Integrand size = 21, antiderivative size = 169 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2}{32 d (1-\cos (c+d x))^4}+\frac {11 a^2}{48 d (1-\cos (c+d x))^3}-\frac {3 a^2}{4 d (1-\cos (c+d x))^2}+\frac {51 a^2}{32 d (1-\cos (c+d x))}-\frac {a^2}{64 d (1+\cos (c+d x))^2}+\frac {9 a^2}{64 d (1+\cos (c+d x))}+\frac {99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac {29 a^2 \log (1+\cos (c+d x))}{128 d} \]
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Time = 0.13 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3964, 90} \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {51 a^2}{32 d (1-\cos (c+d x))}+\frac {9 a^2}{64 d (\cos (c+d x)+1)}-\frac {3 a^2}{4 d (1-\cos (c+d x))^2}-\frac {a^2}{64 d (\cos (c+d x)+1)^2}+\frac {11 a^2}{48 d (1-\cos (c+d x))^3}-\frac {a^2}{32 d (1-\cos (c+d x))^4}+\frac {99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac {29 a^2 \log (\cos (c+d x)+1)}{128 d} \]
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Rule 90
Rule 3964
Rubi steps \begin{align*} \text {integral}& = -\frac {a^{10} \text {Subst}\left (\int \frac {x^7}{(a-a x)^5 (a+a x)^3} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^{10} \text {Subst}\left (\int \left (-\frac {1}{8 a^8 (-1+x)^5}-\frac {11}{16 a^8 (-1+x)^4}-\frac {3}{2 a^8 (-1+x)^3}-\frac {51}{32 a^8 (-1+x)^2}-\frac {99}{128 a^8 (-1+x)}-\frac {1}{32 a^8 (1+x)^3}+\frac {9}{64 a^8 (1+x)^2}-\frac {29}{128 a^8 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^2}{32 d (1-\cos (c+d x))^4}+\frac {11 a^2}{48 d (1-\cos (c+d x))^3}-\frac {3 a^2}{4 d (1-\cos (c+d x))^2}+\frac {51 a^2}{32 d (1-\cos (c+d x))}-\frac {a^2}{64 d (1+\cos (c+d x))^2}+\frac {9 a^2}{64 d (1+\cos (c+d x))}+\frac {99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac {29 a^2 \log (1+\cos (c+d x))}{128 d} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.86 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (-1224 \csc ^2\left (\frac {1}{2} (c+d x)\right )+288 \csc ^4\left (\frac {1}{2} (c+d x)\right )-44 \csc ^6\left (\frac {1}{2} (c+d x)\right )+3 \csc ^8\left (\frac {1}{2} (c+d x)\right )-6 \left (116 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+396 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+18 \sec ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^4\left (\frac {1}{2} (c+d x)\right )\right )\right )}{6144 d} \]
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Time = 2.89 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} \cos \left (d x +c \right )^{8}}{8 \sin \left (d x +c \right )^{8}}+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{9}}{8 \sin \left (d x +c \right )^{8}}+\frac {\cos \left (d x +c \right )^{9}}{48 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{9}}{64 \sin \left (d x +c \right )^{4}}+\frac {5 \cos \left (d x +c \right )^{9}}{128 \sin \left (d x +c \right )^{2}}+\frac {5 \cos \left (d x +c \right )^{7}}{128}+\frac {7 \cos \left (d x +c \right )^{5}}{128}+\frac {35 \cos \left (d x +c \right )^{3}}{384}+\frac {35 \cos \left (d x +c \right )}{128}+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{8}}{8}+\frac {\cot \left (d x +c \right )^{6}}{6}-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(213\) |
default | \(\frac {-\frac {a^{2} \cos \left (d x +c \right )^{8}}{8 \sin \left (d x +c \right )^{8}}+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{9}}{8 \sin \left (d x +c \right )^{8}}+\frac {\cos \left (d x +c \right )^{9}}{48 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{9}}{64 \sin \left (d x +c \right )^{4}}+\frac {5 \cos \left (d x +c \right )^{9}}{128 \sin \left (d x +c \right )^{2}}+\frac {5 \cos \left (d x +c \right )^{7}}{128}+\frac {7 \cos \left (d x +c \right )^{5}}{128}+\frac {35 \cos \left (d x +c \right )^{3}}{384}+\frac {35 \cos \left (d x +c \right )}{128}+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{8}}{8}+\frac {\cot \left (d x +c \right )^{6}}{6}-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(213\) |
risch | \(-i a^{2} x -\frac {2 i a^{2} c}{d}-\frac {a^{2} \left (279 \,{\mathrm e}^{11 i \left (d x +c \right )}-156 \,{\mathrm e}^{10 i \left (d x +c \right )}-1141 \,{\mathrm e}^{9 i \left (d x +c \right )}+2080 \,{\mathrm e}^{8 i \left (d x +c \right )}+670 \,{\mathrm e}^{7 i \left (d x +c \right )}-2696 \,{\mathrm e}^{6 i \left (d x +c \right )}+670 \,{\mathrm e}^{5 i \left (d x +c \right )}+2080 \,{\mathrm e}^{4 i \left (d x +c \right )}-1141 \,{\mathrm e}^{3 i \left (d x +c \right )}-156 \,{\mathrm e}^{2 i \left (d x +c \right )}+279 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{96 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4}}+\frac {99 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d}+\frac {29 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d}\) | \(215\) |
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Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (145) = 290\).
Time = 0.29 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.91 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {558 \, a^{2} \cos \left (d x + c\right )^{5} - 156 \, a^{2} \cos \left (d x + c\right )^{4} - 1268 \, a^{2} \cos \left (d x + c\right )^{3} + 676 \, a^{2} \cos \left (d x + c\right )^{2} + 686 \, a^{2} \cos \left (d x + c\right ) - 448 \, a^{2} - 87 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} + 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 297 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} + 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{384 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} + 4 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {87 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) + 297 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (279 \, a^{2} \cos \left (d x + c\right )^{5} - 78 \, a^{2} \cos \left (d x + c\right )^{4} - 634 \, a^{2} \cos \left (d x + c\right )^{3} + 338 \, a^{2} \cos \left (d x + c\right )^{2} + 343 \, a^{2} \cos \left (d x + c\right ) - 224 \, a^{2}\right )}}{\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}}{384 \, d} \]
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Time = 0.44 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.41 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {1188 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 1536 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {96 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {6 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (3 \, a^{2} + \frac {32 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {174 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {768 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2475 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}}{1536 \, d} \]
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Time = 14.46 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.88 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {99\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (32\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {29\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {a^2}{8}\right )}{64\,d}-\frac {a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
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