Integrand size = 27, antiderivative size = 303 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {10 b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {5 \left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{3 a^5 b d}+\frac {5 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))} \]
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Time = 0.83 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2975, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {10 b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}+\frac {5 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}-\frac {5 \left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{3 a^5 b d}-\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))} \]
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Rule 210
Rule 632
Rule 2739
Rule 2975
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-2 b^2 \left (27 a^2-20 b^2\right )-4 a b \left (6 a^2-b^2\right ) \sin (c+d x)+6 b^2 \left (4 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{24 a^2 b^2} \\ & = -\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-30 b^2 \left (5 a^4-9 a^2 b^2+4 b^4\right )-2 a b \left (12 a^4-17 a^2 b^2+5 b^4\right ) \sin (c+d x)+16 b^2 \left (6 a^4-11 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^3 b^2 \left (a^2-b^2\right )} \\ & = \frac {5 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-16 b \left (3 a^6-23 a^4 b^2+35 a^2 b^4-15 b^6\right )+2 a b^2 \left (21 a^4-41 a^2 b^2+20 b^4\right ) \sin (c+d x)-30 b^3 \left (5 a^4-9 a^2 b^2+4 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{48 a^4 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{3 a^5 b d}+\frac {5 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (30 b^2 \left (3 a^6-15 a^4 b^2+20 a^2 b^4-8 b^6\right )-30 a b^3 \left (5 a^4-9 a^2 b^2+4 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{48 a^5 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{3 a^5 b d}+\frac {5 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {\left (5 b \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^6}+\frac {\left (5 \left (3 a^4-12 a^2 b^2+8 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^6} \\ & = -\frac {5 \left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{3 a^5 b d}+\frac {5 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {\left (10 b \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 d} \\ & = -\frac {5 \left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{3 a^5 b d}+\frac {5 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\left (20 b \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 d} \\ & = -\frac {10 b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {5 \left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{3 a^5 b d}+\frac {5 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac {5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))} \\ \end{align*}
Time = 6.43 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.61 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {10 b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}+\frac {\left (-7 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )+6 b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{3 a^5 d}+\frac {3 \left (3 a^2-4 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^4 d}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{12 a^3 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d}-\frac {5 \left (3 a^4-12 a^2 b^2+8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^6 d}+\frac {5 \left (3 a^4-12 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^6 d}-\frac {3 \left (3 a^2-4 b^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^4 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (7 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-6 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {a^4 \cos (c+d x)-2 a^2 b^2 \cos (c+d x)+b^4 \cos (c+d x)}{a^5 d (a+b \sin (c+d x))}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{12 a^3 d} \]
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Time = 0.83 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{5}}-\frac {1}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+12 b^{2}}{32 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-120 a^{2} b^{2}+80 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{6}}+\frac {b}{12 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (9 a^{2}-8 b^{2}\right )}{4 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 \left (\frac {-\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{5}}{2}+a^{3} b^{2}-\frac {a \,b^{4}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {5 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6}}}{d}\) | \(370\) |
default | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{5}}-\frac {1}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+12 b^{2}}{32 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-120 a^{2} b^{2}+80 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{6}}+\frac {b}{12 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (9 a^{2}-8 b^{2}\right )}{4 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 \left (\frac {-\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{5}}{2}+a^{3} b^{2}-\frac {a \,b^{4}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {5 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6}}}{d}\) | \(370\) |
risch | \(\frac {i \left (-180 i a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}+245 i a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}+600 i a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-720 i a \,b^{4} {\mathrm e}^{5 i \left (d x +c \right )}-770 i a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+960 i a^{3} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-160 a^{2} b^{3}+120 b^{5}+24 a^{4} b +360 i a \,b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-60 i a \,b^{4} {\mathrm e}^{9 i \left (d x +c \right )}-510 i a^{3} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+75 i a^{3} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+720 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-480 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+120 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}-480 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}-24 i a^{5} {\mathrm e}^{i \left (d x +c \right )}+96 i a^{5} {\mathrm e}^{3 i \left (d x +c \right )}-1000 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+680 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+150 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-150 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}-144 i a^{5} {\mathrm e}^{5 i \left (d x +c \right )}+600 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-30 b \,a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-90 b \,a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-120 a^{2} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-24 i a^{5} {\mathrm e}^{9 i \left (d x +c \right )}+96 i a^{5} {\mathrm e}^{7 i \left (d x +c \right )}\right )}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} b \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{5} d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{2}}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 a^{4} d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{a^{6} d}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{2}}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 a^{4} d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{a^{6} d}+\frac {5 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{4}}-\frac {5 i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{6}}-\frac {5 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{4}}+\frac {5 i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{6}}\) | \(850\) |
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Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (286) = 572\).
Time = 0.71 (sec) , antiderivative size = 1576, normalized size of antiderivative = 5.20 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.38 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.57 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {120 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} - \frac {1920 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{6}} + \frac {384 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{6}} + \frac {3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 432 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 384 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{8}} - \frac {750 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3000 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2000 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 432 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
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Time = 11.83 (sec) , antiderivative size = 1117, normalized size of antiderivative = 3.69 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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