Integrand size = 27, antiderivative size = 228 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}+\frac {2 a \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 d}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d} \]
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Time = 0.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2944, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 a \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}-\frac {x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{16 b^7}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2944
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac {\int \frac {\cos ^4(c+d x) \left (-a b-\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 b^2} \\ & = -\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}+\frac {\int \frac {\cos ^2(c+d x) \left (3 a b \left (2 a^2-3 b^2\right )+3 \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 b^4} \\ & = -\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}+\frac {\int \frac {-3 a b \left (8 a^4-18 a^2 b^2+11 b^4\right )-3 \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{48 b^6} \\ & = -\frac {\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}+\frac {\left (a \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^7} \\ & = -\frac {\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}+\frac {\left (2 a \left (a^2-b^2\right )^3\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 {\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}-\frac {\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}-\frac {\left (4 a \left (a^2-b^2\right )^3\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 {\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}+\frac {2 a \left (a^2-b^2\right )^{5/2} \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-5 b \sin (c+d x))}{30 b^2 d}+\frac {\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac {\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d} \\ \end{align*}
Time = 1.51 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.21 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-960 a^6 c+2400 a^4 b^2 c-1800 a^2 b^4 c+300 b^6 c-960 a^6 d x+2400 a^4 b^2 d x-1800 a^2 b^4 d x+300 b^6 d x+1920 a \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-120 a b \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x)+20 \left (4 a^3 b^3-7 a b^5\right ) \cos (3 (c+d x))-12 a b^5 \cos (5 (c+d x))+240 a^4 b^2 \sin (2 (c+d x))-480 a^2 b^4 \sin (2 (c+d x))+225 b^6 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))+45 b^6 \sin (4 (c+d x))+5 b^6 \sin (6 (c+d x))}{960 b^7 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(523\) vs. \(2(215)=430\).
Time = 1.02 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.30
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (\frac {1}{2} a^{4} b^{2}-\frac {9}{8} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{5} b -3 a^{3} b^{3}+3 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{2} a^{4} b^{2}-\frac {19}{8} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{5} b -13 a^{3} b^{3}+9 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{4} b^{2}-\frac {5}{4} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{5} b -\frac {70}{3} a^{3} b^{3}+\frac {46}{3} a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{4} b^{2}+\frac {5}{4} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{5} b -22 a^{3} b^{3}+14 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{2} a^{4} b^{2}+\frac {19}{8} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{5} b -11 a^{3} b^{3}+\frac {31}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{4} b^{2}+\frac {9}{8} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{5} b -\frac {7 a^{3} b^{3}}{3}+\frac {23 a \,b^{5}}{15}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (16 a^{6}-40 a^{4} b^{2}+30 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{7}}+\frac {2 a \left (a^{6}-3 a^{4} b^{2}+3 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 )}{b^{7} \sqrt {a^{2}-b^{2}}}}{d}\) | \(524\) |
default | \(\frac {-\frac {2 \left (\frac {\left (\frac {1}{2} a^{4} b^{2}-\frac {9}{8} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{5} b -3 a^{3} b^{3}+3 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{2} a^{4} b^{2}-\frac {19}{8} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{5} b -13 a^{3} b^{3}+9 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{4} b^{2}-\frac {5}{4} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{5} b -\frac {70}{3} a^{3} b^{3}+\frac {46}{3} a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{4} b^{2}+\frac {5}{4} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{5} b -22 a^{3} b^{3}+14 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{2} a^{4} b^{2}+\frac {19}{8} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{5} b -11 a^{3} b^{3}+\frac {31}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{4} b^{2}+\frac {9}{8} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{5} b -\frac {7 a^{3} b^{3}}{3}+\frac {23 a \,b^{5}}{15}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (16 a^{6}-40 a^{4} b^{2}+30 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{7}}+\frac {2 a \left (a^{6}-3 a^{4} b^{2}+3 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 )}{b^{7} \sqrt {a^{2}-b^{2}}}}{d}\) | \(524\) |
risch | \(-\frac {2 i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}+\frac {i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{7}}+\frac {2 i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}+\frac {5 x}{16 b}-\frac {i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{7}}-\frac {11 a \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b^{2} d}-\frac {\sin \left (2 d x +2 c \right ) a^{2}}{2 b^{3} d}-\frac {a \cos \left (5 d x +5 c \right )}{80 d \,b^{2}}-\frac {\sin \left (4 d x +4 c \right ) a^{2}}{32 b^{3} d}+\frac {a^{3} \cos \left (3 d x +3 c \right )}{12 d \,b^{4}}-\frac {7 a \cos \left (3 d x +3 c \right )}{48 d \,b^{2}}+\frac {\sin \left (2 d x +2 c \right ) a^{4}}{4 b^{5} d}+\frac {\sin \left (6 d x +6 c \right )}{192 b d}+\frac {3 \sin \left (4 d x +4 c \right )}{64 b d}-\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{3}}-\frac {11 a \,{\mathrm e}^{i \left (d x +c \right )}}{16 d \,b^{2}}-\frac {x \,a^{6}}{b^{7}}+\frac {5 x \,a^{4}}{2 b^{5}}-\frac {15 x \,a^{2}}{8 b^{3}}+\frac {15 \sin \left (2 d x +2 c \right )}{64 b d}-\frac {a^{5} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{6} d}+\frac {9 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 b^{4} d}+\frac {9 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 b^{4} d}-\frac {a^{5} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{6} d}\) | \(638\) |
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Time = 0.49 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.50 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{6} - 40 \, a^{4} b^{2} + 30 \, a^{2} b^{4} - 5 \, b^{6}\right )} d x - 120 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 240 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 5 \, {\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + 5 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}, -\frac {48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{6} - 40 \, a^{4} b^{2} + 30 \, a^{2} b^{4} - 5 \, b^{6}\right )} d x + 240 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 240 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 5 \, {\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + 5 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}\right ] \]
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Timed out. \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (214) = 428\).
Time = 0.34 (sec) , antiderivative size = 735, normalized size of antiderivative = 3.22 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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Time = 14.54 (sec) , antiderivative size = 3683, normalized size of antiderivative = 16.15 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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