Integrand size = 21, antiderivative size = 423 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) x-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d} \]
[Out]
Time = 0.82 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2771, 2941, 2748, 2715, 8} \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \sin (c+d x) \cos (c+d x)}{256 d}+\frac {1}{256} x \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d} \]
[In]
[Out]
Rule 8
Rule 2715
Rule 2748
Rule 2771
Rule 2941
Rubi steps \begin{align*} \text {integral}& = -\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{10} \int \cos ^2(c+d x) (a+b \sin (c+d x))^6 \left (10 a^2+7 b^2+17 a b \sin (c+d x)\right ) \, dx \\ & = -\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{90} \int \cos ^2(c+d x) (a+b \sin (c+d x))^5 \left (15 a \left (6 a^2+11 b^2\right )+3 b \left (64 a^2+21 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{720} \int \cos ^2(c+d x) (a+b \sin (c+d x))^4 \left (15 \left (48 a^4+152 a^2 b^2+21 b^4\right )+15 a b \left (112 a^2+109 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {\int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \left (15 a \left (336 a^4+1512 a^2 b^2+583 b^4\right )+15 b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \sin (c+d x)\right ) \, dx}{5040} \\ & = -\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {\int \cos ^2(c+d x) (a+b \sin (c+d x))^2 \left (45 \left (672 a^6+3808 a^4 b^2+2666 a^2 b^4+147 b^6\right )+585 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \sin (c+d x)\right ) \, dx}{30240} \\ & = -\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {\int \cos ^2(c+d x) (a+b \sin (c+d x)) \left (45 a \left (3360 a^6+21952 a^4 b^2+22378 a^2 b^4+3361 b^6\right )+45 b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \sin (c+d x)\right ) \, dx}{151200} \\ & = -\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {\int \cos ^2(c+d x) \left (4725 \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right )+495 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \sin (c+d x)\right ) \, dx}{604800} \\ & = -\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{128} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac {1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \int 1 \, dx \\ & = \frac {1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) x-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d} \\ \end{align*}
Time = 2.06 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.08 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {322560 a^8 c+2257920 a^6 b^2 c+2822400 a^4 b^4 c+705600 a^2 b^6 c+17640 b^8 c+322560 a^8 d x+2257920 a^6 b^2 d x+2822400 a^4 b^4 d x+705600 a^2 b^6 d x+17640 b^8 d x-40320 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \cos (c+d x)-26880 \left (16 a^7 b+28 a^5 b^3+7 a^3 b^5\right ) \cos (3 (c+d x))+451584 a^5 b^3 \cos (5 (c+d x))+338688 a^3 b^5 \cos (5 (c+d x))+32256 a b^7 \cos (5 (c+d x))-80640 a^3 b^5 \cos (7 (c+d x))-14400 a b^7 \cos (7 (c+d x))+2240 a b^7 \cos (9 (c+d x))+161280 a^8 \sin (2 (c+d x))-705600 a^4 b^4 \sin (2 (c+d x))-282240 a^2 b^6 \sin (2 (c+d x))-8820 b^8 \sin (2 (c+d x))-564480 a^6 b^2 \sin (4 (c+d x))-705600 a^4 b^4 \sin (4 (c+d x))-141120 a^2 b^6 \sin (4 (c+d x))-2520 b^8 \sin (4 (c+d x))+235200 a^4 b^4 \sin (6 (c+d x))+94080 a^2 b^6 \sin (6 (c+d x))+2730 b^8 \sin (6 (c+d x))-17640 a^2 b^6 \sin (8 (c+d x))-945 b^8 \sin (8 (c+d x))+126 b^8 \sin (10 (c+d x))}{645120 d} \]
[In]
[Out]
Time = 2.15 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(\frac {\left (161280 a^{8}-705600 a^{4} b^{4}-282240 a^{2} b^{6}-8820 b^{8}\right ) \sin \left (2 d x +2 c \right )+\left (-564480 a^{6} b^{2}-705600 a^{4} b^{4}-141120 a^{2} b^{6}-2520 b^{8}\right ) \sin \left (4 d x +4 c \right )+\left (-430080 a^{7} b -752640 a^{5} b^{3}-188160 a^{3} b^{5}\right ) \cos \left (3 d x +3 c \right )+\left (451584 a^{5} b^{3}+338688 a^{3} b^{5}+32256 a \,b^{7}\right ) \cos \left (5 d x +5 c \right )+\left (235200 a^{4} b^{4}+94080 a^{2} b^{6}+2730 b^{8}\right ) \sin \left (6 d x +6 c \right )+\left (-80640 a^{3} b^{5}-14400 a \,b^{7}\right ) \cos \left (7 d x +7 c \right )+\left (-17640 a^{2} b^{6}-945 b^{8}\right ) \sin \left (8 d x +8 c \right )+126 b^{8} \sin \left (10 d x +10 c \right )+2240 a \,b^{7} \cos \left (9 d x +9 c \right )-1290240 \left (\frac {7}{32} b^{6}+a^{6}+\frac {7}{2} a^{4} b^{2}+\frac {35}{16} a^{2} b^{4}\right ) a b \cos \left (d x +c \right )+322560 a^{8} d x +2257920 a^{6} b^{2} d x +2822400 a^{4} b^{4} d x +705600 a^{2} b^{6} d x +17640 b^{8} d x -1720320 a^{7} b -4816896 a^{5} b^{3}-2752512 a^{3} b^{5}-262144 a \,b^{7}}{645120 d}\) | \(367\) |
