Integrand size = 27, antiderivative size = 231 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}-\frac {2 \left (a^2-b^2\right )^{3/2} \left (6 a^2-b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 d}+\frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac {\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac {\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d} \]
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Time = 0.37 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2942, 2944, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 \left (a^2-b^2\right )^{3/2} \left (6 a^2-b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d}-\frac {\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac {a x \left (24 a^4-40 a^2 b^2+15 b^4\right )}{4 b^7}+\frac {\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}+\frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2942
Rule 2944
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac {\int \frac {\cos ^4(c+d x) (-b-6 a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{b^2} \\ & = \frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac {\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}-\frac {\int \frac {\cos ^2(c+d x) \left (2 b \left (3 a^2-2 b^2\right )+2 a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^4} \\ & = \frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac {\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac {\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}-\frac {\int \frac {-2 b \left (12 a^4-17 a^2 b^2+4 b^4\right )-2 a \left (24 a^4-40 a^2 b^2+15 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8 b^6} \\ & = \frac {a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}+\frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac {\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac {\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}-\frac {\left (\left (a^2-b^2\right )^2 \left (6 a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^7} \\ & = \frac {a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}+\frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac {\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac {\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}-\frac {\left (2 \left (a^2-b^2\right )^2 \left (6 a^2-b^2\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 )}{b^7 d} \\ & = \frac {a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}+\frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac {\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac {\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}+\frac {\left (4 \left (a^2-b^2\right )^2 \left (6 a^2-b^2\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 )}{b^7 d} \\ & = \frac {a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}-\frac {2 \left (a^2-b^2\right )^{3/2} \left (6 a^2-b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 d}+\frac {\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac {\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac {\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d} \\ \end{align*}
Time = 2.85 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.61 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-960 \left (a^2-b^2\right )^{3/2} \left (6 a^2-b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+\frac {2880 a^6 c-4800 a^4 b^2 c+1800 a^2 b^4 c+2880 a^6 d x-4800 a^4 b^2 d x+1800 a^2 b^4 d x+60 a b \left (48 a^4-74 a^2 b^2+23 b^4\right ) \cos (c+d x)+5 \left (24 a^3 b^3-31 a b^5\right ) \cos (3 (c+d x))-9 a b^5 \cos (5 (c+d x))+2880 a^5 b c \sin (c+d x)-4800 a^3 b^3 c \sin (c+d x)+1800 a b^5 c \sin (c+d x)+2880 a^5 b d x \sin (c+d x)-4800 a^3 b^3 d x \sin (c+d x)+1800 a b^5 d x \sin (c+d x)+720 a^4 b^2 \sin (2 (c+d x))-1080 a^2 b^4 \sin (2 (c+d x))+295 b^6 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))+32 b^6 \sin (4 (c+d x))+3 b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}}{480 b^7 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(467\) vs. \(2(214)=428\).
Time = 2.44 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.03
method | result | size |
derivativedivides | \(\frac {-\frac {4 \left (\frac {-\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{5} b}{2}+a^{3} b^{3}-\frac {a \,b^{5}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (6 a^{6}-13 a^{4} b^{2}+8 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}+\frac {\frac {4 \left (\left (a^{3} b^{2}-\frac {9}{8} a \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{4} b -\frac {9}{2} a^{2} b^{3}+\frac {3}{2} b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{3} b^{2}-\frac {5}{4} a \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{4} b -15 a^{2} b^{3}+3 b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b -20 a^{2} b^{3}+\frac {14}{3} b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{3} b^{2}+\frac {5}{4} a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{4} b -13 a^{2} b^{3}+\frac {7}{3} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{3} b^{2}+\frac {9}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 a^{4} b}{2}-\frac {7 a^{2} b^{3}}{2}+\frac {23 b^{5}}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {a \left (24 a^{4}-40 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{b^{7}}}{d}\) | \(468\) |
