Integrand size = 21, antiderivative size = 153 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=a^2 x+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+\frac {2 a b \csc (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^7(c+d x)}{7 d} \]
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Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3971, 3554, 8, 2686, 200, 2687, 30} \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {a^2 \cot ^7(c+d x)}{7 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x-\frac {2 a b \csc ^7(c+d x)}{7 d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {2 a b \csc (c+d x)}{d}-\frac {b^2 \cot ^7(c+d x)}{7 d} \]
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Rule 8
Rule 30
Rule 200
Rule 2686
Rule 2687
Rule 3554
Rule 3971
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cot ^8(c+d x)+2 a b \cot ^7(c+d x) \csc (c+d x)+b^2 \cot ^6(c+d x) \csc ^2(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^8(c+d x) \, dx+(2 a b) \int \cot ^7(c+d x) \csc (c+d x) \, dx+b^2 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx \\ & = -\frac {a^2 \cot ^7(c+d x)}{7 d}-a^2 \int \cot ^6(c+d x) \, dx-\frac {(2 a b) \text {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^2 \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+a^2 \int \cot ^4(c+d x) \, dx-\frac {(2 a b) \text {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+\frac {2 a b \csc (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}-a^2 \int \cot ^2(c+d x) \, dx \\ & = \frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+\frac {2 a b \csc (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}+a^2 \int 1 \, dx \\ & = a^2 x+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+\frac {2 a b \csc (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^7(c+d x)}{7 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.61 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {b \left (-5 b \cot ^7(c+d x)+2 a \csc (c+d x) \left (35-35 \csc ^2(c+d x)+21 \csc ^4(c+d x)-5 \csc ^6(c+d x)\right )\right )-5 a^2 \cot ^7(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(c+d x)\right )}{35 d} \]
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Time = 3.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )-\frac {b^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(187\) |
default | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )-\frac {b^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(187\) |
risch | \(a^{2} x +\frac {2 i \left (210 a b \,{\mathrm e}^{13 i \left (d x +c \right )}+420 a^{2} {\mathrm e}^{12 i \left (d x +c \right )}+105 b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-420 a b \,{\mathrm e}^{11 i \left (d x +c \right )}-1260 a^{2} {\mathrm e}^{10 i \left (d x +c \right )}+1806 a b \,{\mathrm e}^{9 i \left (d x +c \right )}+3080 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+525 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-1272 a b \,{\mathrm e}^{7 i \left (d x +c \right )}-3080 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+1806 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+2436 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+315 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-420 a b \,{\mathrm e}^{3 i \left (d x +c \right )}-812 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+210 a b \,{\mathrm e}^{i \left (d x +c \right )}+176 a^{2}+15 b^{2}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\) | \(254\) |
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Time = 0.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.35 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {210 \, a b \cos \left (d x + c\right )^{6} + {\left (176 \, a^{2} + 15 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 406 \, a^{2} \cos \left (d x + c\right )^{5} - 420 \, a b \cos \left (d x + c\right )^{4} + 350 \, a^{2} \cos \left (d x + c\right )^{3} + 336 \, a b \cos \left (d x + c\right )^{2} - 105 \, a^{2} \cos \left (d x + c\right ) - 96 \, a b + 105 \, {\left (a^{2} d x \cos \left (d x + c\right )^{6} - 3 \, a^{2} d x \cos \left (d x + c\right )^{4} + 3 \, a^{2} d x \cos \left (d x + c\right )^{2} - a^{2} d x\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.76 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a^{2} + \frac {6 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a b}{\sin \left (d x + c\right )^{7}} - \frac {15 \, b^{2}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (141) = 282\).
Time = 0.40 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.39 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 294 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1295 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1470 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13440 \, {\left (d x + c\right )} a^{2} - 9765 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7350 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 525 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {9765 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 7350 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 525 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1295 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1470 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 315 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 294 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} - 30 \, a b - 15 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]
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Time = 13.75 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.69 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=a^2\,x+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\left (a-b\right )}^2}{896\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2}{32}-\frac {5\,a\,b}{48}+\frac {b^2}{48}+\frac {{\left (a-b\right )}^2}{384}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a^2}{80}-\frac {3\,a\,b}{160}+\frac {b^2}{160}+\frac {{\left (a-b\right )}^2}{640}\right )}{d}-\frac {\frac {2\,a\,b}{7}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {37\,a^2}{3}+14\,a\,b+3\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (93\,a^2+70\,a\,b+5\,b^2\right )+\frac {a^2}{7}+\frac {b^2}{7}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {9\,a^2}{5}+\frac {14\,a\,b}{5}+b^2\right )}{128\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {23\,a^2}{32}-\frac {17\,a\,b}{32}+\frac {b^2}{32}+\frac {{\left (a-b\right )}^2}{128}\right )}{d} \]
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