Integrand size = 21, antiderivative size = 128 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{128 a^3 d}-\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {a^2}{40 d (a+a \cos (c+d x))^5}+\frac {3 a}{64 d (a+a \cos (c+d x))^4}-\frac {1}{64 a d (a+a \cos (c+d x))^2}-\frac {3}{128 d \left (a^3+a^3 \cos (c+d x)\right )} \]
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Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2915, 12, 90, 212} \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{128 a^3 d}-\frac {3}{128 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {a^2}{40 d (a \cos (c+d x)+a)^5}+\frac {3 a}{64 d (a \cos (c+d x)+a)^4}-\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {1}{64 a d (a \cos (c+d x)+a)^2} \]
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Rule 12
Rule 90
Rule 212
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = \frac {a^5 \text {Subst}\left (\int \frac {x^3}{a^3 (-a-x)^3 (-a+x)^6} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^2 \text {Subst}\left (\int \frac {x^3}{(-a-x)^3 (-a+x)^6} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^2 \text {Subst}\left (\int \left (-\frac {1}{8 (a-x)^6}+\frac {3}{16 a (a-x)^5}-\frac {1}{32 a^3 (a-x)^3}-\frac {3}{128 a^4 (a-x)^2}+\frac {1}{64 a^3 (a+x)^3}-\frac {3}{128 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {a^2}{40 d (a+a \cos (c+d x))^5}+\frac {3 a}{64 d (a+a \cos (c+d x))^4}-\frac {1}{64 a d (a+a \cos (c+d x))^2}-\frac {3}{128 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {3 \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,-a \cos (c+d x)\right )}{128 a^2 d} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{128 a^3 d}-\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {a^2}{40 d (a+a \cos (c+d x))^5}+\frac {3 a}{64 d (a+a \cos (c+d x))^4}-\frac {1}{64 a d (a+a \cos (c+d x))^2}-\frac {3}{128 d \left (a^3+a^3 \cos (c+d x)\right )} \\ \end{align*}
Time = 3.78 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\left (4-15 \cos ^2\left (\frac {1}{2} (c+d x)\right )+60 \cos ^8\left (\frac {1}{2} (c+d x)\right )+10 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (2+\cot ^4\left (\frac {1}{2} (c+d x)\right )\right )-120 \cos ^{10}\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x)}{640 a^3 d (1+\sec (c+d x))^3} \]
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Time = 0.99 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {-\frac {1}{40 \left (\cos \left (d x +c \right )+1\right )^{5}}+\frac {3}{64 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {1}{64 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {3}{128 \left (\cos \left (d x +c \right )+1\right )}+\frac {3 \ln \left (\cos \left (d x +c \right )+1\right )}{256}-\frac {1}{128 \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {3 \ln \left (\cos \left (d x +c \right )-1\right )}{256}}{d \,a^{3}}\) | \(91\) |
default | \(\frac {-\frac {1}{40 \left (\cos \left (d x +c \right )+1\right )^{5}}+\frac {3}{64 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {1}{64 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {3}{128 \left (\cos \left (d x +c \right )+1\right )}+\frac {3 \ln \left (\cos \left (d x +c \right )+1\right )}{256}-\frac {1}{128 \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {3 \ln \left (\cos \left (d x +c \right )-1\right )}{256}}{d \,a^{3}}\) | \(91\) |
parallelrisch | \(\frac {-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-10 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-60 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-20 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5120 a^{3} d}\) | \(113\) |
norman | \(\frac {-\frac {1}{512 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{1024 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{1280 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{256 d a}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{256 d a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{512 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{256 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{2}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{3} d}\) | \(158\) |
risch | \(-\frac {15 \,{\mathrm e}^{13 i \left (d x +c \right )}+90 \,{\mathrm e}^{12 i \left (d x +c \right )}+170 \,{\mathrm e}^{11 i \left (d x +c \right )}-30 \,{\mathrm e}^{10 i \left (d x +c \right )}+1521 \,{\mathrm e}^{9 i \left (d x +c \right )}+1476 \,{\mathrm e}^{8 i \left (d x +c \right )}+3756 \,{\mathrm e}^{7 i \left (d x +c \right )}+1476 \,{\mathrm e}^{6 i \left (d x +c \right )}+1521 \,{\mathrm e}^{5 i \left (d x +c \right )}-30 \,{\mathrm e}^{4 i \left (d x +c \right )}+170 \,{\mathrm e}^{3 i \left (d x +c \right )}+90 \,{\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}}{320 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 a^{3} d}\) | \(220\) |
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Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (117) = 234\).
Time = 0.27 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.48 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {30 \, \cos \left (d x + c\right )^{6} + 90 \, \cos \left (d x + c\right )^{5} + 40 \, \cos \left (d x + c\right )^{4} - 120 \, \cos \left (d x + c\right )^{3} + 122 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 126 \, \cos \left (d x + c\right ) + 32}{1280 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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\[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\csc ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.47 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{6} + 45 \, \cos \left (d x + c\right )^{5} + 20 \, \cos \left (d x + c\right )^{4} - 60 \, \cos \left (d x + c\right )^{3} + 61 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right ) + 16\right )}}{a^{3} \cos \left (d x + c\right )^{7} + 3 \, a^{3} \cos \left (d x + c\right )^{6} + a^{3} \cos \left (d x + c\right )^{5} - 5 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}} - \frac {15 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {15 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{1280 \, d} \]
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Time = 0.45 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.81 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {10 \, {\left (\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {60 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {\frac {60 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {30 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {20 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{15}}}{5120 \, d} \]
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Time = 13.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.35 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3\,\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{128\,a^3\,d}-\frac {\frac {3\,{\cos \left (c+d\,x\right )}^6}{128}+\frac {9\,{\cos \left (c+d\,x\right )}^5}{128}+\frac {{\cos \left (c+d\,x\right )}^4}{32}-\frac {3\,{\cos \left (c+d\,x\right )}^3}{32}+\frac {61\,{\cos \left (c+d\,x\right )}^2}{640}+\frac {63\,\cos \left (c+d\,x\right )}{640}+\frac {1}{40}}{d\,\left (a^3\,{\cos \left (c+d\,x\right )}^7+3\,a^3\,{\cos \left (c+d\,x\right )}^6+a^3\,{\cos \left (c+d\,x\right )}^5-5\,a^3\,{\cos \left (c+d\,x\right )}^4-5\,a^3\,{\cos \left (c+d\,x\right )}^3+a^3\,{\cos \left (c+d\,x\right )}^2+3\,a^3\,\cos \left (c+d\,x\right )+a^3\right )} \]
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