default | \(\frac {-\frac {4 \left (\frac {-\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{5} b}{2}+a^{3} b^{3}-\frac {a \,b^{5}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (6 a^{6}-13 a^{4} b^{2}+8 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}+\frac {\frac {4 \left (\left (a^{3} b^{2}-\frac {9}{8} a \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{4} b -\frac {9}{2} a^{2} b^{3}+\frac {3}{2} b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{3} b^{2}-\frac {5}{4} a \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{4} b -15 a^{2} b^{3}+3 b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b -20 a^{2} b^{3}+\frac {14}{3} b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{3} b^{2}+\frac {5}{4} a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{4} b -13 a^{2} b^{3}+\frac {7}{3} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{3} b^{2}+\frac {9}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 a^{4} b}{2}-\frac {7 a^{2} b^{3}}{2}+\frac {23 b^{5}}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {a \left (24 a^{4}-40 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{b^{7}}}{d}\) | \(468\) |
risch | \(\frac {6 a^{5} x}{b^{7}}-\frac {10 a^{3} x}{b^{5}}+\frac {15 a x}{4 b^{3}}+\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{2 b^{5} d}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{3}}+\frac {5 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 d \,b^{6}}-\frac {27 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{8 d \,b^{4}}+\frac {11 \,{\mathrm e}^{i \left (d x +c \right )}}{16 b^{2} d}+\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 d \,b^{6}}-\frac {27 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{8 d \,b^{4}}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 d \,b^{2}}+\frac {2 i a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{7} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {6 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right ) a^{4}}{d \,b^{7}}+\frac {7 i \sqrt {a^{2}-b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{3}}-\frac {6 i \sqrt {a^{2}-b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{7}}-\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{2 b^{3} d}-\frac {7 i \sqrt {a^{2}-b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}+\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{2 b^{3} d}-\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 b^{5} d}+\frac {\cos \left (5 d x +5 c \right )}{80 b^{2} d}+\frac {a \sin \left (4 d x +4 c \right )}{16 b^{3} d}-\frac {\cos \left (3 d x +3 c \right ) a^{2}}{4 b^{4} d}+\frac {7 \cos \left (3 d x +3 c \right )}{48 b^{2} d}\) | \(681\) |
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Time = 0.43 (sec) , antiderivative size = 697, normalized size of antiderivative = 3.02 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [-\frac {18 \, a b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (12 \, a^{3} b^{3} - 11 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (24 \, a^{6} - 40 \, a^{4} b^{2} + 15 \, a^{2} b^{4}\right )} d x - 30 \, {\left (6 \, a^{5} - 7 \, a^{3} b^{2} + a b^{4} + {\left (6 \, a^{4} b - 7 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 15 \, {\left (24 \, a^{5} b - 40 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right ) - {\left (12 \, b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (3 \, a^{2} b^{4} - 2 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (24 \, a^{5} b - 40 \, a^{3} b^{3} + 15 \, a b^{5}\right )} d x + 15 \, {\left (12 \, a^{4} b^{2} - 17 \, a^{2} b^{4} + 4 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{60 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}}, -\frac {18 \, a b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (12 \, a^{3} b^{3} - 11 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (24 \, a^{6} - 40 \, a^{4} b^{2} + 15 \, a^{2} b^{4}\right )} d x - 60 \, {\left (6 \, a^{5} - 7 \, a^{3} b^{2} + a b^{4} + {\left (6 \, a^{4} b - 7 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 15 \, {\left (24 \, a^{5} b - 40 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right ) - {\left (12 \, b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (3 \, a^{2} b^{4} - 2 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (24 \, a^{5} b - 40 \, a^{3} b^{3} + 15 \, a b^{5}\right )} d x + 15 \, {\left (12 \, a^{4} b^{2} - 17 \, a^{2} b^{4} + 4 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{60 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}}\right ] \]
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Timed out. \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (214) = 428\).
Time = 0.34 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.57 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {15 \, {\left (24 \, a^{5} - 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{b^{7}} - \frac {120 \, {\left (6 \, a^{6} - 13 \, a^{4} b^{2} + 8 \, 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}} b^{7}} + \frac {120 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, 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^{5} - 2 \, 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 )} b^{6}} + \frac {2 \, {\left (120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 300 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 540 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 180 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 240 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1200 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1800 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 360 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1800 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2400 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, 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} + 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1200 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1560 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 280 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 300 \, a^{4} - 420 \, a^{2} b^{2} + 92 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} b^{6}}}{60 \, d} \]
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Time = 14.48 (sec) , antiderivative size = 3135, normalized size of antiderivative = 13.57 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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