Integrand size = 29, antiderivative size = 274 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \csc (c+d x)}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {\left (24 a^2-57 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}+\frac {b^8 \log (a+b \sin (c+d x))}{a^3 \left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\sin (c+d x))^2}+\frac {9 a+11 b}{16 (a+b)^2 d (1-\sin (c+d x))}+\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}+\frac {9 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))} \]
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Time = 0.34 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 908} \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac {\left (24 a^2-57 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac {b \csc (c+d x)}{a^2 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 d}+\frac {b^8 \log (a+b \sin (c+d x))}{a^3 d \left (a^2-b^2\right )^3}+\frac {9 a+11 b}{16 d (a+b)^2 (1-\sin (c+d x))}+\frac {9 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac {1}{16 d (a+b) (1-\sin (c+d x))^2}+\frac {1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac {\csc ^2(c+d x)}{2 a d} \]
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Rule 12
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {b^3}{x^3 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^8 \text {Subst}\left (\int \frac {1}{x^3 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^8 \text {Subst}\left (\int \left (\frac {1}{8 b^6 (a+b) (b-x)^3}+\frac {9 a+11 b}{16 b^7 (a+b)^2 (b-x)^2}+\frac {24 a^2+57 a b+35 b^2}{16 b^8 (a+b)^3 (b-x)}+\frac {1}{a b^6 x^3}-\frac {1}{a^2 b^6 x^2}+\frac {3 a^2+b^2}{a^3 b^8 x}+\frac {1}{a^3 (a-b)^3 (a+b)^3 (a+x)}+\frac {1}{8 b^6 (-a+b) (b+x)^3}+\frac {-9 a+11 b}{16 (a-b)^2 b^7 (b+x)^2}+\frac {24 a^2-57 a b+35 b^2}{16 b^8 (-a+b)^3 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b \csc (c+d x)}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {\left (24 a^2-57 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}+\frac {b^8 \log (a+b \sin (c+d x))}{a^3 \left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\sin (c+d x))^2}+\frac {9 a+11 b}{16 (a+b)^2 d (1-\sin (c+d x))}+\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}+\frac {9 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))} \\ \end{align*}
Time = 6.17 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b^8 \left (\frac {\csc (c+d x)}{a^2 b^7}-\frac {\csc ^2(c+d x)}{2 a b^8}-\frac {\left (24 a^2+57 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 b^8 (a+b)^3}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{a^3 b^8}-\frac {\left (24 a^2-57 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 b^8}+\frac {\log (a+b \sin (c+d x))}{a^3 (a-b)^3 (a+b)^3}+\frac {1}{16 b^6 (a+b) (b-b \sin (c+d x))^2}+\frac {9 a+11 b}{16 b^7 (a+b)^2 (b-b \sin (c+d x))}+\frac {1}{16 (a-b) b^6 (b+b \sin (c+d x))^2}+\frac {9 a-11 b}{16 (a-b)^2 b^7 (b+b \sin (c+d x))}\right )}{d} \]
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Time = 1.52 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {9 a +11 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-24 a^{2}-57 a b -35 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 a \sin \left (d x +c \right )^{2}}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{3}}+\frac {b}{a^{2} \sin \left (d x +c \right )}+\frac {b^{8} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a^{3}}+\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-9 a +11 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-24 a^{2}+57 a b -35 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}}{d}\) | \(238\) |
default | \(\frac {\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {9 a +11 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-24 a^{2}-57 a b -35 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 a \sin \left (d x +c \right )^{2}}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{3}}+\frac {b}{a^{2} \sin \left (d x +c \right )}+\frac {b^{8} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a^{3}}+\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-9 a +11 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-24 a^{2}+57 a b -35 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}}{d}\) | \(238\) |
