Optimal. Leaf size=286 \[ \frac {4199 a^8 x}{1024}-\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac {4199 a^8 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {4199 a^2 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^3}{6336 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d} \]
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Rubi [A]
time = 0.28, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2757, 2748,
2715, 8} \begin {gather*} -\frac {4199 a^8 \cos ^5(c+d x)}{1920 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 ^3(c+d x)}{1536 d}+\frac {4199 a^8 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {4199 a^8 x}{1024}-\frac {4199 \cos ^5(c+d x) \left (a^4 \sin (c+d x)+a^4\right )^2}{4032 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 {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2748
Rule 2757
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*}
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Mathematica [A]
time = 1.98, size = 211, normalized size = 0.74 \begin {gather*} -\frac {a^8 \cos ^5(c+d x) \left (-29099070 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (-22470656+11469281 \sin (c+d x)+13958687 \sin ^2(c+d x)+20459158 \sin ^3(c+d x)+14283114 \sin ^4(c+d x)-8321928 \sin ^5(c+d x)-26346616 \sin ^6(c+d x)-20428112 \sin ^7(c+d x)-1239728 \sin ^8(c+d x)+9086336 \sin ^9(c+d x)+6969984 \sin ^{10}(c+d x)+2284800 \sin ^{11}(c+d x)+295680 \sin ^{12}(c+d x)\right )\right )}{3548160 d (-1+\sin (c+d x))^3 (1+\sin (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(534\) vs.
\(2(264)=528\).
time = 0.93, size = 535, normalized size = 1.87 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 339, normalized size = 1.19 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 150, normalized size = 0.52 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1280 vs.
\(2 (270) = 540\).
time = 4.04, size = 1280, normalized size = 4.48 \begin {gather*} \begin {cases} \frac {7 a^{8} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {21 a^{8} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {21 a^{8} x \sin ^{10}{\left (c + d x \right )}}{64} + \frac {105 a^{8} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {105 a^{8} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{64} + \frac {105 a^{8} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {35 a^{8} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {105 a^{8} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {105 a^{8} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {7 a^{8} x \sin ^{6}{\left (c + d x \right )}}{4} + \frac {105 a^{8} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {105 a^{8} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {315 a^{8} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {21 a^{8} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{8} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {21 a^{8} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {105 a^{8} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{64} + \frac {105 a^{8} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {21 a^{8} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4} + \frac {3 a^{8} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {7 a^{8} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {21 a^{8} x \cos ^{10}{\left (c + d x \right )}}{64} + \frac {105 a^{8} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {7 a^{8} x \cos ^{6}{\left (c + d x \right )}}{4} + \frac {3 a^{8} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {7 a^{8} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {119 a^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3072 d} + \frac {21 a^{8} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} - \frac {281 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{2560 d} + \frac {49 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{32 d} + \frac {105 a^{8} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} - \frac {8 a^{8} \sin ^{6}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {231 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{2560 d} - \frac {14 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {385 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {7 a^{8} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {48 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {56 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {119 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{3072 d} - \frac {49 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{32 d} - \frac {385 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} + \frac {14 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {3 a^{8} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {64 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{105 d} - \frac {32 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{5 d} - \frac {56 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a^{8} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} - \frac {21 a^{8} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{64 d} - \frac {105 a^{8} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {7 a^{8} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{4 d} + \frac {5 a^{8} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {128 a^{8} \cos ^{11}{\left (c + d x \right )}}{1155 d} - \frac {64 a^{8} \cos ^{9}{\left (c + d x \right )}}{45 d} - \frac {16 a^{8} \cos ^{7}{\left (c + d x \right )}}{5 d} - \frac {8 a^{8} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{8} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.15, size = 208, normalized size = 0.73 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.05, size = 684, normalized size = 2.39 \begin {gather*} \frac {4199\,a^8\,x}{1024}-\frac {\frac {1543\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{512}-\frac {1068767\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2560}-\frac {3297279\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2560}-\frac {168283\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3840}+\frac {256139\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{256}-\frac {256139\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{256}+\frac {168283\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{3840}+\frac {3297279\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{2560}+\frac {1068767\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2560}-\frac {1543\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{512}-\frac {3175\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{23}}{512}+a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}-\frac {43888}{3465}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}\,\left (12\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {12597\,c}{256}+\frac {12597\,d\,x}{256}-16\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (12\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {12597\,c}{256}+\frac {12597\,d\,x}{256}-\frac {157072}{1155}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}\,\left (66\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {138567\,c}{512}+\frac {138567\,d\,x}{512}-336\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (66\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {138567\,c}{512}+\frac {138567\,d\,x}{512}-\frac {52496}{105}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (220\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {230945\,c}{256}+\frac {230945\,d\,x}{256}-\frac {5584}{3}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (220\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {230945\,c}{256}+\frac {230945\,d\,x}{256}-\frac {58288}{63}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (792\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {415701\,c}{128}+\frac {415701\,d\,x}{128}-\frac {17696}{5}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (792\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {415701\,c}{128}+\frac {415701\,d\,x}{128}-\frac {227232}{35}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (924\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {969969\,c}{256}+\frac {969969\,d\,x}{256}-\frac {87776}{15}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (495\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {2078505\,c}{1024}+\frac {2078505\,d\,x}{1024}-3504\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (495\,a^8\,\left (\frac {4199\,c}{1024}+\frac {4199\,d\,x}{1024}\right )-a^8\,\left (\frac {2078505\,c}{1024}+\frac {2078505\,d\,x}{1024}-\frac {19360}{7}\right )\right )+\frac {3175\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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