Integrand size = 21, antiderivative size = 313 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {a^2 b^3}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {\left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^4 d}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {3 a (a-3 b) \log (1-\cos (c+d x))}{16 (a+b)^5 d}-\frac {3 a (a+3 b) \log (1+\cos (c+d x))}{16 (a-b)^5 d}+\frac {3 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^5 d} \]
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Time = 1.46 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2916, 12, 1661, 1643} \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {3 a^2 b^2 \left (a^2+b^2\right )}{d \left (a^2-b^2\right )^4 (a \cos (c+d x)+b)}+\frac {\csc ^4(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{4 d \left (a^2-b^2\right )^3}-\frac {a^2 b^3}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)^2}+\frac {3 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^5}+\frac {\csc ^2(c+d x) \left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right )}{8 d \left (a^2-b^2\right )^4}+\frac {3 a (a-3 b) \log (1-\cos (c+d x))}{16 d (a+b)^5}-\frac {3 a (a+3 b) \log (\cos (c+d x)+1)}{16 d (a-b)^5} \]
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Rule 12
Rule 1643
Rule 1661
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(-b-a \cos (c+d x))^3} \, dx \\ & = \frac {a^5 \text {Subst}\left (\int \frac {x^3}{a^3 (-b+x)^3 \left (a^2-x^2\right )^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^2 \text {Subst}\left (\int \frac {x^3}{(-b+x)^3 \left (a^2-x^2\right )^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^4 b^3 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3}-\frac {a^2 b^2 \left (3 a^4-3 a^2 b^2-4 b^4\right ) x}{\left (a^2-b^2\right )^3}+\frac {a^4 b \left (3 a^2-23 b^2\right ) x^2}{\left (a^2-b^2\right )^3}+\frac {3 a^4 \left (a^2+3 b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(-b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{4 d} \\ & = \frac {\left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^4 d}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^4 b^3 \left (5 a^4+34 a^2 b^2+9 b^4\right )}{\left (a^2-b^2\right )^4}-\frac {3 a^4 b^2 \left (5 a^4+18 a^2 b^2-7 b^4\right ) x}{\left (a^2-b^2\right )^4}+\frac {a^4 b \left (15 a^4-26 a^2 b^2-37 b^4\right ) x^2}{\left (a^2-b^2\right )^4}+\frac {3 a^4 \left (a^4+10 a^2 b^2+5 b^4\right ) x^3}{\left (a^2-b^2\right )^4}}{(-b+x)^3 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{8 a^2 d} \\ & = \frac {\left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^4 d}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \left (\frac {3 a^3 (a+3 b)}{2 (a-b)^5 (a-x)}-\frac {8 a^4 b^3}{\left (a^2-b^2\right )^3 (b-x)^3}+\frac {24 a^4 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 (b-x)^2}-\frac {24 a^4 b \left (a^4+5 a^2 b^2+2 b^4\right )}{\left (a^2-b^2\right )^5 (b-x)}+\frac {3 a^3 (a-3 b)}{2 (a+b)^5 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{8 a^2 d} \\ & = -\frac {a^2 b^3}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {\left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^4 d}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {3 a (a-3 b) \log (1-\cos (c+d x))}{16 (a+b)^5 d}-\frac {3 a (a+3 b) \log (1+\cos (c+d x))}{16 (a-b)^5 d}+\frac {3 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^5 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.51 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.58 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \left (\frac {32 a^2 b^3}{(-a+b)^3 (a+b)^3}+\frac {192 a^2 (a-i b) (a+i b) b^2 (b+a \cos (c+d x))}{(a-b)^4 (a+b)^4}-\frac {384 i a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) (c+d x) (b+a \cos (c+d x))^2}{(a-b)^5 (a+b)^5}-\frac {24 i a (a-3 b) \arctan (\tan (c+d x)) (b+a \cos (c+d x))^2}{(a+b)^5}+\frac {24 i a (a+3 b) \arctan (\tan (c+d x)) (b+a \cos (c+d x))^2}{(a-b)^5}+\frac {6 (-a+b) (b+a \cos (c+d x))^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^4}-\frac {(b+a \cos (c+d x))^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}-\frac {12 a (a+3 b) (b+a \cos (c+d x))^2 \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^5}+\frac {192 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) (b+a \cos (c+d x))^2 \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^5}+\frac {12 a (a-3 b) (b+a \cos (c+d x))^2 \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^5}+\frac {6 (a+b) (b+a \cos (c+d x))^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{(a-b)^4}+\frac {(b+a \cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{(a-b)^3}\right ) \sec ^3(c+d x)}{64 d (a+b \sec (c+d x))^3} \]