parallelrisch | \(\frac {256 b^{8} \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-768 \left (a^{2}+\frac {19}{8} a b +\frac {35}{24} b^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right )^{3} a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-54 \left (\frac {128 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{2} \left (a^{2}-\frac {19}{8} a b +\frac {35}{24} b^{2}\right ) a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{9}+\left (-\frac {128 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a^{2}+\frac {b^{2}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (a^{4}-\frac {89}{54} a^{2} b^{2}+\frac {11}{18} b^{4}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {2}{9} a^{4}-\frac {11}{27} a^{2} b^{2}+\frac {1}{9} b^{4}\right ) \cos \left (4 d x +4 c \right )+\left (-\frac {1}{9} a^{4}+\frac {1}{6} a^{2} b^{2}-\frac {1}{54} b^{4}\right ) \cos \left (6 d x +6 c \right )-\frac {13 a^{2} b^{2}}{27}+\frac {13 b^{4}}{27}+\frac {2 a^{4}}{27}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {80 b \left (\left (a^{4}-\frac {9}{5} a^{2} b^{2}+\frac {4}{5} b^{4}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {3}{8} a^{4}-\frac {27}{40} a^{2} b^{2}+\frac {1}{5} b^{4}\right ) \cos \left (4 d x +4 c \right )+\frac {9 a^{4}}{40}-\frac {29 a^{2} b^{2}}{40}+\frac {3 b^{4}}{5}\right )}{27}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right ) a \right ) \left (a -b \right )\right ) \left (a +b \right )}{64 \left (a -b \right )^{3} \left (a +b \right )^{3} a^{3} d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(507\) |
norman | \(\frac {-\frac {1}{8 a d}-\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2} d}+\frac {b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} d}+\frac {2 \left (-5 a^{4}+8 a^{2} b^{2}-b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (-57 a^{4}+82 a^{2} b^{2}-9 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (-57 a^{4}+82 a^{2} b^{2}-9 b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (15 a^{4}-25 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (15 a^{4}-25 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (5 a^{4}-13 a^{2} b^{2}+4 b^{4}\right ) b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (5 a^{4}-13 a^{2} b^{2}+4 b^{4}\right ) b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}+\frac {b^{8} \ln \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 )}{a^{3} d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {\left (24 a^{2}-57 a b +35 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {\left (24 a^{2}+57 a b +35 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}\) | \(683\) |
risch | \(\text {Expression too large to display}\) | \(1418\) |
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Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (256) = 512\).
Time = 3.31 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.34 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {4 \, a^{8} - 8 \, a^{6} b^{2} + 4 \, a^{4} b^{4} - 8 \, {\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 5 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{2} - 16 \, {\left (b^{8} \cos \left (d x + c\right )^{6} - b^{8} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 16 \, {\left ({\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - b^{8}\right )} \cos \left (d x + c\right )^{6} - {\left (3 \, a^{8} - 8 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - b^{8}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) + {\left ({\left (24 \, a^{8} + 15 \, a^{7} b - 64 \, a^{6} b^{2} - 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} + 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{6} - {\left (24 \, a^{8} + 15 \, a^{7} b - 64 \, a^{6} b^{2} - 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} + 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (24 \, a^{8} - 15 \, a^{7} b - 64 \, a^{6} b^{2} + 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} - 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{6} - {\left (24 \, a^{8} - 15 \, a^{7} b - 64 \, a^{6} b^{2} + 42 \, a^{5} b^{3} + 48 \, a^{4} b^{4} - 35 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a^{7} b - 4 \, a^{5} b^{3} + 2 \, a^{3} b^{5} - {\left (15 \, a^{7} b - 42 \, a^{5} b^{3} + 35 \, a^{3} b^{5} - 8 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (5 \, a^{7} b - 14 \, a^{5} b^{3} + 9 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{6} - {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
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Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.54 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {16 \, b^{8} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}} - \frac {{\left (24 \, a^{2} - 57 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (24 \, a^{2} + 57 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left ({\left (15 \, a^{4} b - 27 \, a^{2} b^{3} + 8 \, b^{5}\right )} \sin \left (d x + c\right )^{5} - 4 \, a^{5} + 8 \, a^{3} b^{2} - 4 \, a b^{4} - 4 \, {\left (3 \, a^{5} - 5 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{4} - {\left (25 \, a^{4} b - 45 \, a^{2} b^{3} + 16 \, b^{5}\right )} \sin \left (d x + c\right )^{3} + 2 \, {\left (9 \, a^{5} - 15 \, a^{3} b^{2} + 4 \, a b^{4}\right )} \sin \left (d x + c\right )^{2} + 8 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{6} - 2 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{4} + {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{2}} + \frac {16 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{16 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (256) = 512\).
