Integrand size = 21, antiderivative size = 127 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {10 \cot ^7(c+d x)}{7 a^3 d}+\frac {11 \cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac {3 \csc ^7(c+d x)}{7 a^3 d}+\frac {7 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d} \]
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Time = 0.71 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2954, 2952, 2687, 276, 2686, 14} \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {11 \cot ^9(c+d x)}{9 a^3 d}+\frac {10 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac {7 \csc ^9(c+d x)}{9 a^3 d}-\frac {3 \csc ^7(c+d x)}{7 a^3 d} \]
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Rule 14
Rule 276
Rule 2686
Rule 2687
Rule 2952
Rule 2954
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot ^3(c+d x) \csc ^3(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = -\frac {\int (-a+a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^9(c+d x) \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^6(c+d x) \csc ^6(c+d x)+3 a^3 \cot ^5(c+d x) \csc ^7(c+d x)-3 a^3 \cot ^4(c+d x) \csc ^8(c+d x)+a^3 \cot ^3(c+d x) \csc ^9(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^6(c+d x) \csc ^6(c+d x) \, dx}{a^3}+\frac {\int \cot ^3(c+d x) \csc ^9(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^5(c+d x) \csc ^7(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^4(c+d x) \csc ^8(c+d x) \, dx}{a^3} \\ & = -\frac {\text {Subst}\left (\int x^8 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {Subst}\left (\int \left (-x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = \frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {10 \cot ^7(c+d x)}{7 a^3 d}+\frac {11 \cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac {3 \csc ^7(c+d x)}{7 a^3 d}+\frac {7 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d} \\ \end{align*}
Time = 1.41 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.76 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\csc (c) \csc ^5(c+d x) \sec ^3(c+d x) (-3886080 \sin (c)+563200 \sin (d x)+524150 \sin (c+d x)+314490 \sin (2 (c+d x))-162010 \sin (3 (c+d x))-238250 \sin (4 (c+d x))-47650 \sin (5 (c+d x))+47650 \sin (6 (c+d x))+28590 \sin (7 (c+d x))+4765 \sin (8 (c+d x))-2027520 \sin (2 c+d x)+1486848 \sin (c+2 d x)-2365440 \sin (3 c+2 d x)+452608 \sin (2 c+3 d x)+665600 \sin (3 c+4 d x)+133120 \sin (4 c+5 d x)-133120 \sin (5 c+6 d x)-79872 \sin (6 c+7 d x)-13312 \sin (7 c+8 d x))}{56770560 a^3 d (1+\sec (c+d x))^3} \]
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Time = 0.90 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(\frac {-315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-770 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+990 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+4158 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-693 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-2310 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-20790 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+6930 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{887040 a^{3} d}\) | \(110\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d \,a^{3}}\) | \(112\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d \,a^{3}}\) | \(112\) |
risch | \(-\frac {16 i \left (2310 \,{\mathrm e}^{10 i \left (d x +c \right )}+1980 \,{\mathrm e}^{9 i \left (d x +c \right )}+3795 \,{\mathrm e}^{8 i \left (d x +c \right )}+550 \,{\mathrm e}^{7 i \left (d x +c \right )}+1452 \,{\mathrm e}^{6 i \left (d x +c \right )}+442 \,{\mathrm e}^{5 i \left (d x +c \right )}+650 \,{\mathrm e}^{4 i \left (d x +c \right )}+130 \,{\mathrm e}^{3 i \left (d x +c \right )}-130 \,{\mathrm e}^{2 i \left (d x +c \right )}-78 \,{\mathrm e}^{i \left (d x +c \right )}-13\right )}{3465 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5}}\) | \(148\) |
norman | \(\frac {-\frac {1}{1280 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{896 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{1152 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{2816 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{384 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{128 d a}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{128 d a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{640 d a}}{a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}\) | \(158\) |
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Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.50 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {104 \, \cos \left (d x + c\right )^{8} + 312 \, \cos \left (d x + c\right )^{7} + 52 \, \cos \left (d x + c\right )^{6} - 676 \, \cos \left (d x + c\right )^{5} - 585 \, \cos \left (d x + c\right )^{4} + 325 \, \cos \left (d x + c\right )^{3} - 25 \, \cos \left (d x + c\right )^{2} - 150 \, \cos \left (d x + c\right ) - 50}{3465 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\csc ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.37 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {\frac {20790 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4158 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {990 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{3}} + \frac {231 \, {\left (\frac {10 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {30 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{887040 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {231 \, {\left (30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {315 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 770 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 990 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4158 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20790 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{33}}}{887040 \, d} \]
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Time = 13.93 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.58 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {693\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+2310\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+20790\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-4158\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-990\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+315\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{887040\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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