derivativedivides | \(\frac {a^{8} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 a^{7} b \left (\cos ^{3}\left (d x +c \right )\right )}{3}+28 a^{6} b^{2} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+56 a^{5} b^{3} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+70 a^{4} b^{4} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+56 a^{3} b^{5} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+28 a^{2} b^{6} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {5 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{128}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+8 a \,b^{7} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{21}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{105}-\frac {16 \left (\cos ^{3}\left (d x +c \right )\right )}{315}\right )+b^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{10}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{80}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{96}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{128}+\frac {7 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{256}+\frac {7 d x}{256}+\frac {7 c}{256}\right )}{d}\) | \(497\) |
default | \(\frac {a^{8} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 a^{7} b \left (\cos ^{3}\left (d x +c \right )\right )}{3}+28 a^{6} b^{2} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+56 a^{5} b^{3} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+70 a^{4} b^{4} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+56 a^{3} b^{5} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+28 a^{2} b^{6} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {5 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{128}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+8 a \,b^{7} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{21}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{105}-\frac {16 \left (\cos ^{3}\left (d x +c \right )\right )}{315}\right )+b^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{10}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{80}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{96}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{128}+\frac {7 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{256}+\frac {7 d x}{256}+\frac {7 c}{256}\right )}{d}\) | \(497\) |
risch | \(\frac {a \,b^{7} \cos \left (9 d x +9 c \right )}{288 d}-\frac {35 \sin \left (2 d x +2 c \right ) a^{4} b^{4}}{32 d}-\frac {7 \sin \left (2 d x +2 c \right ) a^{2} b^{6}}{16 d}-\frac {3 \sin \left (8 d x +8 c \right ) b^{8}}{2048 d}+\frac {13 \sin \left (6 d x +6 c \right ) b^{8}}{3072 d}-\frac {\sin \left (4 d x +4 c \right ) b^{8}}{256 d}+\frac {\sin \left (2 d x +2 c \right ) a^{8}}{4 d}-\frac {7 \sin \left (2 d x +2 c \right ) b^{8}}{512 d}+\frac {a^{8} x}{2}+\frac {7 b^{8} x}{256}+\frac {b^{8} \sin \left (10 d x +10 c \right )}{5120 d}+\frac {7 x \,a^{6} b^{2}}{2}+\frac {35 x \,a^{4} b^{4}}{8}+\frac {35 x \,a^{2} b^{6}}{32}-\frac {2 a^{7} b \cos \left (d x +c \right )}{d}-\frac {7 a^{5} b^{3} \cos \left (d x +c \right )}{d}-\frac {35 a^{3} b^{5} \cos \left (d x +c \right )}{8 d}-\frac {7 a \,b^{7} \cos \left (d x +c \right )}{16 d}-\frac {7 \sin \left (8 d x +8 c \right ) a^{2} b^{6}}{256 d}-\frac {a^{3} b^{5} \cos \left (7 d x +7 c \right )}{8 d}-\frac {5 a \,b^{7} \cos \left (7 d x +7 c \right )}{224 d}+\frac {35 \sin \left (6 d x +6 c \right ) a^{4} b^{4}}{96 d}+\frac {7 \sin \left (6 d x +6 c \right ) a^{2} b^{6}}{48 d}+\frac {7 a^{5} b^{3} \cos \left (5 d x +5 c \right )}{10 d}+\frac {21 a^{3} b^{5} \cos \left (5 d x +5 c \right )}{40 d}+\frac {a \,b^{7} \cos \left (5 d x +5 c \right )}{20 d}-\frac {7 \sin \left (4 d x +4 c \right ) a^{6} b^{2}}{8 d}-\frac {35 \sin \left (4 d x +4 c \right ) a^{4} b^{4}}{32 d}-\frac {7 \sin \left (4 d x +4 c \right ) a^{2} b^{6}}{32 d}-\frac {2 a^{7} b \cos \left (3 d x +3 c \right )}{3 d}-\frac {7 a^{5} b^{3} \cos \left (3 d x +3 c \right )}{6 d}-\frac {7 a^{3} b^{5} \cos \left (3 d x +3 c \right )}{24 d}\) | \(539\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {71680 \, a b^{7} \cos \left (d x + c\right )^{9} - 92160 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{7} + 129024 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{5} - 215040 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{3} + 315 \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} d x + 21 \, {\left (384 \, b^{8} \cos \left (d x + c\right )^{9} - 48 \, {\left (280 \, a^{2} b^{6} + 31 \, b^{8}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (5600 \, a^{4} b^{4} + 4760 \, a^{2} b^{6} + 263 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (2688 \, a^{6} b^{2} + 7840 \, a^{4} b^{4} + 3304 \, a^{2} b^{6} + 121 \, b^{8}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (415) = 830\).
