Optimal. Leaf size=201 \[ -\frac {3003 a^8 x}{16}+\frac {1001 a^8 \cos ^5(c+d x)}{10 d}-\frac {3003 a^8 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {1001 a^8 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac {286 a^{14} \cos ^9(c+d x)}{3 d \left (a^2-a^2 \sin (c+d x)\right )^3}+\frac {143 a^{16} \cos ^7(c+d x)}{2 d \left (a^8-a^8 \sin (c+d x)\right )} \]
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Rubi [A]
time = 0.24, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2749, 2759,
2758, 2761, 2715, 8} \begin {gather*} \frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac {1001 a^8 \cos ^5(c+d x)}{10 d}-\frac {1001 a^8 \sin (c+d x) \cos ^3(c+d x)}{8 d}-\frac {3003 a^8 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {3003 a^8 x}{16}+\frac {143 a^{16} \cos ^7(c+d x)}{2 d \left (a^8-a^8 \sin (c+d x)\right )}+\frac {286 a^{14} \cos ^9(c+d x)}{3 d \left (a^2-a^2 \sin (c+d x)\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2749
Rule 2758
Rule 2759
Rule 2761
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^8 \, dx &=a^{16} \int \frac {\cos ^{14}(c+d x)}{(a-a \sin (c+d x))^8} \, dx\\ &=\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}-\left (13 a^{14}\right ) \int \frac {\cos ^{12}(c+d x)}{(a-a \sin (c+d x))^6} \, dx\\ &=\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}-\left (143 a^{12}\right ) \int \frac {\cos ^{10}(c+d x)}{(a-a \sin (c+d x))^4} \, dx\\ &=\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac {286 a^{11} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^3}-\left (429 a^{10}\right ) \int \frac {\cos ^8(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac {286 a^{11} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^3}+\frac {143 a^{10} \cos ^7(c+d x)}{2 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {1}{2} \left (1001 a^9\right ) \int \frac {\cos ^6(c+d x)}{a-a \sin (c+d x)} \, dx\\ &=\frac {1001 a^8 \cos ^5(c+d x)}{10 d}+\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac {286 a^{11} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^3}+\frac {143 a^{10} \cos ^7(c+d x)}{2 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {1}{2} \left (1001 a^8\right ) \int \cos ^4(c+d x) \, dx\\ &=\frac {1001 a^8 \cos ^5(c+d x)}{10 d}-\frac {1001 a^8 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac {286 a^{11} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^3}+\frac {143 a^{10} \cos ^7(c+d x)}{2 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {1}{8} \left (3003 a^8\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {1001 a^8 \cos ^5(c+d x)}{10 d}-\frac {3003 a^8 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {1001 a^8 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac {286 a^{11} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^3}+\frac {143 a^{10} \cos ^7(c+d x)}{2 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {1}{16} \left (3003 a^8\right ) \int 1 \, dx\\ &=-\frac {3003 a^8 x}{16}+\frac {1001 a^8 \cos ^5(c+d x)}{10 d}-\frac {3003 a^8 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {1001 a^8 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac {26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac {286 a^{11} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^3}+\frac {143 a^{10} \cos ^7(c+d x)}{2 d \left (a^2-a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 55, normalized size = 0.27 \begin {gather*} \frac {128 \sqrt {2} a^8 \, _2F_1\left (-\frac {13}{2},-\frac {1}{2};\frac {1}{2};\frac {1}{2} (1-\sin (c+d x))\right ) \sec (c+d x) \sqrt {1+\sin (c+d x)}}{d} \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. \(388\) vs.
