Integrand size = 27, antiderivative size = 157 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 a b x}{8}-\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {b^2 \cos ^7(c+d x)}{7 d}+\frac {5 a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{3 d} \]
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Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2990, 2715, 8, 14, 213} \[ \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^2 \cos ^5(c+d x)}{5 d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a b \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac {5 a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac {5 a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac {5 a b x}{8}-\frac {b^2 \cos ^7(c+d x)}{7 d} \]
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Rule 8
Rule 14
Rule 213
Rule 2715
Rule 2990
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \cos ^6(c+d x) \, dx+\int \cos ^5(c+d x) \cot (c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {a b \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac {1}{3} (5 a b) \int \cos ^4(c+d x) \, dx-\frac {\text {Subst}\left (\int x^6 \left (b^2-\frac {a^2}{-1+x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac {1}{4} (5 a b) \int \cos ^2(c+d x) \, dx-\frac {\text {Subst}\left (\int \left (-a^2-a^2 x^2-a^2 x^4+b^2 x^6-\frac {a^2}{-1+x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {b^2 \cos ^7(c+d x)}{7 d}+\frac {5 a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac {1}{8} (5 a b) \int 1 \, dx+\frac {a^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {5 a b x}{8}-\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {b^2 \cos ^7(c+d x)}{7 d}+\frac {5 a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.06 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {4200 a b c+4200 a b d x+105 \left (88 a^2-5 b^2\right ) \cos (c+d x)+35 \left (28 a^2-9 b^2\right ) \cos (3 (c+d x))+84 a^2 \cos (5 (c+d x))-105 b^2 \cos (5 (c+d x))-15 b^2 \cos (7 (c+d x))-6720 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+6720 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3150 a b \sin (2 (c+d x))+630 a b \sin (4 (c+d x))+70 a b \sin (6 (c+d x))}{6720 d} \]
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Time = 0.62 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+2 a b \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\) | \(113\) |
default | \(\frac {a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+2 a b \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\) | \(113\) |
parallelrisch | \(\frac {4200 a b x d -15 b^{2} \cos \left (7 d x +7 c \right )+70 a b \sin \left (6 d x +6 c \right )+630 a b \sin \left (4 d x +4 c \right )+3150 a b \sin \left (2 d x +2 c \right )+84 \cos \left (5 d x +5 c \right ) a^{2}-105 \cos \left (5 d x +5 c \right ) b^{2}+980 \cos \left (3 d x +3 c \right ) a^{2}-315 \cos \left (3 d x +3 c \right ) b^{2}+9240 \cos \left (d x +c \right ) a^{2}-525 \cos \left (d x +c \right ) b^{2}+6720 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10304 a^{2}-960 b^{2}}{6720 d}\) | \(169\) |
risch | \(\frac {5 a b x}{8}+\frac {11 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {5 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {11 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{128 d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {\cos \left (7 d x +7 c \right ) b^{2}}{448 d}+\frac {a b \sin \left (6 d x +6 c \right )}{96 d}+\frac {\cos \left (5 d x +5 c \right ) a^{2}}{80 d}-\frac {\cos \left (5 d x +5 c \right ) b^{2}}{64 d}+\frac {3 a b \sin \left (4 d x +4 c \right )}{32 d}+\frac {7 \cos \left (3 d x +3 c \right ) a^{2}}{48 d}-\frac {3 \cos \left (3 d x +3 c \right ) b^{2}}{64 d}+\frac {15 a b \sin \left (2 d x +2 c \right )}{32 d}\) | \(247\) |
norman | \(\frac {\frac {\left (6 a^{2}-2 b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {322 a^{2}-30 b^{2}}{105 d}+\frac {5 a b x}{8}+\frac {24 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {176 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {232 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {\left (146 a^{2}-30 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {\left (202 a^{2}-30 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {11 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {7 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {85 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {85 a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {7 a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {11 a b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {175 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {175 a b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a b x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a b x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {5 a b x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(412\) |
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Time = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {120 \, b^{2} \cos \left (d x + c\right )^{7} - 168 \, a^{2} \cos \left (d x + c\right )^{5} - 280 \, a^{2} \cos \left (d x + c\right )^{3} - 525 \, a b d x - 840 \, a^{2} \cos \left (d x + c\right ) + 420 \, a^{2} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 420 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 35 \, {\left (8 \, a b \cos \left (d x + c\right )^{5} + 10 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \]
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Timed out. \[ \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.78 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {480 \, b^{2} \cos \left (d x + c\right )^{7} - 112 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b}{3360 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (143) = 286\).
Time = 0.37 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.85 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {525 \, {\left (d x + c\right )} a b + 840 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (1155 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 840 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 980 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 10080 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 2975 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 20440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 4200 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 24640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2975 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16968 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2520 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 980 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6496 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1155 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1288 \, a^{2} + 120 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{840 \, d} \]
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Time = 13.41 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.64 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (6\,a^2-2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {146\,a^2}{3}-10\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {202\,a^2}{5}-6\,b^2\right )+\frac {232\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {176\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+24\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {46\,a^2}{15}-\frac {2\,b^2}{7}+\frac {7\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {85\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}-\frac {85\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}-\frac {7\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}-\frac {11\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {11\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {5\,a\,b\,\mathrm {atan}\left (\frac {25\,a^2\,b^2}{16\,\left (\frac {5\,a^3\,b}{2}-\frac {25\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {5\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {5\,a^3\,b}{2}-\frac {25\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d} \]
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