Integrand size = 19, antiderivative size = 119 \[ \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {a \cos (c+d x)}{d}+\frac {3 b \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {3 b \cos ^4(c+d x)}{4 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {b \cos ^6(c+d x)}{6 d}+\frac {a \cos ^7(c+d x)}{7 d}-\frac {b \log (\cos (c+d x))}{d} \]
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Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3957, 2916, 12, 780} \[ \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {a \cos ^7(c+d x)}{7 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {a \cos (c+d x)}{d}+\frac {b \cos ^6(c+d x)}{6 d}-\frac {3 b \cos ^4(c+d x)}{4 d}+\frac {3 b \cos ^2(c+d x)}{2 d}-\frac {b \log (\cos (c+d x))}{d} \]
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Rule 12
Rule 780
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x)) \sin ^6(c+d x) \tan (c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a (-b+x) \left (a^2-x^2\right )^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-b+x) \left (a^2-x^2\right )^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^6 d} \\ & = \frac {\text {Subst}\left (\int \left (a^6-\frac {a^6 b}{x}+3 a^4 b x-3 a^4 x^2-3 a^2 b x^3+3 a^2 x^4+b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d} \\ & = -\frac {a \cos (c+d x)}{d}+\frac {3 b \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {3 b \cos ^4(c+d x)}{4 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {b \cos ^6(c+d x)}{6 d}+\frac {a \cos ^7(c+d x)}{7 d}-\frac {b \log (\cos (c+d x))}{d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95 \[ \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {35 a \cos (c+d x)}{64 d}+\frac {7 a \cos (3 (c+d x))}{64 d}-\frac {7 a \cos (5 (c+d x))}{320 d}+\frac {a \cos (7 (c+d x))}{448 d}-\frac {b \left (-\frac {3}{2} \cos ^2(c+d x)+\frac {3}{4} \cos ^4(c+d x)-\frac {1}{6} \cos ^6(c+d x)+\log (\cos (c+d x))\right )}{d} \]
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Time = 1.86 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}+b \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(87\) |
default | \(\frac {-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}+b \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(87\) |
parts | \(-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7 d}+\frac {b \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(89\) |
parallelrisch | \(\frac {-3675 a \cos \left (d x +c \right )+15 a \cos \left (7 d x +7 c \right )-147 a \cos \left (5 d x +5 c \right )+735 a \cos \left (3 d x +3 c \right )+35 b \cos \left (6 d x +6 c \right )-420 b \cos \left (4 d x +4 c \right )+3045 \cos \left (2 d x +2 c \right ) b +6720 b \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-6720 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-6720 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-3072 a -2660 b}{6720 d}\) | \(139\) |
risch | \(i b x +\frac {29 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {29 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}+\frac {2 i b c}{d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {35 a \cos \left (d x +c \right )}{64 d}+\frac {a \cos \left (7 d x +7 c \right )}{448 d}+\frac {b \cos \left (6 d x +6 c \right )}{192 d}-\frac {7 a \cos \left (5 d x +5 c \right )}{320 d}-\frac {b \cos \left (4 d x +4 c \right )}{16 d}+\frac {7 a \cos \left (3 d x +3 c \right )}{64 d}\) | \(150\) |
norman | \(\frac {-\frac {32 a}{35 d}-\frac {128 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}-\frac {14 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {\left (96 a +70 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d}-\frac {\left (96 a +128 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {\left (32 a +10 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {b \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(200\) |
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Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.78 \[ \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, b \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, b \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, b \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, b \log \left (-\cos \left (d x + c\right )\right )}{420 \, d} \]
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Timed out. \[ \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, b \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, b \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, b \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, b \log \left (\cos \left (d x + c\right )\right )}{420 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (109) = 218\).
Time = 0.34 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.66 \[ \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {420 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {384 \, a + 1089 \, b - \frac {2688 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8463 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {8064 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28749 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {13440 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {56035 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {28749 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1089 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}}{420 \, d} \]
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Time = 14.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {a\,\cos \left (c+d\,x\right )-a\,{\cos \left (c+d\,x\right )}^3+\frac {3\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {a\,{\cos \left (c+d\,x\right )}^7}{7}-\frac {3\,b\,{\cos \left (c+d\,x\right )}^2}{2}+\frac {3\,b\,{\cos \left (c+d\,x\right )}^4}{4}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{6}+b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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