Integrand size = 29, antiderivative size = 185 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {9 x}{256 a^2}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{256 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2954, 2952, 2648, 2715, 8, 2645, 276} \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \cos ^9(c+d x)}{9 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {\sin ^5(c+d x) \cos ^5(c+d x)}{10 a^2 d}-\frac {3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {3 \sin (c+d x) \cos ^5(c+d x)}{32 a^2 d}+\frac {3 \sin (c+d x) \cos ^3(c+d x)}{128 a^2 d}+\frac {9 \sin (c+d x) \cos (c+d x)}{256 a^2 d}+\frac {9 x}{256 a^2} \]
[In]
[Out]
Rule 8
Rule 276
Rule 2645
Rule 2648
Rule 2715
Rule 2952
Rule 2954
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^4(c+d x) \sin ^4(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \cos ^4(c+d x) \sin ^4(c+d x)-2 a^2 \cos ^4(c+d x) \sin ^5(c+d x)+a^2 \cos ^4(c+d x) \sin ^6(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{a^2}+\frac {\int \cos ^4(c+d x) \sin ^6(c+d x) \, dx}{a^2}-\frac {2 \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx}{a^2} \\ & = -\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{8 a^2}+\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{2 a^2}+\frac {2 \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^2 d} \\ & = -\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {\int \cos ^4(c+d x) \, dx}{16 a^2}+\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{16 a^2}+\frac {2 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d} \\ & = \frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {\int \cos ^4(c+d x) \, dx}{32 a^2}+\frac {3 \int \cos ^2(c+d x) \, dx}{64 a^2} \\ & = \frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{128 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int 1 \, dx}{128 a^2}+\frac {3 \int \cos ^2(c+d x) \, dx}{128 a^2} \\ & = \frac {3 x}{128 a^2}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{256 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int 1 \, dx}{256 a^2} \\ & = \frac {9 x}{256 a^2}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{256 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(585\) vs. \(2(185)=370\).
Time = 5.64 (sec) , antiderivative size = 585, normalized size of antiderivative = 3.16 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-2520 (187 c-18 d x) \cos \left (\frac {c}{2}\right )+30240 \cos \left (\frac {c}{2}+d x\right )+30240 \cos \left (\frac {3 c}{2}+d x\right )-1260 \cos \left (\frac {3 c}{2}+2 d x\right )+1260 \cos \left (\frac {5 c}{2}+2 d x\right )+6720 \cos \left (\frac {5 c}{2}+3 d x\right )+6720 \cos \left (\frac {7 c}{2}+3 d x\right )-7560 \cos \left (\frac {7 c}{2}+4 d x\right )+7560 \cos \left (\frac {9 c}{2}+4 d x\right )-4032 \cos \left (\frac {9 c}{2}+5 d x\right )-4032 \cos \left (\frac {11 c}{2}+5 d x\right )+630 \cos \left (\frac {11 c}{2}+6 d x\right )-630 \cos \left (\frac {13 c}{2}+6 d x\right )-720 \cos \left (\frac {13 c}{2}+7 d x\right )-720 \cos \left (\frac {15 c}{2}+7 d x\right )+945 \cos \left (\frac {15 c}{2}+8 d x\right )-945 \cos \left (\frac {17 c}{2}+8 d x\right )+560 \cos \left (\frac {17 c}{2}+9 d x\right )+560 \cos \left (\frac {19 c}{2}+9 d x\right )-126 \cos \left (\frac {19 c}{2}+10 d x\right )+126 \cos \left (\frac {21 c}{2}+10 d x\right )+327180 \sin \left (\frac {c}{2}\right )-471240 c \sin \left (\frac {c}{2}\right )+45360 d x \sin \left (\frac {c}{2}\right )-30240 \sin \left (\frac {c}{2}+d x\right )+30240 \sin \left (\frac {3 c}{2}+d x\right )-1260 \sin \left (\frac {3 c}{2}+2 d x\right )-1260 \sin \left (\frac {5 c}{2}+2 d x\right )-6720 \sin \left (\frac {5 c}{2}+3 d x\right )+6720 \sin \left (\frac {7 c}{2}+3 d x\right )-7560 \sin \left (\frac {7 c}{2}+4 d x\right )-7560 \sin \left (\frac {9 c}{2}+4 d x\right )+4032 \sin \left (\frac {9 c}{2}+5 d x\right )-4032 \sin \left (\frac {11 c}{2}+5 d x\right )+630 \sin \left (\frac {11 c}{2}+6 d x\right )+630 \sin \left (\frac {13 c}{2}+6 d x\right )+720 \sin \left (\frac {13 c}{2}+7 d x\right )-720 \sin \left (\frac {15 c}{2}+7 d x\right )+945 \sin \left (\frac {15 c}{2}+8 d x\right )+945 \sin \left (\frac {17 c}{2}+8 d x\right )-560 \sin \left (\frac {17 c}{2}+9 d x\right )+560 \sin \left (\frac {19 c}{2}+9 d x\right )-126 \sin \left (\frac {19 c}{2}+10 d x\right )-126 \sin \left (\frac {21 c}{2}+10 d x\right )}{1290240 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
[In]
[Out]
Time = 0.43 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.66
method | result | size |
parallelrisch | \(\frac {22680 d x -126 \sin \left (10 d x +10 c \right )+560 \cos \left (9 d x +9 c \right )+945 \sin \left (8 d x +8 c \right )+630 \sin \left (6 d x +6 c \right )-7560 \sin \left (4 d x +4 c \right )-1260 \sin \left (2 d x +2 c \right )-720 \cos \left (7 d x +7 c \right )-4032 \cos \left (5 d x +5 c \right )+6720 \cos \left (3 d x +3 c \right )+30240 \cos \left (d x +c \right )+32768}{645120 d \,a^{2}}\) | \(122\) |
risch | \(\frac {9 x}{256 a^{2}}+\frac {3 \cos \left (d x +c \right )}{64 a^{2} d}-\frac {\sin \left (10 d x +10 c \right )}{5120 d \,a^{2}}+\frac {\cos \left (9 d x +9 c \right )}{1152 d \,a^{2}}+\frac {3 \sin \left (8 d x +8 c \right )}{2048 d \,a^{2}}-\frac {\cos \left (7 d x +7 c \right )}{896 d \,a^{2}}+\frac {\sin \left (6 d x +6 c \right )}{1024 d \,a^{2}}-\frac {\cos \left (5 d x +5 c \right )}{160 d \,a^{2}}-\frac {3 \sin \left (4 d x +4 c \right )}{256 d \,a^{2}}+\frac {\cos \left (3 d x +3 c \right )}{96 d \,a^{2}}-\frac {\sin \left (2 d x +2 c \right )}{512 d \,a^{2}}\) | \(175\) |
