Integrand size = 27, antiderivative size = 340 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 \sqrt {a^2-b^2} d}-\frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}-\frac {b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))} \]
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Time = 0.98 (sec) , antiderivative size = 340, 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 = {2969, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}-\frac {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^7 d \sqrt {a^2-b^2}}-\frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2} \]
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Rule 210
Rule 632
Rule 2739
Rule 2969
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\int \frac {\csc ^4(c+d x) \left (6 \left (2 a^2-5 b^2\right )-2 a b \sin (c+d x)-8 \left (a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{8 a^2 b} \\ & = \frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (12 \left (3 a^4-13 a^2 b^2+10 b^4\right )-6 a b \left (a^2-b^2\right ) \sin (c+d x)-6 \left (4 a^4-19 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{8 a^3 b \left (a^2-b^2\right )} \\ & = -\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-18 b \left (7 a^4-27 a^2 b^2+20 b^4\right )+30 a b^2 \left (a^2-b^2\right ) \sin (c+d x)+24 b \left (3 a^4-13 a^2 b^2+10 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^4 b \left (a^2-b^2\right )} \\ & = \frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (24 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right )+6 a b \left (3 a^4-23 a^2 b^2+20 b^4\right ) \sin (c+d x)-18 b^2 \left (7 a^4-27 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{48 a^5 b \left (a^2-b^2\right )} \\ & = -\frac {b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (18 b \left (a^6-25 a^4 b^2+64 a^2 b^4-40 b^6\right )-18 a b^2 \left (7 a^4-27 a^2 b^2+20 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{48 a^6 b \left (a^2-b^2\right )} \\ & = -\frac {b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}-\frac {\left (3 b \left (2 a^4-11 a^2 b^2+10 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^7}+\frac {\left (3 \left (a^4-24 a^2 b^2+40 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^7} \\ & = -\frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}-\frac {b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}-\frac {\left (3 b \left (2 a^4-11 a^2 b^2+10 b^4\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 {3 \left (a^4-24 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}-\frac {b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac {\left (6 b \left (2 a^4-11 a^2 b^2+10 b^4\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 {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 \sqrt {a^2-b^2} d}-\frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}-\frac {b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^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}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))} \\ \end{align*}
Time = 4.43 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {384 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+48 \left (a^4-24 a^2 b^2+40 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-48 \left (a^4-24 a^2 b^2+40 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \cot (c+d x) \csc ^5(c+d x) \left (-4 a^5+289 a^3 b^2-540 a b^4+4 \left (5 a^5-93 a^3 b^2+180 a b^4\right ) \cos (2 (c+d x))+\left (83 a^3 b^2-180 a b^4\right ) \cos (4 (c+d x))+100 a^4 b \sin (c+d x)+20 a^2 b^3 \sin (c+d x)-600 b^5 \sin (c+d x)-44 a^4 b \sin (3 (c+d x))-50 a^2 b^3 \sin (3 (c+d x))+300 b^5 \sin (3 (c+d x))+26 a^2 b^3 \sin (5 (c+d x))-60 b^5 \sin (5 (c+d x))\right )}{(b+a \csc (c+d x))^2}}{128 a^7 d} \]
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Time = 1.04 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{6}}-\frac {1}{64 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+24 b^{2}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-144 a^{2} b^{2}+240 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{7}}+\frac {b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 b \left (3 a^{2}-8 b^{2}\right )}{8 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\frac {\left (\frac {7}{2} a^{3} b^{2}-6 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (6 a^{4}+a^{2} b^{2}-22 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (17 a^{2}-32 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (6 a^{2}-11 b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 \left (2 a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7}}}{d}\) | \(431\) |
default | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{6}}-\frac {1}{64 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+24 b^{2}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-144 a^{2} b^{2}+240 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{7}}+\frac {b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 b \left (3 a^{2}-8 b^{2}\right )}{8 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\frac {\left (\frac {7}{2} a^{3} b^{2}-6 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (6 a^{4}+a^{2} b^{2}-22 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (17 a^{2}-32 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (6 a^{2}-11 b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 \left (2 a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7}}}{d}\) | \(431\) |
risch | \(\text {Expression too large to display}\) | \(1128\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1254 vs. \(2 (321) = 642\).
Time = 0.74 (sec) , antiderivative size = 2592, normalized size of antiderivative = 7.62 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.41 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.62 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {24 \, {\left (a^{4} - 24 \, a^{2} b^{2} + 40 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{7}} - \frac {192 \, {\left (2 \, a^{4} b - 11 \, a^{2} b^{3} + 10 \, 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 (7 \, 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} + 6 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 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} + 17 \, 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 ) + 6 \, 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 {50 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1200 \, 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} + 120 \, 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} - 8 \, 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} - 8 \, 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} + 120 \, 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 = 13.23 (sec) , antiderivative size = 1487, normalized size of antiderivative = 4.37 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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