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Time = 1.85 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {-\frac {1}{16 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {-3 a +3 b}{16 \left (a +b \right )^{4} \left (\cos \left (d x +c \right )-1\right )}+\frac {3 a \left (a -3 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{5}}-\frac {b^{3} a^{2}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a -b \right )^{5} \left (a +b \right )^{5}}+\frac {3 a^{2} b^{2} \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}+\frac {1}{16 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-3 a -3 b}{16 \left (a -b \right )^{4} \left (\cos \left (d x +c \right )+1\right )}-\frac {3 a \left (a +3 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{5}}}{d}\) | \(255\) |
default | \(\frac {-\frac {1}{16 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {-3 a +3 b}{16 \left (a +b \right )^{4} \left (\cos \left (d x +c \right )-1\right )}+\frac {3 a \left (a -3 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{5}}-\frac {b^{3} a^{2}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a -b \right )^{5} \left (a +b \right )^{5}}+\frac {3 a^{2} b^{2} \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}+\frac {1}{16 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-3 a -3 b}{16 \left (a -b \right )^{4} \left (\cos \left (d x +c \right )+1\right )}-\frac {3 a \left (a +3 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{5}}}{d}\) | \(255\) |
norman | \(\frac {-\frac {1}{64 d \left (a +b \right )}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{64 d \left (a -b \right )}-\frac {\left (3 a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{32 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (3 a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{32 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (15 a^{7}+459 a^{5} b^{2}+621 b^{4} a^{3}+57 a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 \left (a^{8}-2 a^{7} b -2 a^{6} b^{2}+6 a^{5} b^{3}-6 a^{3} b^{5}+2 a^{2} b^{6}+2 a \,b^{7}-b^{8}\right ) d}-\frac {\left (15 a^{7}+45 b \,a^{6}+459 a^{5} b^{2}+249 a^{4} b^{3}+621 b^{4} a^{3}+87 b^{5} a^{2}+57 a \,b^{6}+3 b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}+\frac {3 a \left (a -3 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right )}+\frac {3 b \,a^{2} \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \left (a^{10}-5 a^{8} b^{2}+10 a^{6} b^{4}-10 b^{6} a^{4}+5 b^{8} a^{2}-b^{10}\right )}\) | \(519\) |
parallelrisch | \(\frac {3 b \,a^{2} \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )+\frac {3 a \left (a -3 b \right ) \left (a -b \right )^{5} \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (a +b \right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\left (5 a^{7} b -\frac {313}{16} a^{6} b^{2}-\frac {251}{2} a^{5} b^{3}-\frac {647}{16} a^{4} b^{4}-\frac {359}{2} a^{3} b^{5}-\frac {459}{16} a^{2} b^{6}-24 a \,b^{7}-\frac {5}{16} a^{8}+5 b^{8}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {14}{3} a^{7} b -\frac {317}{24} a^{6} b^{2}+\frac {83}{3} a^{5} b^{3}-\frac {367}{24} a^{4} b^{4}+\frac {137}{3} a^{3} b^{5}+\frac {89}{24} a^{2} b^{6}+\frac {10}{3} a \,b^{7}-\frac {5}{8} a^{8}+\frac {17}{12} b^{8}\right ) \cos \left (4 d x +4 c \right )-\frac {5 \left (a +b \right ) \left (a^{6}+\frac {1}{4} a^{5} b +\frac {611}{20} a^{4} b^{2}+\frac {15}{2} a^{3} b^{3}+\frac {467}{10} a^{2} b^{4}-\frac {31}{4} a \,b^{5}+\frac {163}{20} b^{6}\right ) a \cos \left (3 d x +3 c \right )}{3}+\left (a^{6}-\frac {3}{4} a^{5} b +\frac {241}{12} a^{4} b^{2}-\frac {7}{6} a^{3} b^{3}+\frac {51}{2} a^{2} b^{4}+\frac {23}{12} a \,b^{5}+\frac {17}{12} b^{6}\right ) \left (a +b \right ) a \cos \left (5 d x +5 c \right )+\frac {5 \left (a^{6}+\frac {16}{15} a^{5} b +\frac {443}{15} a^{4} b^{2}+\frac {136}{15} a^{3} b^{3}+\frac {97}{3} a^{2} b^{4}+\frac {8}{3} a \,b^{5}+\frac {17}{15} b^{6}\right ) a^{2} \cos \left (6 d x +6 c \right )}{16}+\left (-10 a^{8}+\frac {25}{2} a^{7} b +64 a^{6} b^{2}+\frac {25}{2} a^{5} b^{3}+34 a^{4} b^{4}+\frac {139}{2} a^{3} b^{5}+8 a^{2} b^{6}+\frac {3}{2} a \,b^{7}\right ) \cos \left (d x +c \right )+\frac {17 b^{8}}{4}-\frac {59 a^{2} b^{6}}{8}+10 a^{7} b +10 a \,b^{7}+\frac {5 a^{8}}{8}+\frac {103 a^{6} b^{2}}{8}+63 a^{5} b^{3}+\frac {621 a^{4} b^{4}}{8}+165 a^{3} b^{5}\right )}{1024}}{\left (a -b \right )^{5} \left (a +b \right )^{5} d \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right )}\) | \(665\) |
risch | \(\text {Expression too large to display}\) | \(1848\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1803 vs. \(2 (303) = 606\).
