Integrand size = 21, antiderivative size = 115 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^4}{4 d (a-a \cos (c+d x))^2}-\frac {5 a^3}{4 d (a-a \cos (c+d x))}+\frac {17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {a^2 \log (1+\cos (c+d x))}{8 d}+\frac {a^2 \sec (c+d x)}{d} \]
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Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 90} \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^4}{4 d (a-a \cos (c+d x))^2}-\frac {5 a^3}{4 d (a-a \cos (c+d x))}+\frac {a^2 \sec (c+d x)}{d}+\frac {17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {a^2 \log (\cos (c+d x)+1)}{8 d} \]
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int (-a-a \cos (c+d x))^2 \csc ^5(c+d x) \sec ^2(c+d x) \, dx \\ & = \frac {a^5 \text {Subst}\left (\int \frac {a^2}{(-a-x)^3 x^2 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^7 \text {Subst}\left (\int \frac {1}{(-a-x)^3 x^2 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^7 \text {Subst}\left (\int \left (\frac {1}{8 a^5 (a-x)}+\frac {1}{a^4 x^2}-\frac {2}{a^5 x}+\frac {1}{2 a^3 (a+x)^3}+\frac {5}{4 a^4 (a+x)^2}+\frac {17}{8 a^5 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^4}{4 d (a-a \cos (c+d x))^2}-\frac {5 a^3}{4 d (a-a \cos (c+d x))}+\frac {17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {a^2 \log (1+\cos (c+d x))}{8 d}+\frac {a^2 \sec (c+d x)}{d} \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (10 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right )+4 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+8 \log (\cos (c+d x))-17 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 \sec (c+d x)\right )\right )}{64 d} \]
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Time = 0.97 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(\frac {17 \left (-\frac {8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )}{17}-\frac {8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )}{17}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (d x +c \right )-\frac {45 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\cos \left (d x +c \right )-\frac {3 \cos \left (2 d x +2 c \right )}{10}-\frac {59}{90}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{136}\right ) a^{2}}{4 d \cos \left (d x +c \right )}\) | \(118\) |
norman | \(\frac {\frac {a^{2}}{16 d}+\frac {11 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 d}-\frac {11 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {17 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(134\) |
risch | \(\frac {a^{2} \left (9 \,{\mathrm e}^{5 i \left (d x +c \right )}-28 \,{\mathrm e}^{4 i \left (d x +c \right )}+34 \,{\mathrm e}^{3 i \left (d x +c \right )}-28 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}+\frac {17 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(152\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(157\) |
default | \(\frac {a^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(157\) |
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Time = 0.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.82 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {18 \, a^{2} \cos \left (d x + c\right )^{2} - 28 \, a^{2} \cos \left (d x + c\right ) + 8 \, a^{2} - 16 \, {\left (a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right )\right ) - {\left (a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 17 \, {\left (a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{8 \, {\left (d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}} \]
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\[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=a^{2} \left (\int 2 \csc ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) - 17 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) + 16 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (9 \, a^{2} \cos \left (d x + c\right )^{2} - 14 \, a^{2} \cos \left (d x + c\right ) + 4 \, a^{2}\right )}}{\cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )}}{8 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.66 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {34 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {{\left (a^{2} - \frac {12 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {51 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} + \frac {32 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{16 \, d} \]
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Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {17\,a^2\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{8\,d}-\frac {a^2\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{8\,d}+\frac {\frac {9\,a^2\,{\cos \left (c+d\,x\right )}^2}{4}-\frac {7\,a^2\,\cos \left (c+d\,x\right )}{2}+a^2}{d\,\left ({\cos \left (c+d\,x\right )}^3-2\,{\cos \left (c+d\,x\right )}^2+\cos \left (c+d\,x\right )\right )}-\frac {2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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