Integrand size = 21, antiderivative size = 177 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {x}{a^3}+\frac {\cot (c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {3 \csc (c+d x)}{a^3 d}+\frac {13 \csc ^3(c+d x)}{3 a^3 d}-\frac {21 \csc ^5(c+d x)}{5 a^3 d}+\frac {15 \csc ^7(c+d x)}{7 a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d} \]
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Time = 0.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}+\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {\cot (c+d x)}{a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d}+\frac {15 \csc ^7(c+d x)}{7 a^3 d}-\frac {21 \csc ^5(c+d x)}{5 a^3 d}+\frac {13 \csc ^3(c+d x)}{3 a^3 d}-\frac {3 \csc (c+d x)}{a^3 d}+\frac {x}{a^3} \]
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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 ^{10}(c+d x) (-a+a \sec (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^{10}(c+d x)+3 a^3 \cot ^9(c+d x) \csc (c+d x)-3 a^3 \cot ^8(c+d x) \csc ^2(c+d x)+a^3 \cot ^7(c+d x) \csc ^3(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^{10}(c+d x) \, dx}{a^3}+\frac {\int \cot ^7(c+d x) \csc ^3(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^9(c+d x) \csc (c+d x) \, dx}{a^3}-\frac {3 \int \cot ^8(c+d x) \csc ^2(c+d x) \, dx}{a^3} \\ & = \frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {\int \cot ^8(c+d x) \, dx}{a^3}-\frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^8 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{a^3 d} \\ & = -\frac {\cot ^7(c+d x)}{7 a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {\int \cot ^6(c+d x) \, dx}{a^3}-\frac {\text {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d} \\ & = \frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {3 \csc (c+d x)}{a^3 d}+\frac {13 \csc ^3(c+d x)}{3 a^3 d}-\frac {21 \csc ^5(c+d x)}{5 a^3 d}+\frac {15 \csc ^7(c+d x)}{7 a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d}+\frac {\int \cot ^4(c+d x) \, dx}{a^3} \\ & = -\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {3 \csc (c+d x)}{a^3 d}+\frac {13 \csc ^3(c+d x)}{3 a^3 d}-\frac {21 \csc ^5(c+d x)}{5 a^3 d}+\frac {15 \csc ^7(c+d x)}{7 a^3 d}-\frac {4 \csc ^9(c+d x)}{9 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 {\cot ^7(c+d x)}{7 a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {3 \csc (c+d x)}{a^3 d}+\frac {13 \csc ^3(c+d x)}{3 a^3 d}-\frac {21 \csc ^5(c+d x)}{5 a^3 d}+\frac {15 \csc ^7(c+d x)}{7 a^3 d}-\frac {4 \csc ^9(c+d x)}{9 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 {\cot ^7(c+d x)}{7 a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {3 \csc (c+d x)}{a^3 d}+\frac {13 \csc ^3(c+d x)}{3 a^3 d}-\frac {21 \csc ^5(c+d x)}{5 a^3 d}+\frac {15 \csc ^7(c+d x)}{7 a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(366\) vs. \(2(177)=354\).
Time = 1.89 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.07 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\csc \left (\frac {c}{2}\right ) \csc ^3(2 (c+d x)) \sec \left (\frac {c}{2}\right ) (181440 d x \cos (d x)-181440 d x \cos (2 c+d x)+136080 d x \cos (c+2 d x)-136080 d x \cos (3 c+2 d x)-10080 d x \cos (2 c+3 d x)+10080 d x \cos (4 c+3 d x)-60480 d x \cos (3 c+4 d x)+60480 d x \cos (5 c+4 d x)-30240 d x \cos (4 c+5 d x)+30240 d x \cos (6 c+5 d x)-5040 d x \cos (5 c+6 d x)+5040 d x \cos (7 c+6 d x)-169344 \sin (c)-338112 \sin (d x)+675036 \sin (c+d x)+506277 \sin (2 (c+d x))-37502 \sin (3 (c+d x))-225012 \sin (4 (c+d x))-112506 \sin (5 (c+d x))-18751 \sin (6 (c+d x))-431424 \sin (2 c+d x)-375552 \sin (c+2 d x)-201600 \sin (3 c+2 d x)-41248 \sin (2 c+3 d x)+84000 \sin (4 c+3 d x)+155712 \sin (3 c+4 d x)+100800 \sin (5 c+4 d x)+98016 \sin (4 c+5 d x)+30240 \sin (6 c+5 d x)+21376 \sin (5 c+6 d x))}{80640 a^3 d (1+\sec (c+d x))^3} \]
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Time = 0.64 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-99 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+128 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d \,a^{3}}\) | \(111\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-99 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+128 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d \,a^{3}}\) | \(111\) |
risch | \(\frac {x}{a^{3}}-\frac {2 i \left (945 \,{\mathrm e}^{11 i \left (d x +c \right )}+3150 \,{\mathrm e}^{10 i \left (d x +c \right )}+2625 \,{\mathrm e}^{9 i \left (d x +c \right )}-6300 \,{\mathrm e}^{8 i \left (d x +c \right )}-13482 \,{\mathrm e}^{7 i \left (d x +c \right )}-5292 \,{\mathrm e}^{6 i \left (d x +c \right )}+10566 \,{\mathrm e}^{5 i \left (d x +c \right )}+11736 \,{\mathrm e}^{4 i \left (d x +c \right )}+1289 \,{\mathrm e}^{3 i \left (d x +c \right )}-4866 \,{\mathrm e}^{2 i \left (d x +c \right )}-3063 \,{\mathrm e}^{i \left (d x +c \right )}-668\right )}{315 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{3}}\) | \(165\) |
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Time = 0.28 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.22 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {668 \, \cos \left (d x + c\right )^{6} + 1059 \, \cos \left (d x + c\right )^{5} - 573 \, \cos \left (d x + c\right )^{4} - 1813 \, \cos \left (d x + c\right )^{3} - 393 \, \cos \left (d x + c\right )^{2} + 315 \, {\left (d x \cos \left (d x + c\right )^{5} + 3 \, d x \cos \left (d x + c\right )^{4} + 2 \, d x \cos \left (d x + c\right )^{3} - 2 \, d x \cos \left (d x + c\right )^{2} - 3 \, d x \cos \left (d x + c\right ) - d x\right )} \sin \left (d x + c\right ) + 789 \, \cos \left (d x + c\right ) + 368}{315 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{4}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {\frac {31185 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6720 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1827 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {360 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{3}} - \frac {40320 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {105 \, {\left (\frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a^{3} \sin \left (d x + c\right )^{3}}}{20160 \, d} \]
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Time = 0.47 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.74 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {20160 \, {\left (d x + c\right )}}{a^{3}} + \frac {105 \, {\left (24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {35 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1827 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6720 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 31185 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{27}}}{20160 \, d} \]
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Time = 14.28 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.16 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (c+d\,x\right )}{a^3\,d\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\right )}-\frac {\frac {668\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{315}-\frac {983\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{210}+\frac {346\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{105}-\frac {2291\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2520}+\frac {173\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{840}-\frac {19\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{672}+\frac {1}{576}}{a^3\,d\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \]
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