Integrand size = 19, antiderivative size = 101 \[ \int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx=\frac {3 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \tan (c+d x)}{d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d} \]
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Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2827, 3852, 3853, 3855} \[ \int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx=\frac {3 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \tan ^5(c+d x)}{5 d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rule 2827
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = a \int \sec ^5(c+d x) \, dx+a \int \sec ^6(c+d x) \, dx \\ & = \frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} (3 a) \int \sec ^3(c+d x) \, dx-\frac {a \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d} \\ & = \frac {a \tan (c+d x)}{d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d}+\frac {1}{8} (3 a) \int \sec (c+d x) \, dx \\ & = \frac {3 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \tan (c+d x)}{d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.64 \[ \int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx=\frac {a \left (45 \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (120+45 \sec (c+d x)+30 \sec ^3(c+d x)+80 \tan ^2(c+d x)+24 \tan ^4(c+d x)\right )\right )}{120 d} \]
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Time = 2.50 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(83\) |
default | \(\frac {a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(83\) |
parts | \(-\frac {a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(85\) |
risch | \(-\frac {i a \left (45 \,{\mathrm e}^{9 i \left (d x +c \right )}+210 \,{\mathrm e}^{7 i \left (d x +c \right )}-640 \,{\mathrm e}^{4 i \left (d x +c \right )}-210 \,{\mathrm e}^{3 i \left (d x +c \right )}-320 \,{\mathrm e}^{2 i \left (d x +c \right )}-45 \,{\mathrm e}^{i \left (d x +c \right )}-64\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(128\) |
norman | \(\frac {-\frac {13 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {137 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}-\frac {167 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}+\frac {17 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {3 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(170\) |
parallelrisch | \(\frac {8 \left (\left (-\frac {45 \cos \left (d x +c \right )}{32}-\frac {45 \cos \left (3 d x +3 c \right )}{64}-\frac {9 \cos \left (5 d x +5 c \right )}{64}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (\frac {45 \cos \left (d x +c \right )}{32}+\frac {45 \cos \left (3 d x +3 c \right )}{64}+\frac {9 \cos \left (5 d x +5 c \right )}{64}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\sin \left (3 d x +3 c \right )+\frac {9 \sin \left (4 d x +4 c \right )}{32}+\frac {\sin \left (5 d x +5 c \right )}{5}+2 \sin \left (d x +c \right )+\frac {21 \sin \left (2 d x +2 c \right )}{16}\right ) a}{3 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(177\) |
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Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx=\frac {45 \, a \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 45 \, a \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (64 \, a \cos \left (d x + c\right )^{4} + 45 \, a \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} + 30 \, a \cos \left (d x + c\right ) + 24 \, a\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
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\[ \int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx=a \left (\int \cos {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \sec ^{6}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx=\frac {16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a - 15 \, a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.23 \[ \int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx=\frac {45 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 45 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 130 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 190 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 195 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
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Time = 19.01 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.56 \[ \int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx=\frac {3\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}-\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{6}+\frac {116\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}-\frac {19\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {13\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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