Integrand size = 27, antiderivative size = 355 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {15 b \left (a^2-2 b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}-\frac {15 \left (a^4-8 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}+\frac {\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))} \]
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Time = 1.15 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2975, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {15 b \left (a^2-2 b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}-\frac {15 \left (a^4-8 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}+\frac {\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2} \]
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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)}{2 b d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^3(c+d x) \left (-18 b^2 \left (9 a^2-10 b^2\right )-18 a b \left (2 a^2-b^2\right ) \sin (c+d x)+36 b^2 \left (3 a^2-4 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{72 a^2 b^2} \\ & = -\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^3(c+d x) \left (-36 b^2 \left (17 a^4-37 a^2 b^2+20 b^4\right )-72 a b \left (a^2-b^2\right )^2 \sin (c+d x)+108 b^2 \left (4 a^4-9 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{144 a^3 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-540 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2-36 a b \left (2 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+144 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right )^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{144 a^4 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-144 b \left (a^2-b^2\right )^2 \left (a^4-25 a^2 b^2+30 b^4\right )+36 a b^2 \left (11 a^2-20 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-540 b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{288 a^5 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (540 b^2 \left (a^2-b^2\right )^2 \left (a^4-8 a^2 b^2+8 b^4\right )-540 a b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{288 a^6 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {\left (15 b \left (a^2-2 b^2\right ) \left (a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^7}+\frac {\left (15 \left (a^4-8 a^2 b^2+8 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^7} \\ & = -\frac {15 \left (a^4-8 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}+\frac {\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {\left (15 b \left (a^2-2 b^2\right ) \left (a^2-b^2\right )\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^7 d} \\ & = -\frac {15 \left (a^4-8 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}+\frac {\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\left (30 b \left (a^2-2 b^2\right ) \left (a^2-b^2\right )\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^7 d} \\ & = -\frac {15 b \left (a^2-2 b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}-\frac {15 \left (a^4-8 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}+\frac {\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.02 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {1920 b \left (a^4-3 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-240 \left (a^4-8 a^2 b^2+8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+240 \left (a^4-8 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \cot (c+d x) \csc ^5(c+d x) \left (44 a^5-505 a^3 b^2+540 a b^4+\left (-68 a^5+660 a^3 b^2-720 a b^4\right ) \cos (2 (c+d x))+\left (8 a^5-155 a^3 b^2+180 a b^4\right ) \cos (4 (c+d x))-176 a^4 b \sin (c+d x)-260 a^2 b^3 \sin (c+d x)+600 b^5 \sin (c+d x)+66 a^4 b \sin (3 (c+d x))+170 a^2 b^3 \sin (3 (c+d x))-300 b^5 \sin (3 (c+d x))+2 a^4 b \sin (5 (c+d x))-50 a^2 b^3 \sin (5 (c+d x))+60 b^5 \sin (5 (c+d x))\right )}{(b+a \csc (c+d x))^2}}{128 a^7 d} \]
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Time = 1.26 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+54 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{6}}-\frac {2 \left (\frac {\left (-\frac {3}{2} a^{5} b +\frac {15}{2} a^{3} b^{3}-6 a \,b^{5}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{6}+\frac {9}{2} a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}-11 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a b \left (5 a^{4}-37 a^{2} b^{2}+32 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{2} \left (2 a^{4}-13 a^{2} b^{2}+11 b^{4}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {15 b \left (a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7}}-\frac {1}{64 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+24 b^{2}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-240 a^{2} b^{2}+240 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{7}}+\frac {b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (27 a^{2}-40 b^{2}\right )}{8 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(455\) |
default | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+54 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{6}}-\frac {2 \left (\frac {\left (-\frac {3}{2} a^{5} b +\frac {15}{2} a^{3} b^{3}-6 a \,b^{5}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{6}+\frac {9}{2} a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}-11 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a b \left (5 a^{4}-37 a^{2} b^{2}+32 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{2} \left (2 a^{4}-13 a^{2} b^{2}+11 b^{4}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {15 b \left (a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7}}-\frac {1}{64 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+24 b^{2}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-240 a^{2} b^{2}+240 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{7}}+\frac {b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (27 a^{2}-40 b^{2}\right )}{8 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(455\) |
risch | \(\text {Expression too large to display}\) | \(1022\) |
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Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (334) = 668\).
Time = 0.62 (sec) , antiderivative size = 2022, normalized size of antiderivative = 5.70 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.40 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.70 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {120 \, {\left (a^{4} - 8 \, 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^{7}} - \frac {960 \, {\left (a^{4} b - 3 \, a^{2} b^{3} + 2 \, 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^{7}} + \frac {64 \, {\left (3 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 22 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 37 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{6} - 13 \, a^{4} b^{2} + 11 \, a^{2} 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 )}^{2} a^{7}} - \frac {250 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2000 \, 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} + 216 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{4}}{a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 216 \, a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 320 \, a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{64 \, d} \]
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Time = 12.03 (sec) , antiderivative size = 1275, normalized size of antiderivative = 3.59 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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