Integrand size = 21, antiderivative size = 259 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {a^3 b^2}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^3 d}+\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac {\left (3 a^2-4 a b-b^2\right ) \log (1-\cos (c+d x))}{16 (a+b)^4 d}-\frac {\left (3 a^2+4 a b-b^2\right ) \log (1+\cos (c+d x))}{16 (a-b)^4 d}+\frac {2 a^3 b \left (a^2+2 b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d} \]
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Time = 0.96 (sec) , antiderivative size = 259, 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} \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\left (3 a^2-4 a b-b^2\right ) \log (1-\cos (c+d x))}{16 d (a+b)^4}-\frac {\left (3 a^2+4 a b-b^2\right ) \log (\cos (c+d x)+1)}{16 d (a-b)^4}+\frac {\csc ^4(c+d x) \left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right )}{4 d \left (a^2-b^2\right )^2}+\frac {\csc ^2(c+d x) \left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right )}{8 d \left (a^2-b^2\right )^3}+\frac {a^3 b^2}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac {2 a^3 b \left (a^2+2 b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4} \]
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Rule 12
Rule 1643
Rule 1661
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{(-b-a \cos (c+d x))^2} \, dx \\ & = \frac {a^5 \text {Subst}\left (\int \frac {x^2}{a^2 (-b+x)^2 \left (a^2-x^2\right )^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^3 \text {Subst}\left (\int \frac {x^2}{(-b+x)^2 \left (a^2-x^2\right )^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac {a \text {Subst}\left (\int \frac {-\frac {a^2 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^2}+\frac {2 a^2 b \left (a^2-3 b^2\right ) x}{\left (a^2-b^2\right )^2}+\frac {3 a^2 \left (a^2+b^2\right ) x^2}{\left (a^2-b^2\right )^2}}{(-b+x)^2 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{4 d} \\ & = \frac {\left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^3 d}+\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {-\frac {a^2 b^2 \left (5 a^4+12 a^2 b^2-b^4\right )}{\left (a^2-b^2\right )^3}+\frac {2 a^2 b \left (5 a^2+b^2\right ) x}{\left (a^2-b^2\right )^2}+\frac {a^2 \left (3 a^4+12 a^2 b^2+b^4\right ) x^2}{\left (a^2-b^2\right )^3}}{(-b+x)^2 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{8 a d} \\ & = \frac {\left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^3 d}+\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac {\text {Subst}\left (\int \left (\frac {a \left (3 a^2+4 a b-b^2\right )}{2 (a-b)^4 (a-x)}+\frac {8 a^4 b^2}{\left (a^2-b^2\right )^3 (b-x)^2}-\frac {16 a^4 b \left (a^2+2 b^2\right )}{\left (a^2-b^2\right )^4 (b-x)}+\frac {a \left (3 a^2-4 a b-b^2\right )}{2 (a+b)^4 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{8 a d} \\ & = \frac {a^3 b^2}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^3 d}+\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac {\left (3 a^2-4 a b-b^2\right ) \log (1-\cos (c+d x))}{16 (a+b)^4 d}-\frac {\left (3 a^2+4 a b-b^2\right ) \log (1+\cos (c+d x))}{16 (a-b)^4 d}+\frac {2 a^3 b \left (a^2+2 b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d} \\ \end{align*}
Time = 1.62 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.24 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {(b+a \cos (c+d x)) \left (\frac {64 a^3 b^2}{(a-b)^3 (a+b)^3}+\frac {2 (-3 a+b) (b+a \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}-\frac {(b+a \cos (c+d x)) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b)^2}+\frac {8 \left (-3 a^2-4 a b+b^2\right ) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^4}+\frac {128 a^3 b \left (a^2+2 b^2\right ) (b+a \cos (c+d x)) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4}+\frac {8 \left (3 a^2-4 a b-b^2\right ) (b+a \cos (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^4}+\frac {2 (3 a+b) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{(a-b)^3}+\frac {(b+a \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )}{(a-b)^2}\right ) \sec ^2(c+d x)}{64 d (a+b \sec (c+d x))^2} \]
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Time = 1.47 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} b^{2}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{16 \left (a +b \right )^{2} \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {-3 a +b}{16 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}+\frac {1}{16 \left (a -b \right )^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-3 a -b}{16 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-3 a^{2}-4 a b +b^{2}\right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{4}}}{d}\) | \(221\) |
default | \(\frac {\frac {a^{3} b^{2}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{16 \left (a +b \right )^{2} \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {-3 a +b}{16 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}+\frac {1}{16 \left (a -b \right )^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-3 a -b}{16 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-3 a^{2}-4 a b +b^{2}\right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{4}}}{d}\) | \(221\) |
