Integrand size = 21, antiderivative size = 103 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^7(c+d x)}{a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {3 \csc ^5(c+d x)}{5 a^3 d}+\frac {\csc ^7(c+d x)}{a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d} \]
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Time = 0.63 (sec) , antiderivative size = 103, 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, 14, 2686, 276} \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {4 \cot ^9(c+d x)}{9 a^3 d}+\frac {\cot ^7(c+d x)}{a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d}+\frac {\csc ^7(c+d x)}{a^3 d}-\frac {3 \csc ^5(c+d x)}{5 a^3 d} \]
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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 (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 ^7(c+d x) \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^6(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)-3 a^3 \cot ^4(c+d x) \csc ^6(c+d x)+a^3 \cot ^3(c+d x) \csc ^7(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^6(c+d x) \csc ^4(c+d x) \, dx}{a^3}+\frac {\int \cot ^3(c+d x) \csc ^7(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^5(c+d x) \csc ^5(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx}{a^3} \\ & = -\frac {\text {Subst}\left (\int x^6 \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 ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^4 \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 )^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = \frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^7(c+d x)}{a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {3 \csc ^5(c+d x)}{5 a^3 d}+\frac {\csc ^7(c+d x)}{a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d} \\ \end{align*}
Time = 1.35 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.70 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\csc (c) \csc ^3(2 (c+d x)) (5376 \sin (c)-1152 \sin (d x)-1764 \sin (c+d x)-1323 \sin (2 (c+d x))+98 \sin (3 (c+d x))+588 \sin (4 (c+d x))+294 \sin (5 (c+d x))+49 \sin (6 (c+d x))+3456 \sin (2 c+d x)-1152 \sin (c+2 d x)+2880 \sin (3 c+2 d x)-128 \sin (2 c+3 d x)-768 \sin (3 c+4 d x)-384 \sin (4 c+5 d x)-64 \sin (5 c+6 d x))}{5760 a^3 d (1+\sec (c+d x))^3} \]
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Time = 0.81 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58
method | result | size |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d \,a^{3}}\) | \(60\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d \,a^{3}}\) | \(60\) |
parallelrisch | \(\frac {-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-135 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-15}{2880 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}\) | \(61\) |
norman | \(\frac {-\frac {1}{192 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{576 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{64 d a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{320 d a}}{a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}\) | \(82\) |
risch | \(\frac {4 i \left (45 \,{\mathrm e}^{8 i \left (d x +c \right )}+54 \,{\mathrm e}^{7 i \left (d x +c \right )}+84 \,{\mathrm e}^{6 i \left (d x +c \right )}+18 \,{\mathrm e}^{5 i \left (d x +c \right )}+18 \,{\mathrm e}^{4 i \left (d x +c \right )}+2 \,{\mathrm e}^{3 i \left (d x +c \right )}+12 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{45 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{3}}\) | \(126\) |
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Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.42 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {2 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} - 7 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right ) + 2}{45 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\csc ^{4}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.89 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {\frac {135 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {27 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{3}} + \frac {15 \, {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a^{3} \sin \left (d x + c\right )^{3}}}{2880 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {15}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {5 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 27 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 135 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{27}}}{2880 \, d} \]
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Time = 13.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+135\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-27\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{2880\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]
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