\(2(189)=378\).
time = 0.13, size = 389, normalized size = 1.94
method | result | size |
risch | \(-\frac {3003 a^{8} x}{16}+\frac {173 a^{8} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {173 a^{8} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {256 a^{8}}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}+\frac {a^{8} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{8} \cos \left (5 d x +5 c \right )}{10 d}-\frac {61 a^{8} \sin \left (4 d x +4 c \right )}{64 d}-\frac {37 a^{8} \cos \left (3 d x +3 c \right )}{6 d}+\frac {2063 a^{8} \sin \left (2 d x +2 c \right )}{64 d}\) | \(149\) |
derivativedivides | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+8 a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+56 a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+56 a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+28 a^{8} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {8 a^{8}}{\cos \left (d x +c \right )}+a^{8} \tan \left (d x +c \right )}{d}\) | \(389\) |
default | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+8 a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+56 a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+56 a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+28 a^{8} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {8 a^{8}}{\cos \left (d x +c \right )}+a^{8} \tan \left (d x +c \right )}{d}\) | \(389\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 331, normalized size = 1.65 \begin {gather*} \frac {384 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} + \frac {5}{\cos \left (d x + c\right )} + 15 \, \cos \left (d x + c\right )\right )} a^{8} - 4480 \, {\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{8} - 5 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} a^{8} - 840 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{8} - 8400 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{8} - 6720 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{8} + 13440 \, a^{8} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 240 \, a^{8} \tan \left (d x + c\right ) + \frac {1920 \, a^{8}}{\cos \left (d x + c\right )}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 231, normalized size = 1.15 \begin {gather*} \frac {40 \, a^{8} \cos \left (d x + c\right )^{7} + 384 \, a^{8} \cos \left (d x + c\right )^{6} - 1526 \, a^{8} \cos \left (d x + c\right )^{5} - 6400 \, a^{8} \cos \left (d x + c\right )^{4} + 11865 \, a^{8} \cos \left (d x + c\right )^{3} - 45045 \, a^{8} d x + 46080 \, a^{8} \cos \left (d x + c\right )^{2} + 30720 \, a^{8} - 15 \, {\left (3003 \, a^{8} d x - 4027 \, a^{8}\right )} \cos \left (d x + c\right ) + {\left (40 \, a^{8} \cos \left (d x + c\right )^{6} - 344 \, a^{8} \cos \left (d x + c\right )^{5} - 1870 \, a^{8} \cos \left (d x + c\right )^{4} + 4530 \, a^{8} \cos \left (d x + c\right )^{3} + 45045 \, a^{8} d x + 16395 \, a^{8} \cos \left (d x + c\right )^{2} - 29685 \, a^{8} \cos \left (d x + c\right ) + 30720 \, a^{8}\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.49, size = 231, normalized size = 1.15 \begin {gather*} -\frac {45045 \, {\left (d x + c\right )} a^{8} + \frac {61440 \, a^{8}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} + \frac {2 \, {\left (14565 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 28800 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 50855 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 174720 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 36930 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 400640 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 36930 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 426240 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 50855 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 211584 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 14565 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 40064 \, a^{8}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.73, size = 513, normalized size = 2.55 \begin {gather*} -\frac {3003\,a^8\,x}{16}-\frac {\frac {3003\,a^8\,\left (c+d\,x\right )}{16}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3003\,a^8\,\left (c+d\,x\right )}{16}-\frac {a^8\,\left (45045\,c+45045\,d\,x-50998\right )}{240}\right )-\frac {a^8\,\left (45045\,c+45045\,d\,x-141568\right )}{240}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {3003\,a^8\,\left (c+d\,x\right )}{16}-\frac {a^8\,\left (45045\,c+45045\,d\,x-90570\right )}{240}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {9009\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (270270\,c+270270\,d\,x-86730\right )}{240}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {9009\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (270270\,c+270270\,d\,x-321458\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {9009\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (270270\,c+270270\,d\,x-527950\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {9009\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (270270\,c+270270\,d\,x-762678\right )}{240}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {45045\,a^8\,\left (c+d\,x\right )}{16}-\frac {a^8\,\left (675675\,c+675675\,d\,x-451150\right )}{240}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {45045\,a^8\,\left (c+d\,x\right )}{16}-\frac {a^8\,\left (675675\,c+675675\,d\,x-778620\right )}{240}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {15015\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (900900\,c+900900\,d\,x-875140\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {45045\,a^8\,\left (c+d\,x\right )}{16}-\frac {a^8\,\left (675675\,c+675675\,d\,x-1344900\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {45045\,a^8\,\left (c+d\,x\right )}{16}-\frac {a^8\,\left (675675\,c+675675\,d\,x-1672370\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {15015\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (900900\,c+900900\,d\,x-1956220\right )}{240}\right )}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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