Optimal. Leaf size=179 \[ \frac {1155 a^8 x}{8}-\frac {385 a^8 \cos ^3(c+d x)}{4 d}+\frac {1155 a^8 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac {66 a^{14} \cos ^7(c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )^3}-\frac {231 a^{16} \cos ^5(c+d x)}{4 d \left (a^8-a^8 \sin (c+d x)\right )} \]
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Rubi [A]
time = 0.21, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac {385 a^8 \cos ^3(c+d x)}{4 d}+\frac {1155 a^8 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1155 a^8 x}{8}-\frac {231 a^{16} \cos ^5(c+d x)}{4 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {66 a^{14} \cos ^7(c+d x)}{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 ^4(c+d x) (a+a \sin (c+d x))^8 \, dx &=a^{16} \int \frac {\cos ^{12}(c+d x)}{(a-a \sin (c+d x))^8} \, dx\\ &=\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {1}{3} \left (11 a^{14}\right ) \int \frac {\cos ^{10}(c+d x)}{(a-a \sin (c+d x))^6} \, dx\\ &=\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}+\left (33 a^{12}\right ) \int \frac {\cos ^8(c+d x)}{(a-a \sin (c+d x))^4} \, dx\\ &=\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac {66 a^{11} \cos ^7(c+d x)}{d (a-a \sin (c+d x))^3}+\left (231 a^{10}\right ) \int \frac {\cos ^6(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac {66 a^{11} \cos ^7(c+d x)}{d (a-a \sin (c+d x))^3}-\frac {231 a^{10} \cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}+\frac {1}{4} \left (1155 a^9\right ) \int \frac {\cos ^4(c+d x)}{a-a \sin (c+d x)} \, dx\\ &=-\frac {385 a^8 \cos ^3(c+d x)}{4 d}+\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac {66 a^{11} \cos ^7(c+d x)}{d (a-a \sin (c+d x))^3}-\frac {231 a^{10} \cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}+\frac {1}{4} \left (1155 a^8\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {385 a^8 \cos ^3(c+d x)}{4 d}+\frac {1155 a^8 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac {66 a^{11} \cos ^7(c+d x)}{d (a-a \sin (c+d x))^3}-\frac {231 a^{10} \cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}+\frac {1}{8} \left (1155 a^8\right ) \int 1 \, dx\\ &=\frac {1155 a^8 x}{8}-\frac {385 a^8 \cos ^3(c+d x)}{4 d}+\frac {1155 a^8 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac {66 a^{11} \cos ^7(c+d x)}{d (a-a \sin (c+d x))^3}-\frac {231 a^{10} \cos ^5(c+d x)}{4 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 = 59, normalized size = 0.33 \begin {gather*} \frac {64 \sqrt {2} a^8 \, _2F_1\left (-\frac {11}{2},-\frac {3}{2};-\frac {1}{2};\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^3(c+d x) (1+\sin (c+d x))^{3/2}}{3 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. \(477\) vs.
\(2(167)=334\).
time = 0.20, size = 478, normalized size = 2.67
method | result | size |
risch | \(\frac {1155 a^{8} x}{8}+\frac {31 i a^{8} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {39 a^{8} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {39 a^{8} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {31 i a^{8} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {128 \left (-15 i a^{8} {\mathrm e}^{i \left (d x +c \right )}+9 a^{8} {\mathrm e}^{2 i \left (d x +c \right )}-8 a^{8}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {a^{8} \sin \left (4 d x +4 c \right )}{32 d}+\frac {2 a^{8} \cos \left (3 d x +3 c \right )}{3 d}\) | \(166\) |
derivativedivides | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \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}{8}+\frac {35 c}{8}\right )+8 a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \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 )}{3}\right )+28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \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 )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+56 a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\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 {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {28 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {8 a^{8}}{3 \cos \left (d x +c \right )^{3}}-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(478\) |
default | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \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}{8}+\frac {35 c}{8}\right )+8 a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \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 )}{3}\right )+28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \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 )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+56 a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\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 {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {28 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {8 a^{8}}{3 \cos \left (d x +c \right )^{3}}-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(478\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 311, normalized size = 1.74 \begin {gather*} \frac {224 \, a^{8} \tan \left (d x + c\right )^{3} + 64 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a^{8} + {\left (8 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - \frac {3 \, {\left (13 \, \tan \left (d x + c\right )^{3} + 11 \, \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 72 \, \tan \left (d x + c\right )\right )} a^{8} + 112 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{8} + 560 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{8} + 8 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{8} - 448 \, a^{8} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {448 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{8}}{\cos \left (d x + c\right )^{3}} + \frac {64 \, a^{8}}{\cos \left (d x + c\right )^{3}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 247, normalized size = 1.38 \begin {gather*} -\frac {6 \, a^{8} \cos \left (d x + c\right )^{6} - 52 \, a^{8} \cos \left (d x + c\right )^{5} - 317 \, a^{8} \cos \left (d x + c\right )^{4} + 1286 \, a^{8} \cos \left (d x + c\right )^{3} + 6930 \, a^{8} d x + 512 \, a^{8} - {\left (3465 \, a^{8} d x + 5641 \, a^{8}\right )} \cos \left (d x + c\right )^{2} + {\left (3465 \, a^{8} d x - 6674 \, a^{8}\right )} \cos \left (d x + c\right ) - {\left (6 \, a^{8} \cos \left (d x + c\right )^{5} + 58 \, a^{8} \cos \left (d x + c\right )^{4} - 259 \, a^{8} \cos \left (d x + c\right )^{3} + 6930 \, a^{8} d x - 1545 \, a^{8} \cos \left (d x + c\right )^{2} - 512 \, a^{8} + {\left (3465 \, a^{8} d x - 7186 \, a^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, 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 = 6.88, size = 200, normalized size = 1.12 \begin {gather*} \frac {3465 \, {\left (d x + c\right )} a^{8} + \frac {1024 \, {\left (6 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{8}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {2 \, {\left (369 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1728 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 393 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5568 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 393 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5696 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 369 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1856 \, a^{8}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.13, size = 437, normalized size = 2.44 \begin {gather*} \frac {1155\,a^8\,x}{8}+\frac {\frac {1155\,a^8\,\left (c+d\,x\right )}{8}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3465\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (10395\,c+10395\,d\,x-25758\right )}{24}\right )-\frac {a^8\,\left (3465\,c+3465\,d\,x-10880\right )}{24}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {3465\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (10395\,c+10395\,d\,x-6882\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {8085\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (24255\,c+24255\,d\,x-21030\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {8085\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (24255\,c+24255\,d\,x-55130\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {15015\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (45045\,c+45045\,d\,x-45112\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {15015\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (45045\,c+45045\,d\,x-96328\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {10395\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (62370\,c+62370\,d\,x-86040\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {10395\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (62370\,c+62370\,d\,x-109800\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {12705\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (76230\,c+76230\,d\,x-103972\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {12705\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (76230\,c+76230\,d\,x-135388\right )}{24}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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