a^{3} n^{7} + 53 \, a^{3} n^{6} + 566 \, a^{3} n^{5} + 3155 \, a^{3} n^{4} + 9908 \, a^{3} n^{3} + 17492 \, a^{3} n^{2} + 15984 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{4} + 44288 \, a^{3} n + 30720 \, a^{3} + 8 \,{\left (2 \, a^{3} n^{6} + 47 \, a^{3} n^{5} + 425 \, a^{3} n^{4} + 1880 \, a^{3} n^{3} + 4268 \, a^{3} n^{2} + 4688 \, a^{3} n + 1920 \, a^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 36 \, d n^{7} + 546 \, d n^{6} + 4536 \, d n^{5} + 22449 \, d n^{4} + 67284 \, d n^{3} + 118124 \, d n^{2} + 109584 \, d n + 40320 \, d} \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 [B] time = 1.23205, size = 1038, normalized size = 5.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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