Integrand size = 27, antiderivative size = 292 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 b \left (2 a^4-7 a^2 b^2+5 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))} \]
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Time = 0.71 (sec) , antiderivative size = 292, 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 = {2969, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}-\frac {2 b \left (2 a^4-7 a^2 b^2+5 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}}-\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))} \]
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Rule 210
Rule 632
Rule 2739
Rule 2969
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 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))}+\frac {\int \frac {\csc ^4(c+d x) \left (4 \left (3 a^2-5 b^2\right )-a b \sin (c+d x)-\left (8 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^2 b} \\ & = -\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 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))}+\frac {\int \frac {\csc ^3(c+d x) \left (-3 b \left (13 a^2-20 b^2\right )+5 a b^2 \sin (c+d x)+8 b \left (3 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^3 b} \\ & = \frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 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))}+\frac {\int \frac {\csc ^2(c+d x) \left (8 b^2 \left (11 a^2-15 b^2\right )+a b \left (9 a^2-20 b^2\right ) \sin (c+d x)-3 b^2 \left (13 a^2-20 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^4 b} \\ & = -\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 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))}+\frac {\int \frac {\csc (c+d x) \left (3 b \left (3 a^4-36 a^2 b^2+40 b^4\right )-3 a b^2 \left (13 a^2-20 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^5 b} \\ & = -\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 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))}-\frac {\left (b \left (2 a^4-7 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^6}+\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \int \csc (c+d x) \, dx}{8 a^6} \\ & = -\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 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))}-\frac {\left (2 b \left (2 a^4-7 a^2 b^2+5 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^6 d} \\ & = -\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 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))}+\frac {\left (4 b \left (2 a^4-7 a^2 b^2+5 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^6 d} \\ & = -\frac {2 b \left (2 a^4-7 a^2 b^2+5 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 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))} \\ \end{align*}
Time = 7.02 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.70 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 b \left (2 a^4-7 a^2 b^2+5 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {2 \left (2 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )-3 b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{3 a^5 d}+\frac {\left (5 a^2-12 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^4 d}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{12 a^3 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d}+\frac {\left (-3 a^4+36 a^2 b^2-40 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^6 d}+\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^6 d}+\frac {\left (-5 a^2+12 b^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^4 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d}+\frac {2 \sec \left (\frac {1}{2} (c+d x)\right ) \left (2 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {-a^2 b^2 \cos (c+d x)+b^4 \cos (c+d x)}{a^5 d (a+b \sin (c+d x))}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{12 a^3 d} \]
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Time = 0.87 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{5}}-\frac {1}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+12 b^{2}}{32 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-72 a^{2} b^{2}+80 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{6}}+\frac {b}{12 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{2}-8 b^{2}\right )}{4 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 b \left (\frac {\frac {b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {b a \left (a^{2}-b^{2}\right )}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{4}-7 a^{2} b^{2}+5 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6}}}{d}\) | \(365\) |
default | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{5}}-\frac {1}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+12 b^{2}}{32 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-72 a^{2} b^{2}+80 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{6}}+\frac {b}{12 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{2}-8 b^{2}\right )}{4 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 b \left (\frac {\frac {b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {b a \left (a^{2}-b^{2}\right )}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{4}-7 a^{2} b^{2}+5 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6}}}{d}\) | \(365\) |
risch | \(-\frac {-39 i a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-137 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-600 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+410 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+720 i a \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-528 i a^{3} b \,{\mathrm e}^{5 i \left (d x +c \right )}+294 i a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}+180 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-360 i a \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+60 i a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+88 a^{2} b^{2}-120 b^{4}+48 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-312 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+568 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-392 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-120 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+480 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-720 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+480 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+30 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+18 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+18 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+30 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{5} d}+\frac {2 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 {2 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 {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{2}}+\frac {9 \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 {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{2}}-\frac {9 \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}\) | \(753\) |
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Leaf count of result is larger than twice the leaf count of optimal. 747 vs. \(2 (275) = 550\).
Time = 0.52 (sec) , antiderivative size = 1578, normalized size of antiderivative = 5.40 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.49 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.58 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {24 \, {\left (3 \, a^{4} - 36 \, 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^{6}} - \frac {384 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} + 5 \, 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^{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^{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} - 24 \, 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} + 240 \, 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 {150 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, 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} + 240 \, 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} - 24 \, 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 = 10.64 (sec) , antiderivative size = 1158, normalized size of antiderivative = 3.97 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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