Integrand size = 29, antiderivative size = 244 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b^2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}+\frac {b \left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}-\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \]
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Time = 0.71 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2972, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {2 b^2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}-\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \]
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Rule 210
Rule 632
Rule 2739
Rule 2972
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\int \frac {\csc ^4(c+d x) \left (4 \left (6 a^2-5 b^2\right )-a b \sin (c+d x)-5 \left (4 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{20 a^2} \\ & = \frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\int \frac {\csc ^3(c+d x) \left (-15 b \left (5 a^2-4 b^2\right )-a \left (12 a^2-5 b^2\right ) \sin (c+d x)+8 b \left (6 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{60 a^3} \\ & = -\frac {b \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 ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (-8 \left (3 a^4-20 a^2 b^2+15 b^4\right )+a b \left (21 a^2-20 b^2\right ) \sin (c+d x)-15 b^2 \left (5 a^2-4 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^4} \\ & = -\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \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 ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\int \frac {\csc (c+d x) \left (15 b \left (3 a^4-12 a^2 b^2+8 b^4\right )-15 a b^2 \left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^5} \\ & = -\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \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 ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\left (b^2 \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^6}-\frac {\left (b \left (3 a^4-12 a^2 b^2+8 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^6} \\ & = \frac {b \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)}{15 a^5 d}-\frac {b \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 ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\left (2 b^2 \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 {b \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)}{15 a^5 d}-\frac {b \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 ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\left (4 b^2 \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 {2 b^2 \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 {b \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)}{15 a^5 d}-\frac {b \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 ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(507\) vs. \(2(244)=488\).
Time = 1.98 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.08 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {1920 b^2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-32 \left (3 a^5-20 a^3 b^2+15 a b^4\right ) \cot \left (\frac {1}{2} (c+d x)\right )-150 a^4 b \csc ^2\left (\frac {1}{2} (c+d x)\right )+120 a^2 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+15 a^4 b \csc ^4\left (\frac {1}{2} (c+d x)\right )+360 a^4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-1440 a^2 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+960 b^5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-360 a^4 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1440 a^2 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-960 b^5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+150 a^4 b \sec ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )-15 a^4 b \sec ^4\left (\frac {1}{2} (c+d x)\right )-336 a^5 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+320 a^3 b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+21 a^5 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-20 a^3 b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-3 a^5 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+96 a^5 \tan \left (\frac {1}{2} (c+d x)\right )-640 a^3 b^2 \tan \left (\frac {1}{2} (c+d x)\right )+480 a b^4 \tan \left (\frac {1}{2} (c+d x)\right )+6 a^5 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{960 a^6 d} \]
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Time = 0.72 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.55
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}-a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+4 a^{3} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{5}}-\frac {1}{160 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-3 a^{2}+4 b^{2}}{96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {2 a^{4}-20 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (a^{2}-b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (3 a^{4}-12 a^{2} b^{2}+8 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{6}}+\frac {2 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6} \sqrt {a^{2}-b^{2}}}}{d}\) | \(379\) |
default | \(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}-a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+4 a^{3} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{5}}-\frac {1}{160 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-3 a^{2}+4 b^{2}}{96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {2 a^{4}-20 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (a^{2}-b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (3 a^{4}-12 a^{2} b^{2}+8 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{6}}+\frac {2 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6} \sqrt {a^{2}-b^{2}}}}{d}\) | \(379\) |
risch | \(\frac {880 i a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-720 i a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-120 i a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+75 a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-60 a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-120 i b^{4}+240 i a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-30 a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}+120 b^{3} a \,{\mathrm e}^{7 i \left (d x +c \right )}+160 i a^{2} b^{2}+480 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+480 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-560 i a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-720 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+30 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-120 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}-240 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-24 i a^{4}-120 i b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-75 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+60 \,{\mathrm e}^{i \left (d x +c \right )} b^{3} a}{60 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a^{2} d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{6} d}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{4}}+\frac {\sqrt {-a^{2}+b^{2}}\, b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{6}}+\frac {\sqrt {-a^{2}+b^{2}}\, b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{4}}-\frac {\sqrt {-a^{2}+b^{2}}\, b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{6}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a^{2} d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{4} d}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{6} d}\) | \(674\) |
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Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (227) = 454\).
Time = 0.60 (sec) , antiderivative size = 1051, normalized size of antiderivative = 4.31 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (227) = 454\).
Time = 0.45 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.98 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 30 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 60 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} - \frac {120 \, {\left (3 \, a^{4} b - 12 \, a^{2} b^{3} + 8 \, b^{5}\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} - 2 \, a^{2} b^{4} + b^{6}\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 {822 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3288 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2192 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 600 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 480 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 40 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{5}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
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Time = 12.50 (sec) , antiderivative size = 1082, normalized size of antiderivative = 4.43 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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