Integrand size = 21, antiderivative size = 91 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a^2 d}-\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {2 \csc ^5(c+d x)}{5 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d} \]
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Time = 0.43 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2954, 2952, 2687, 14, 2686, 276} \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d}-\frac {2 \csc ^5(c+d x)}{5 a^2 d} \]
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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 ^2(c+d x) \csc ^2(c+d x)}{(-a-a \cos (c+d x))^2} \, dx \\ & = \frac {\int (-a+a \cos (c+d x))^2 \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a^4} \\ & = \frac {\int \left (a^2 \cot ^4(c+d x) \csc ^4(c+d x)-2 a^2 \cot ^3(c+d x) \csc ^5(c+d x)+a^2 \cot ^2(c+d x) \csc ^6(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \cot ^4(c+d x) \csc ^4(c+d x) \, dx}{a^2}+\frac {\int \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^3(c+d x) \csc ^5(c+d x) \, dx}{a^2} \\ & = \frac {\text {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \text {Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \text {Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d} \\ & = -\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a^2 d}-\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {2 \csc ^5(c+d x)}{5 a^2 d}+\frac {2 \csc ^7(c+d x)}{7 a^2 d} \\ \end{align*}
Time = 1.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.64 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\csc (c) \csc ^3(c+d x) \sec ^2(c+d x) (1344 \sin (c)-1456 \sin (d x)-714 \sin (c+d x)-408 \sin (2 (c+d x))+153 \sin (3 (c+d x))+204 \sin (4 (c+d x))+51 \sin (5 (c+d x))+1680 \sin (2 c+d x)+128 \sin (c+2 d x)-48 \sin (2 c+3 d x)-64 \sin (3 c+4 d x)-16 \sin (4 c+5 d x))}{13440 a^2 d (1+\sec (c+d x))^2} \]
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Time = 0.64 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+21 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-70 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-35 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-210 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-105 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{3360 a^{2} d}\) | \(84\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{32 d \,a^{2}}\) | \(86\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{32 d \,a^{2}}\) | \(86\) |
risch | \(\frac {4 i \left (105 \,{\mathrm e}^{6 i \left (d x +c \right )}+84 \,{\mathrm e}^{5 i \left (d x +c \right )}+91 \,{\mathrm e}^{4 i \left (d x +c \right )}-8 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{105 a^{2} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{3}}\) | \(104\) |
norman | \(\frac {-\frac {1}{96 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{32 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{48 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{160 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{224 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}\) | \(120\) |
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Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {2 \, \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 6 \, \cos \left (d x + c\right )^{2} + 24 \, \cos \left (d x + c\right ) + 12}{105 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\csc ^{4}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.47 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {\frac {210 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {70 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{2}} + \frac {35 \, {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a^{2} \sin \left (d x + c\right )^{3}}}{3360 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {35 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {15 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 70 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{14}}}{3360 \, d} \]
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Time = 13.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.33 \[ \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-96\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+54\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-15}{3360\,a^2\,d\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \]
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