Time = 0.56 (sec) , antiderivative size = 1803, normalized size of antiderivative = 5.76 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (303) = 606\).
Time = 0.23 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.26 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {48 \, {\left (a^{6} b + 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}} - \frac {3 \, {\left (a^{2} + 3 \, a b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} + \frac {3 \, {\left (a^{2} - 3 \, a b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} + \frac {2 \, {\left (38 \, a^{4} b^{3} + 56 \, a^{2} b^{5} + 2 \, b^{7} + 3 \, {\left (a^{7} + 18 \, a^{5} b^{2} + 13 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{5} - 6 \, {\left (a^{6} b - 8 \, a^{4} b^{3} - 9 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} - {\left (5 \, a^{7} + 103 \, a^{5} b^{2} + 91 \, a^{3} b^{4} - 7 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (2 \, a^{6} b - 23 \, a^{4} b^{3} - 26 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (55 \, a^{5} b^{2} + 46 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )\right )}}{a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10} + {\left (a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \cos \left (d x + c\right )^{5} - {\left (2 \, a^{10} - 9 \, a^{8} b^{2} + 16 \, a^{6} b^{4} - 14 \, a^{4} b^{6} + 6 \, a^{2} b^{8} - b^{10}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{10} - 6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} - 16 \, a^{4} b^{6} + 9 \, a^{2} b^{8} - 2 \, b^{10}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \cos \left (d x + c\right )}}{16 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1551 vs. \(2 (303) = 606\).
Time = 0.48 (sec) , antiderivative size = 1551, normalized size of antiderivative = 4.96 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
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Time = 14.98 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.15 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3}{16\,{\left (a+b\right )}^3}-\frac {15\,b}{16\,{\left (a+b\right )}^4}+\frac {3\,b^2}{4\,{\left (a+b\right )}^5}\right )}{d}+\frac {\frac {19\,a^4\,b^3+28\,a^2\,b^5+b^7}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (-2\,a^6\,b+23\,a^4\,b^3+26\,a^2\,b^5+b^7\right )}{2\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,{\cos \left (c+d\,x\right )}^4\,\left (-a^6\,b+8\,a^4\,b^3+9\,a^2\,b^5\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,{\cos \left (c+d\,x\right )}^5\,\left (a^7+18\,a^5\,b^2+13\,a^3\,b^4\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {a\,\cos \left (c+d\,x\right )\,\left (55\,a^4\,b^2+46\,a^2\,b^4-5\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {a\,{\cos \left (c+d\,x\right )}^3\,\left (5\,a^6+103\,a^4\,b^2+91\,a^2\,b^4-7\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2\,\left (a^2-2\,b^2\right )-{\cos \left (c+d\,x\right )}^4\,\left (2\,a^2-b^2\right )+b^2+a^2\,{\cos \left (c+d\,x\right )}^6+2\,a\,b\,\cos \left (c+d\,x\right )-4\,a\,b\,{\cos \left (c+d\,x\right )}^3+2\,a\,b\,{\cos \left (c+d\,x\right )}^5\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{4\,{\left (a-b\right )}^5}+\frac {15\,b}{16\,{\left (a-b\right )}^4}+\frac {3}{16\,{\left (a-b\right )}^3}\right )}{d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (3\,a^6\,b+15\,a^4\,b^3+6\,a^2\,b^5\right )}{d\,\left (a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}\right )} \]
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