derivativedivides | \(\frac {\frac {32 \left (\frac {1}{315}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63}-\frac {87 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {553 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5120}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {491 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}+\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2555 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}+\frac {2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {2555 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {491 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}+\frac {2 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {553 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5120}+\frac {87 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}+\frac {9 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{d \,a^{2}}\) | \(257\) |
default | \(\frac {\frac {32 \left (\frac {1}{315}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63}-\frac {87 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {553 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5120}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {491 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}+\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2555 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}+\frac {2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {2555 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {491 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}+\frac {2 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {553 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5120}+\frac {87 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}+\frac {9 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{d \,a^{2}}\) | \(257\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.54 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {17920 \, \cos \left (d x + c\right )^{9} - 46080 \, \cos \left (d x + c\right )^{7} + 32256 \, \cos \left (d x + c\right )^{5} + 2835 \, d x - 63 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 496 \, \cos \left (d x + c\right )^{7} + 488 \, \cos \left (d x + c\right )^{5} - 30 \, \cos \left (d x + c\right )^{3} - 45 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, a^{2} d} \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (167) = 334\).
Time = 0.32 (sec) , antiderivative size = 605, normalized size of antiderivative = 3.27 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {2835 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40960 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {27405 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {184320 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {139356 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {368640 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {618660 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1290240 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {1609650 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {516096 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {1609650 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {430080 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {618660 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {860160 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {139356 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {27405 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {2835 \, \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} - 4096}{a^{2} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {120 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {210 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {252 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {210 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {120 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {45 \, a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {a^{2} \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}}} - \frac {2835 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{40320 \, d} \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.39 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {2835 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (2835 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 27405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 139356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 860160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 618660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 430080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 1609650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 516096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 1609650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1290240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 618660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 368640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 139356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 184320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 27405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2835 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4096\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{10} a^{2}}}{80640 \, d} \]
[In]
[Out]
Time = 12.41 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {9\,x}{256\,a^2}+\frac {\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {87\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {553\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}-\frac {491\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {2555\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}-\frac {2555\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {491\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {553\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {87\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {32}{315}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]
[In]
[Out]