Integrand size = 21, antiderivative size = 91 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}-\frac {\csc ^9(c+d x)}{9 a d} \]
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Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2918, 2686, 30, 2687, 276} \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cot ^9(c+d x)}{9 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^9(c+d x)}{9 a d} \]
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Rule 30
Rule 276
Rule 2686
Rule 2687
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot (c+d x) \csc ^7(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = -\frac {\int \cot ^2(c+d x) \csc ^8(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^9(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int x^8 \, dx,x,\csc (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = -\frac {\csc ^9(c+d x)}{9 a d}-\frac {\text {Subst}\left (\int \left (x^2+3 x^4+3 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = \frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}-\frac {\csc ^9(c+d x)}{9 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(200\) vs. \(2(91)=182\).
Time = 2.07 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.20 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\csc (c) \csc ^7(c+d x) \sec (c+d x) (645120 \sin (c)-143360 \sin (d x)-85750 \sin (c+d x)-17150 \sin (2 (c+d x))+51450 \sin (3 (c+d x))+17150 \sin (4 (c+d x))-17150 \sin (5 (c+d x))-7350 \sin (6 (c+d x))+2450 \sin (7 (c+d x))+1225 \sin (8 (c+d x))-28672 \sin (c+2 d x)+86016 \sin (2 c+3 d x)+28672 \sin (3 c+4 d x)-28672 \sin (4 c+5 d x)-12288 \sin (5 c+6 d x)+4096 \sin (6 c+7 d x)+2048 \sin (7 c+8 d x))}{5160960 a d (1+\sec (c+d x))} \]
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Time = 0.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.23
method | result | size |
parallelrisch | \(\frac {-35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-270 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-45 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-882 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-378 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-1470 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-1470 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-4410 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{80640 d a}\) | \(112\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d a}\) | \(114\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d a}\) | \(114\) |
risch | \(\frac {32 i \left (315 \,{\mathrm e}^{8 i \left (d x +c \right )}+70 \,{\mathrm e}^{7 i \left (d x +c \right )}+14 \,{\mathrm e}^{6 i \left (d x +c \right )}-42 \,{\mathrm e}^{5 i \left (d x +c \right )}-14 \,{\mathrm e}^{4 i \left (d x +c \right )}+14 \,{\mathrm e}^{3 i \left (d x +c \right )}+6 \,{\mathrm e}^{2 i \left (d x +c \right )}-2 \,{\mathrm e}^{i \left (d x +c \right )}-1\right )}{315 a d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7}}\) | \(126\) |
norman | \(\frac {-\frac {1}{1792 a d}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{640 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{896 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{2304 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{640 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{384 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{128 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{384 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}\) | \(155\) |
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Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (81) = 162\).
Time = 0.31 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.95 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {16 \, \cos \left (d x + c\right )^{8} + 16 \, \cos \left (d x + c\right )^{7} - 56 \, \cos \left (d x + c\right )^{6} - 56 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )^{2} - 35 \, \cos \left (d x + c\right ) - 35}{315 \, {\left (a d \cos \left (d x + c\right )^{7} + a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{5} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{3} + 3 \, a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\csc ^{8}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (81) = 162\).
Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.93 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {\frac {1470 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {882 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a} + \frac {3 \, {\left (\frac {126 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {490 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1470 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 15\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a \sin \left (d x + c\right )^{7}}}{80640 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.45 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {3 \, {\left (1470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 490 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 126 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} + \frac {35 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 270 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 882 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1470 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a^{9}}}{80640 \, d} \]
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Time = 13.93 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.21 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+378\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1470\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4410\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1470\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+882\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+270\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{80640\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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