Optimal. Leaf size=286 \[ -\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac{4199 a^8 \sin (c+d x) \cos ^3(c+d x)}{1536 d}-\frac{323 a^3 \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{132 d}-\frac{4199 a^2 \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^3}{6336 d}-\frac{323 \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left (a^8 \sin (c+d x)+a^8\right )}{2688 d}+\frac{4199 a^8 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{4199 a^8 x}{1024}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d} \]
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Rubi [A] time = 0.403442, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2678, 2669, 2635, 8} \[ -\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac{4199 a^8 \sin (c+d x) \cos ^3(c+d x)}{1536 d}-\frac{323 a^3 \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{132 d}-\frac{4199 a^2 \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^3}{6336 d}-\frac{323 \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left (a^8 \sin (c+d x)+a^8\right )}{2688 d}+\frac{4199 a^8 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{4199 a^8 x}{1024}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d} \]
Antiderivative was successfully verified.
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Rule 2678
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx &=-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}+\frac{1}{12} (19 a) \int \cos ^4(c+d x) (a+a \sin (c+d x))^7 \, dx\\ &=-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}+\frac{1}{132} \left (323 a^2\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^6 \, dx\\ &=-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}+\frac{1}{88} \left (323 a^3\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^5 \, dx\\ &=-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac{323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}+\frac{1}{792} \left (4199 a^4\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^4 \, dx\\ &=-\frac{4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac{323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}+\frac{1}{576} \left (4199 a^5\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac{4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac{323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}+\frac{1}{448} \left (4199 a^6\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac{323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}+\frac{1}{384} \left (4199 a^7\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}-\frac{4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac{323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}+\frac{1}{384} \left (4199 a^8\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac{4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac{4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac{323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}+\frac{1}{512} \left (4199 a^8\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac{4199 a^8 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac{4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac{323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}+\frac{\left (4199 a^8\right ) \int 1 \, dx}{1024}\\ &=\frac{4199 a^8 x}{1024}-\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac{4199 a^8 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac{4199 a^5 \cos ^5(c+d x) (a+a \sin (c+d x))^3}{6336 d}-\frac{323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac{323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d}\\ \end{align*}
Mathematica [A] time = 3.16037, size = 211, normalized size = 0.74 \[ -\frac{a^8 \left (\sqrt{\sin (c+d x)+1} \left (295680 \sin ^{12}(c+d x)+2284800 \sin ^{11}(c+d x)+6969984 \sin ^{10}(c+d x)+9086336 \sin ^9(c+d x)-1239728 \sin ^8(c+d x)-20428112 \sin ^7(c+d x)-26346616 \sin ^6(c+d x)-8321928 \sin ^5(c+d x)+14283114 \sin ^4(c+d x)+20459158 \sin ^3(c+d x)+13958687 \sin ^2(c+d x)+11469281 \sin (c+d x)-22470656\right )-29099070 \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right ) \sqrt{1-\sin (c+d x)}\right ) \cos ^5(c+d x)}{3548160 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 535, normalized size = 1.9 \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 [A] time = 0.995742, size = 458, normalized size = 1.6 \begin{align*} -\frac{45416448 \, a^{8} \cos \left (d x + c\right )^{5} - 196608 \,{\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 5046272 \,{\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{8} - 45416448 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 231 \,{\left (384 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 20 \, \sin \left (4 \, d x + 4 \, c\right )^{3} - 840 \, d x - 840 \, c - 15 \, \sin \left (8 \, d x + 8 \, c\right ) + 240 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 77616 \,{\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 4139520 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 1940400 \,{\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 887040 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8}}{28385280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0788, size = 450, normalized size = 1.57 \begin{align*} \frac{2580480 \, a^{8} \cos \left (d x + c\right )^{11} - 31539200 \, a^{8} \cos \left (d x + c\right )^{9} + 97320960 \, a^{8} \cos \left (d x + c\right )^{7} - 90832896 \, a^{8} \cos \left (d x + c\right )^{5} + 14549535 \, a^{8} d x + 231 \,{\left (1280 \, a^{8} \cos \left (d x + c\right )^{11} - 47744 \, a^{8} \cos \left (d x + c\right )^{9} + 253488 \, a^{8} \cos \left (d x + c\right )^{7} - 359624 \, a^{8} \cos \left (d x + c\right )^{5} + 41990 \, a^{8} \cos \left (d x + c\right )^{3} + 62985 \, a^{8} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 86.4144, size = 1280, normalized size = 4.48 \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 [A] time = 1.26822, size = 281, normalized size = 0.98 \begin{align*} \frac{4199}{1024} \, a^{8} x + \frac{a^{8} \cos \left (11 \, d x + 11 \, c\right )}{1408 \, d} - \frac{31 \, a^{8} \cos \left (9 \, d x + 9 \, c\right )}{1152 \, d} + \frac{139 \, a^{8} \cos \left (7 \, d x + 7 \, c\right )}{896 \, d} + \frac{171 \, a^{8} \cos \left (5 \, d x + 5 \, c\right )}{640 \, d} - \frac{323 \, a^{8} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac{323 \, a^{8} \cos \left (d x + c\right )}{64 \, d} + \frac{a^{8} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac{29 \, a^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{673 \, a^{8} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac{361 \, a^{8} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac{8721 \, a^{8} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac{323 \, a^{8} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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