norman | \(\frac {\frac {1}{64 d \left (a +b \right )}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64 d \left (a -b \right )}+\frac {\left (7 a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (7 a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (a^{6}+18 a^{4} b^{2}+5 a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 d \left (a^{7}-b \,a^{6}-3 a^{5} b^{2}+3 a^{4} b^{3}+3 b^{4} a^{3}-3 b^{5} a^{2}-a \,b^{6}+b^{7}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}+\frac {\left (3 a^{2}-4 a b -b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {2 a^{3} b \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}\) | \(388\) |
parallelrisch | \(\frac {8192 b \,a^{3} \left (a^{2}+2 b^{2}\right ) \left (b +a \cos \left (d x +c \right )\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )+1536 \left (b +a \cos \left (d x +c \right )\right ) \left (a^{2}-\frac {4}{3} a b -\frac {1}{3} b^{2}\right ) \left (a -b \right )^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (a +b \right ) \left (\left (-16 a^{6}+8 a^{5} b -116 a^{4} b^{2}-28 a^{3} b^{3}-60 a^{2} b^{4}+20 a \,b^{5}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {31}{2} a^{5} b +\frac {23}{2} a^{3} b^{3}-\frac {53}{2} a^{2} b^{4}+4 a \,b^{5}-\frac {77}{2} a^{4} b^{2}-4 b^{6}-3 a^{6}\right ) \cos \left (3 d x +3 c \right )+\left (6 a^{6}-4 a^{5} b +37 a^{4} b^{2}-a^{3} b^{3}+5 a^{2} b^{4}+5 a \,b^{5}\right ) \cos \left (4 d x +4 c \right )+a^{2} \left (a^{4}-\frac {3}{2} a^{3} b +\frac {39}{2} a^{2} b^{2}+\frac {3}{2} a \,b^{3}+\frac {7}{2} b^{4}\right ) \cos \left (5 d x +5 c \right )+\left (2 a^{6}+49 a^{5} b -13 a^{4} b^{2}-77 a^{3} b^{3}+87 a^{2} b^{4}+28 a \,b^{5}-28 b^{6}\right ) \cos \left (d x +c \right )-22 a^{6}+28 a^{5} b +143 a^{4} b^{2}-35 a^{3} b^{3}+23 a^{2} b^{4}+7 a \,b^{5}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4096 d \left (a -b \right )^{4} \left (a +b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}\) | \(446\) |
risch | \(\text {Expression too large to display}\) | \(1539\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1205 vs. \(2 (251) = 502\).
Time = 0.49 (sec) , antiderivative size = 1205, normalized size of antiderivative = 4.65 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (251) = 502\).
Time = 0.22 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.97 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {32 \, {\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (3 \, a^{2} + 4 \, a b - b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {{\left (3 \, a^{2} - 4 \, a b - b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (20 \, a^{3} b^{2} + 4 \, a b^{4} + {\left (3 \, a^{5} + 20 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{4} - {\left (5 \, a^{4} b - 4 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{5} + 36 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (7 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )\right )}}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{5} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )}}{16 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (251) = 502\).
Time = 0.40 (sec) , antiderivative size = 710, normalized size of antiderivative = 2.74 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (3 \, a^{2} - 4 \, a b - b^{2}\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {128 \, {\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {\frac {8 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (a^{2} + 2 \, a b + b^{2} - \frac {8 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {24 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {128 \, {\left (a^{6} b + a^{4} b^{3} + 2 \, a^{3} b^{4} + \frac {a^{6} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}}{64 \, d} \]
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Time = 14.70 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.73 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3}{16\,{\left (a+b\right )}^2}-\frac {5\,b}{8\,{\left (a+b\right )}^3}+\frac {3\,b^2}{8\,{\left (a+b\right )}^4}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{8\,{\left (a-b\right )}^4}+\frac {5\,b}{8\,{\left (a-b\right )}^3}+\frac {3}{16\,{\left (a-b\right )}^2}\right )}{d}+\frac {\frac {{\cos \left (c+d\,x\right )}^4\,\left (3\,a^5+20\,a^3\,b^2+a\,b^4\right )}{8\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {\cos \left (c+d\,x\right )\,\left (7\,a^2\,b-b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\cos \left (c+d\,x\right )}^3\,\left (5\,a^2\,b+b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (5\,a^5+36\,a^3\,b^2+7\,a\,b^4\right )}{8\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {b\,\left (5\,a^3\,b+a\,b^3\right )}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (a\,{\cos \left (c+d\,x\right )}^5+b\,{\cos \left (c+d\,x\right )}^4-2\,a\,{\cos \left (c+d\,x\right )}^3-2\,b\,{\cos \left (c+d\,x\right )}^2+a\,\cos \left (c+d\,x\right )+b\right )}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^5\,b+4\,a^3\,b^3\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )} \]
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