Time = 1.48 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.64 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.79 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {1720320 \, a^{7} b \cos \left (d x + c\right )^{3} - 161280 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8} - 564480 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{6} b^{2} - 2408448 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{5} b^{3} + 235200 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} b^{4} + 344064 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{3} b^{5} + 5880 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 120 \, d x - 120 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{6} - 16384 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a b^{7} - 21 \, {\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c - 45 \, \sin \left (8 \, d x + 8 \, c\right ) - 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{8}}{645120 \, d} \]
[In]
[Out]
none
Time = 0.55 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {a b^{7} \cos \left (9 \, d x + 9 \, c\right )}{288 \, d} + \frac {b^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {1}{256} \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} x - \frac {{\left (28 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac {{\left (28 \, a^{5} b^{3} + 21 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {{\left (16 \, a^{7} b + 28 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \cos \left (3 \, d x + 3 \, c\right )}{24 \, d} - \frac {{\left (32 \, a^{7} b + 112 \, a^{5} b^{3} + 70 \, a^{3} b^{5} + 7 \, a b^{7}\right )} \cos \left (d x + c\right )}{16 \, d} - \frac {{\left (56 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {{\left (1120 \, a^{4} b^{4} + 448 \, a^{2} b^{6} + 13 \, b^{8}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {{\left (224 \, a^{6} b^{2} + 280 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {{\left (128 \, a^{8} - 560 \, a^{4} b^{4} - 224 \, a^{2} b^{6} - 7 \, b^{8}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
[In]
[Out]
Time = 6.88 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {\frac {2205\,b^8\,\sin \left (2\,c+2\,d\,x\right )}{2}-20160\,a^8\,\sin \left (2\,c+2\,d\,x\right )+315\,b^8\,\sin \left (4\,c+4\,d\,x\right )-\frac {1365\,b^8\,\sin \left (6\,c+6\,d\,x\right )}{4}+\frac {945\,b^8\,\sin \left (8\,c+8\,d\,x\right )}{8}-\frac {63\,b^8\,\sin \left (10\,c+10\,d\,x\right )}{4}+53760\,a^7\,b\,\cos \left (3\,c+3\,d\,x\right )-4032\,a\,b^7\,\cos \left (5\,c+5\,d\,x\right )+1800\,a\,b^7\,\cos \left (7\,c+7\,d\,x\right )-280\,a\,b^7\,\cos \left (9\,c+9\,d\,x\right )+352800\,a^3\,b^5\,\cos \left (c+d\,x\right )+564480\,a^5\,b^3\,\cos \left (c+d\,x\right )+23520\,a^3\,b^5\,\cos \left (3\,c+3\,d\,x\right )+94080\,a^5\,b^3\,\cos \left (3\,c+3\,d\,x\right )-42336\,a^3\,b^5\,\cos \left (5\,c+5\,d\,x\right )-56448\,a^5\,b^3\,\cos \left (5\,c+5\,d\,x\right )+10080\,a^3\,b^5\,\cos \left (7\,c+7\,d\,x\right )+35280\,a^2\,b^6\,\sin \left (2\,c+2\,d\,x\right )+88200\,a^4\,b^4\,\sin \left (2\,c+2\,d\,x\right )+17640\,a^2\,b^6\,\sin \left (4\,c+4\,d\,x\right )+88200\,a^4\,b^4\,\sin \left (4\,c+4\,d\,x\right )+70560\,a^6\,b^2\,\sin \left (4\,c+4\,d\,x\right )-11760\,a^2\,b^6\,\sin \left (6\,c+6\,d\,x\right )-29400\,a^4\,b^4\,\sin \left (6\,c+6\,d\,x\right )+2205\,a^2\,b^6\,\sin \left (8\,c+8\,d\,x\right )+35280\,a\,b^7\,\cos \left (c+d\,x\right )+161280\,a^7\,b\,\cos \left (c+d\,x\right )-40320\,a^8\,d\,x-2205\,b^8\,d\,x-88200\,a^2\,b^6\,d\,x-352800\,a^4\,b^4\,d\,x-282240\,a^6\,b^2\,d\,x}{80640\,d} \]
[In]
[Out]