Integrand size = 21, antiderivative size = 143 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {x}{a^3}-\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {10 \csc ^3(c+d x)}{3 a^3 d}+\frac {11 \csc ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d} \]
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Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3973, 3971, 3554, 8, 2686, 200, 2687, 30, 276} \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {4 \cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d}+\frac {11 \csc ^5(c+d x)}{5 a^3 d}-\frac {10 \csc ^3(c+d x)}{3 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {x}{a^3} \]
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Rule 8
Rule 30
Rule 200
Rule 276
Rule 2686
Rule 2687
Rule 3554
Rule 3971
Rule 3973
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cot ^8(c+d x) (-a+a \sec (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^8(c+d x)+3 a^3 \cot ^7(c+d x) \csc (c+d x)-3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^5(c+d x) \csc ^3(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^8(c+d x) \, dx}{a^3}+\frac {\int \cot ^5(c+d x) \csc ^3(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^7(c+d x) \csc (c+d x) \, dx}{a^3}-\frac {3 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx}{a^3} \\ & = \frac {\cot ^7(c+d x)}{7 a^3 d}+\frac {\int \cot ^6(c+d x) \, dx}{a^3}-\frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{a^3 d} \\ & = -\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^7(c+d x)}{7 a^3 d}-\frac {\int \cot ^4(c+d x) \, dx}{a^3}-\frac {\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d} \\ & = \frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {10 \csc ^3(c+d x)}{3 a^3 d}+\frac {11 \csc ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d}+\frac {\int \cot ^2(c+d x) \, dx}{a^3} \\ & = -\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {10 \csc ^3(c+d x)}{3 a^3 d}+\frac {11 \csc ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d}-\frac {\int 1 \, dx}{a^3} \\ & = -\frac {x}{a^3}-\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {10 \csc ^3(c+d x)}{3 a^3 d}+\frac {11 \csc ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d} \\ \end{align*}
Time = 1.64 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.76 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) (-5880 d x \cos (d x)+5880 d x \cos (2 c+d x)-5880 d x \cos (c+2 d x)+5880 d x \cos (3 c+2 d x)-2520 d x \cos (2 c+3 d x)+2520 d x \cos (4 c+3 d x)-420 d x \cos (3 c+4 d x)+420 d x \cos (5 c+4 d x)+4200 \sin (c)+11032 \sin (d x)-23282 \sin (c+d x)-23282 \sin (2 (c+d x))-9978 \sin (3 (c+d x))-1663 \sin (4 (c+d x))+13720 \sin (2 c+d x)+15512 \sin (c+2 d x)+9240 \sin (3 c+2 d x)+8088 \sin (2 c+3 d x)+2520 \sin (4 c+3 d x)+1768 \sin (3 c+4 d x))}{215040 a^3 d} \]
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Time = 0.64 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59
method | result | size |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{3}}\) | \(85\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{3}}\) | \(85\) |
risch | \(-\frac {x}{a^{3}}+\frac {2 i \left (315 \,{\mathrm e}^{7 i \left (d x +c \right )}+1155 \,{\mathrm e}^{6 i \left (d x +c \right )}+1715 \,{\mathrm e}^{5 i \left (d x +c \right )}+525 \,{\mathrm e}^{4 i \left (d x +c \right )}-1379 \,{\mathrm e}^{3 i \left (d x +c \right )}-1939 \,{\mathrm e}^{2 i \left (d x +c \right )}-1011 \,{\mathrm e}^{i \left (d x +c \right )}-221\right )}{105 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}\) | \(122\) |
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Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {221 \, \cos \left (d x + c\right )^{4} + 348 \, \cos \left (d x + c\right )^{3} - 25 \, \cos \left (d x + c\right )^{2} + 105 \, {\left (d x \cos \left (d x + c\right )^{3} + 3 \, d x \cos \left (d x + c\right )^{2} + 3 \, d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) - 303 \, \cos \left (d x + c\right ) - 136}{105 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.93 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {\frac {2730 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {560 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {126 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} - \frac {3360 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {105 \, {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{3} \sin \left (d x + c\right )}}{1680 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {1680 \, {\left (d x + c\right )}}{a^{3}} + \frac {105}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {15 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 126 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 560 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2730 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{21}}}{1680 \, d} \]
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Time = 14.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {x}{a^3}-\frac {\frac {221\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{105}-\frac {268\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{105}+\frac {257\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{420}-\frac {31\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{280}+\frac {1}{112}}{a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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