Time = 0.37 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.15 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {16 \, b^{9} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}} - \frac {{\left (24 \, a^{2} - 57 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (24 \, a^{2} + 57 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {16 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (4 \, b^{8} \sin \left (d x + c\right )^{6} + 15 \, a^{7} b \sin \left (d x + c\right )^{5} - 42 \, a^{5} b^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{5} \sin \left (d x + c\right )^{5} - 8 \, a b^{7} \sin \left (d x + c\right )^{5} - 12 \, a^{8} \sin \left (d x + c\right )^{4} + 32 \, a^{6} b^{2} \sin \left (d x + c\right )^{4} - 24 \, a^{4} b^{4} \sin \left (d x + c\right )^{4} + 4 \, a^{2} b^{6} \sin \left (d x + c\right )^{4} - 8 \, b^{8} \sin \left (d x + c\right )^{4} - 25 \, a^{7} b \sin \left (d x + c\right )^{3} + 70 \, a^{5} b^{3} \sin \left (d x + c\right )^{3} - 61 \, a^{3} b^{5} \sin \left (d x + c\right )^{3} + 16 \, a b^{7} \sin \left (d x + c\right )^{3} + 18 \, a^{8} \sin \left (d x + c\right )^{2} - 48 \, a^{6} b^{2} \sin \left (d x + c\right )^{2} + 38 \, a^{4} b^{4} \sin \left (d x + c\right )^{2} - 8 \, a^{2} b^{6} \sin \left (d x + c\right )^{2} + 4 \, b^{8} \sin \left (d x + c\right )^{2} + 8 \, a^{7} b \sin \left (d x + c\right ) - 24 \, a^{5} b^{3} \sin \left (d x + c\right ) + 24 \, a^{3} b^{5} \sin \left (d x + c\right ) - 8 \, a b^{7} \sin \left (d x + c\right ) - 4 \, a^{8} + 12 \, a^{6} b^{2} - 12 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right )}}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} {\left (\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )\right )}^{2}}}{16 \, d} \]
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Time = 12.83 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.50 \[ \int \frac {\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )\right )\,\left (3\,a^2+b^2\right )}{a^3\,d}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {b^2}{8\,{\left (a-b\right )}^3}-\frac {9\,b}{16\,{\left (a-b\right )}^2}+\frac {3}{2\,\left (a-b\right )}\right )}{d}-\frac {\frac {1}{2\,a}-\frac {b\,\sin \left (c+d\,x\right )}{a^2}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (3\,a^4-5\,a^2\,b^2+b^4\right )}{2\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (9\,a^4-15\,a^2\,b^2+4\,b^4\right )}{4\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^5\,\left (15\,a^4\,b-27\,a^2\,b^3+8\,b^5\right )}{8\,a^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (25\,a^4\,b-45\,a^2\,b^3+16\,b^5\right )}{8\,a^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^6-2\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {9\,b}{16\,{\left (a+b\right )}^2}+\frac {3}{2\,\left (a+b\right )}+\frac {b^2}{8\,{\left (a+b\right )}^3}\right )}{d}+\frac {b^8\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^9-3\,a^7\,b^2+3\,a^5\,b^4-a^3\,b^6\right )} \]
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