Integrand size = 19, antiderivative size = 127 \[ \int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx=\frac {5 a x}{16}+\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {b \sin (c+d x)}{d}-\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}-\frac {b \sin ^3(c+d x)}{3 d}-\frac {5 a \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {b \sin ^5(c+d x)}{5 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 d} \]
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Time = 0.15 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3957, 2917, 2672, 308, 212, 2715, 8} \[ \int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx=-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a x}{16}+\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {b \sin ^5(c+d x)}{5 d}-\frac {b \sin ^3(c+d x)}{3 d}-\frac {b \sin (c+d x)}{d} \]
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Rule 8
Rule 212
Rule 308
Rule 2672
Rule 2715
Rule 2917
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x)) \sin ^5(c+d x) \tan (c+d x) \, dx \\ & = a \int \sin ^6(c+d x) \, dx+b \int \sin ^5(c+d x) \tan (c+d x) \, dx \\ & = -\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{6} (5 a) \int \sin ^4(c+d x) \, dx+\frac {b \text {Subst}\left (\int \frac {x^6}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {5 a \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{8} (5 a) \int \sin ^2(c+d x) \, dx+\frac {b \text {Subst}\left (\int \left (-1-x^2-x^4+\frac {1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {b \sin (c+d x)}{d}-\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}-\frac {b \sin ^3(c+d x)}{3 d}-\frac {5 a \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {b \sin ^5(c+d x)}{5 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{16} (5 a) \int 1 \, dx+\frac {b \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {5 a x}{16}+\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {b \sin (c+d x)}{d}-\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}-\frac {b \sin ^3(c+d x)}{3 d}-\frac {5 a \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {b \sin ^5(c+d x)}{5 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 d} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx=\frac {5 a (c+d x)}{16 d}+\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {b \sin (c+d x)}{d}-\frac {b \sin ^3(c+d x)}{3 d}-\frac {b \sin ^5(c+d x)}{5 d}-\frac {15 a \sin (2 (c+d x))}{64 d}+\frac {3 a \sin (4 (c+d x))}{64 d}-\frac {a \sin (6 (c+d x))}{192 d} \]
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Time = 1.89 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+b \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(96\) |
default | \(\frac {a \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+b \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(96\) |
parts | \(\frac {a \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {b \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(98\) |
parallelrisch | \(\frac {300 a x d -1320 \sin \left (d x +c \right ) b -5 a \sin \left (6 d x +6 c \right )+45 a \sin \left (4 d x +4 c \right )-225 a \sin \left (2 d x +2 c \right )-12 \sin \left (5 d x +5 c \right ) b +140 \sin \left (3 d x +3 c \right ) b -960 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+960 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{960 d}\) | \(111\) |
risch | \(\frac {5 a x}{16}+\frac {11 i b \,{\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {11 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a \sin \left (6 d x +6 c \right )}{192 d}-\frac {b \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a \sin \left (4 d x +4 c \right )}{64 d}+\frac {7 b \sin \left (3 d x +3 c \right )}{48 d}-\frac {15 a \sin \left (2 d x +2 c \right )}{64 d}\) | \(150\) |
norman | \(\frac {\frac {5 a x}{16}+\frac {15 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {75 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {25 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {75 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {15 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {5 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}+\frac {\left (5 a -16 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}-\frac {\left (5 a +16 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (85 a -304 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (85 a +304 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {\left (165 a -688 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}-\frac {\left (165 a +688 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(284\) |
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Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx=\frac {75 \, a d x + 120 \, b \log \left (\sin \left (d x + c\right ) + 1\right ) - 120 \, b \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (40 \, a \cos \left (d x + c\right )^{5} + 48 \, b \cos \left (d x + c\right )^{4} - 130 \, a \cos \left (d x + c\right )^{3} - 176 \, b \cos \left (d x + c\right )^{2} + 165 \, a \cos \left (d x + c\right ) + 368 \, b\right )} \sin \left (d x + c\right )}{240 \, d} \]
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\[ \int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \sin ^{6}{\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.83 \[ \int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx=\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a - 32 \, {\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} b}{960 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.80 \[ \int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx=\frac {75 \, {\left (d x + c\right )} a + 240 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 240 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (75 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 240 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 425 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1520 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 990 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4128 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 990 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4128 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 425 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1520 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 75 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
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Time = 15.25 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.61 \[ \int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx=\frac {5\,a\,\mathrm {atan}\left (\frac {125\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,\left (\frac {125\,a^3}{64}+20\,a\,b^2\right )}+\frac {20\,a\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\frac {125\,a^3}{64}+20\,a\,b^2}\right )}{8\,d}+\frac {2\,b\,\mathrm {atanh}\left (\frac {64\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\frac {25\,a^2\,b}{4}+64\,b^3}+\frac {25\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {25\,a^2\,b}{4}+64\,b^3\right )}\right )}{d}-\frac {\left (2\,b-\frac {5\,a}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {38\,b}{3}-\frac {85\,a}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {172\,b}{5}-\frac {33\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {33\,a}{4}+\frac {172\,b}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {85\,a}{24}+\frac {38\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,a}{8}+2\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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