Optimal. Leaf size=210 \[ \frac {a^3 \sec ^{11}(c+d x)}{11 d}+\frac {3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac {a^3 \sec ^9(c+d x)}{9 d}-\frac {11 a^3 \sec ^8(c+d x)}{8 d}-\frac {6 a^3 \sec ^7(c+d x)}{7 d}+\frac {7 a^3 \sec ^6(c+d x)}{3 d}+\frac {14 a^3 \sec ^5(c+d x)}{5 d}-\frac {3 a^3 \sec ^4(c+d x)}{2 d}-\frac {11 a^3 \sec ^3(c+d x)}{3 d}-\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac {a^3 \sec ^{11}(c+d x)}{11 d}+\frac {3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac {a^3 \sec ^9(c+d x)}{9 d}-\frac {11 a^3 \sec ^8(c+d x)}{8 d}-\frac {6 a^3 \sec ^7(c+d x)}{7 d}+\frac {7 a^3 \sec ^6(c+d x)}{3 d}+\frac {14 a^3 \sec ^5(c+d x)}{5 d}-\frac {3 a^3 \sec ^4(c+d x)}{2 d}-\frac {11 a^3 \sec ^3(c+d x)}{3 d}-\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^4 (a+a x)^7}{x^{12}} \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^{11}}{x^{12}}+\frac {3 a^{11}}{x^{11}}-\frac {a^{11}}{x^{10}}-\frac {11 a^{11}}{x^9}-\frac {6 a^{11}}{x^8}+\frac {14 a^{11}}{x^7}+\frac {14 a^{11}}{x^6}-\frac {6 a^{11}}{x^5}-\frac {11 a^{11}}{x^4}-\frac {a^{11}}{x^3}+\frac {3 a^{11}}{x^2}+\frac {a^{11}}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \sec ^2(c+d x)}{2 d}-\frac {11 a^3 \sec ^3(c+d x)}{3 d}-\frac {3 a^3 \sec ^4(c+d x)}{2 d}+\frac {14 a^3 \sec ^5(c+d x)}{5 d}+\frac {7 a^3 \sec ^6(c+d x)}{3 d}-\frac {6 a^3 \sec ^7(c+d x)}{7 d}-\frac {11 a^3 \sec ^8(c+d x)}{8 d}-\frac {a^3 \sec ^9(c+d x)}{9 d}+\frac {3 a^3 \sec ^{10}(c+d x)}{10 d}+\frac {a^3 \sec ^{11}(c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 214, normalized size = 1.02 \[ -\frac {a^3 \sec ^{11}(c+d x) (-1613260 \cos (2 (c+d x))+960960 \cos (3 (c+d x))-1131504 \cos (4 (c+d x))+314160 \cos (5 (c+d x))-432894 \cos (6 (c+d x))+145530 \cos (7 (c+d x))-106260 \cos (8 (c+d x))+6930 \cos (9 (c+d x))-20790 \cos (10 (c+d x))+1143450 \cos (3 (c+d x)) \log (\cos (c+d x))+571725 \cos (5 (c+d x)) \log (\cos (c+d x))+190575 \cos (7 (c+d x)) \log (\cos (c+d x))+38115 \cos (9 (c+d x)) \log (\cos (c+d x))+3465 \cos (11 (c+d x)) \log (\cos (c+d x))+462 \cos (c+d x) (3465 \log (\cos (c+d x))+2606)-1151740)}{3548160 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 169, normalized size = 0.80 \[ -\frac {27720 \, a^{3} \cos \left (d x + c\right )^{11} \log \left (-\cos \left (d x + c\right )\right ) - 83160 \, a^{3} \cos \left (d x + c\right )^{10} + 13860 \, a^{3} \cos \left (d x + c\right )^{9} + 101640 \, a^{3} \cos \left (d x + c\right )^{8} + 41580 \, a^{3} \cos \left (d x + c\right )^{7} - 77616 \, a^{3} \cos \left (d x + c\right )^{6} - 64680 \, a^{3} \cos \left (d x + c\right )^{5} + 23760 \, a^{3} \cos \left (d x + c\right )^{4} + 38115 \, a^{3} \cos \left (d x + c\right )^{3} + 3080 \, a^{3} \cos \left (d x + c\right )^{2} - 8316 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3}}{27720 \, d \cos \left (d x + c\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 19.68, size = 367, normalized size = 1.75 \[ \frac {27720 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 27720 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {153343 \, a^{3} + \frac {1742213 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9043705 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28369275 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {59954070 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {67458930 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {57997170 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {36975510 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {16879995 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {5213945 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {976261 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {83711 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{11}}}{27720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.87, size = 351, normalized size = 1.67 \[ \frac {4352 a^{3} \cos \left (d x +c \right )}{3465 d}+\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{99 d \cos \left (d x +c \right )^{9}}-\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{693 d \cos \left (d x +c \right )^{7}}+\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{1155 d \cos \left (d x +c \right )^{5}}-\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{693 d \cos \left (d x +c \right )^{3}}+\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{99 d \cos \left (d x +c \right )}+\frac {a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{11 d \cos \left (d x +c \right )^{11}}+\frac {3 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{10 d \cos \left (d x +c \right )^{10}}+\frac {34 a^{3} \cos \left (d x +c \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{99 d}+\frac {272 a^{3} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{693 d}+\frac {544 a^{3} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{1155 d}+\frac {2176 a^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{3465 d}+\frac {a^{3} \left (\tan ^{8}\left (d x +c \right )\right )}{8 d}-\frac {a^{3} \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}+\frac {a^{3} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 162, normalized size = 0.77 \[ -\frac {27720 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {83160 \, a^{3} \cos \left (d x + c\right )^{10} - 13860 \, a^{3} \cos \left (d x + c\right )^{9} - 101640 \, a^{3} \cos \left (d x + c\right )^{8} - 41580 \, a^{3} \cos \left (d x + c\right )^{7} + 77616 \, a^{3} \cos \left (d x + c\right )^{6} + 64680 \, a^{3} \cos \left (d x + c\right )^{5} - 23760 \, a^{3} \cos \left (d x + c\right )^{4} - 38115 \, a^{3} \cos \left (d x + c\right )^{3} - 3080 \, a^{3} \cos \left (d x + c\right )^{2} + 8316 \, a^{3} \cos \left (d x + c\right ) + 2520 \, a^{3}}{\cos \left (d x + c\right )^{11}}}{27720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.26, size = 337, normalized size = 1.60 \[ \frac {2\,a^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}-22\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+\frac {332\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{3}-\frac {1012\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {10456\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{15}-\frac {5192\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}+\frac {8164\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}-\frac {3676\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {10090\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{63}-\frac {9334\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{315}+\frac {8704\,a^3}{3465}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}-11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+55\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-165\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+330\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-462\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+462\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-330\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+165\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-55\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 51.80, size = 439, normalized size = 2.09 \[ \begin {cases} \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{11 d} + \frac {3 a^{3} \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{3 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )}}{8 d} - \frac {8 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{99 d} - \frac {3 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac {8 a^{3} \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{21 d} - \frac {a^{3} \tan ^{6}{\left (c + d x \right )}}{6 d} + \frac {16 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{231 d} + \frac {3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {16 a^{3} \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {64 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{1155 d} - \frac {3 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac {64 a^{3} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{105 d} - \frac {a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {128 a^{3} \sec ^{3}{\left (c + d x \right )}}{3465 d} + \frac {3 a^{3} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {128 a^{3} \sec {\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right )^{3} \tan ^{9}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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