Integrand size = 29, antiderivative size = 284 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {a \left (9-\frac {20 a^2}{b^2}\right ) x}{2 b^4}+\frac {\left (20 a^4-19 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^6 \sqrt {a^2-b^2} d}-\frac {\left (60 a^2-17 b^2\right ) \cos (c+d x)}{6 b^5 d}+\frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{a b^4 d}-\frac {\left (20 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{6 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (6 a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))} \]
1/2*a*(9-20*a^2/b^2)*x/b^4-1/6*(60*a^2-17*b^2)*cos(d*x+c)/b^5/d+(5*a^2-b^2 )*cos(d*x+c)*sin(d*x+c)/a/b^4/d-1/6*(20*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)^2 /a^2/b^3/d-1/2*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^3/a/b^2/d/(a+b*sin(d*x+c))^ 2+1/2*(6*a^2-b^2)*cos(d*x+c)*sin(d*x+c)^3/a^2/b^2/d/(a+b*sin(d*x+c))+(20*a ^4-19*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^6/ d/(a^2-b^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1030\) vs. \(2(284)=568\).
Time = 5.15 (sec) , antiderivative size = 1030, normalized size of antiderivative = 3.63 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {12 \left (-48 a (c+d x)+\frac {6 \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-16 b \cos (c+d x)+\frac {b \left (8 a^4-8 a^2 b^2+b^4\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}+\frac {a b \left (-40 a^4+72 a^2 b^2-29 b^4\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}\right )}{b^4}+12 \left (\frac {2 \left (2 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 )}{\left (a^2-b^2\right )^{5/2}}+\frac {b \cos (c+d x) \left (4 a^2-b^2+3 a b \sin (c+d x)\right )}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))^2}\right )+\frac {6 \left (-\frac {6 b^2 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {\cos (c+d x) \left (-b \left (2 a^2+b^2\right )+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}\right )}{(a-b)^2 (a+b)^2}-\frac {-\frac {12 \left (640 a^8-1792 a^6 b^2+1680 a^4 b^4-560 a^2 b^6+35 b^8\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {3840 a^9 (c+d x)-6912 a^7 b^2 (c+d x)+1728 a^5 b^4 (c+d x)+1920 a^3 b^6 (c+d x)-576 a b^8 (c+d x)+3840 a^8 b \cos (c+d x)-7872 a^6 b^3 \cos (c+d x)+4256 a^4 b^5 \cos (c+d x)-172 a^2 b^7 \cos (c+d x)-70 b^9 \cos (c+d x)-1920 a^7 b^2 (c+d x) \cos (2 (c+d x))+4416 a^5 b^4 (c+d x) \cos (2 (c+d x))-3072 a^3 b^6 (c+d x) \cos (2 (c+d x))+576 a b^8 (c+d x) \cos (2 (c+d x))-320 a^6 b^3 \cos (3 (c+d x))+696 a^4 b^5 \cos (3 (c+d x))-432 a^2 b^7 \cos (3 (c+d x))+56 b^9 \cos (3 (c+d x))+8 a^4 b^5 \cos (5 (c+d x))-16 a^2 b^7 \cos (5 (c+d x))+8 b^9 \cos (5 (c+d x))+7680 a^8 b (c+d x) \sin (c+d x)-17664 a^6 b^3 (c+d x) \sin (c+d x)+12288 a^4 b^5 (c+d x) \sin (c+d x)-2304 a^2 b^7 (c+d x) \sin (c+d x)+2880 a^7 b^2 \sin (2 (c+d x))-6304 a^5 b^4 \sin (2 (c+d x))+4022 a^3 b^6 \sin (2 (c+d x))-607 a b^8 \sin (2 (c+d x))+40 a^5 b^4 \sin (4 (c+d x))-80 a^3 b^6 \sin (4 (c+d x))+40 a b^8 \sin (4 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}}{b^6}}{384 d} \]
((-12*(-48*a*(c + d*x) + (6*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*Arc Tan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 16*b*Co s[c + d*x] + (b*(8*a^4 - 8*a^2*b^2 + b^4)*Cos[c + d*x])/((a - b)*(a + b)*( a + b*Sin[c + d*x])^2) + (a*b*(-40*a^4 + 72*a^2*b^2 - 29*b^4)*Cos[c + d*x] )/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))))/b^4 + 12*((2*(2*a^2 + b^2)* ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b*C os[c + d*x]*(4*a^2 - b^2 + 3*a*b*Sin[c + d*x]))/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^2)) + (6*((-6*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(-(b*(2*a^2 + b^2)) + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) - (( -12*(640*a^8 - 1792*a^6*b^2 + 1680*a^4*b^4 - 560*a^2*b^6 + 35*b^8)*ArcTan[ (b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (3840*a^9*( c + d*x) - 6912*a^7*b^2*(c + d*x) + 1728*a^5*b^4*(c + d*x) + 1920*a^3*b^6* (c + d*x) - 576*a*b^8*(c + d*x) + 3840*a^8*b*Cos[c + d*x] - 7872*a^6*b^3*C os[c + d*x] + 4256*a^4*b^5*Cos[c + d*x] - 172*a^2*b^7*Cos[c + d*x] - 70*b^ 9*Cos[c + d*x] - 1920*a^7*b^2*(c + d*x)*Cos[2*(c + d*x)] + 4416*a^5*b^4*(c + d*x)*Cos[2*(c + d*x)] - 3072*a^3*b^6*(c + d*x)*Cos[2*(c + d*x)] + 576*a *b^8*(c + d*x)*Cos[2*(c + d*x)] - 320*a^6*b^3*Cos[3*(c + d*x)] + 696*a^4*b ^5*Cos[3*(c + d*x)] - 432*a^2*b^7*Cos[3*(c + d*x)] + 56*b^9*Cos[3*(c + d*x )] + 8*a^4*b^5*Cos[5*(c + d*x)] - 16*a^2*b^7*Cos[5*(c + d*x)] + 8*b^9*C...
Time = 1.61 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.13, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {3042, 3370, 3042, 3528, 25, 3042, 3528, 27, 3042, 3502, 27, 3042, 3214, 3042, 3139, 1083, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^4}{(a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3370 |
| \(\displaystyle -\frac {\int \frac {\sin ^2(c+d x) \left (15 a^2-b \sin (c+d x) a-2 b^2-\left (20 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin (c+d x)^2 \left (15 a^2-b \sin (c+d x) a-2 b^2-\left (20 a^2-3 b^2\right ) \sin (c+d x)^2\right )}{a+b \sin (c+d x)}dx}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {\int -\frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^2-12 \left (5 a^2-b^2\right ) \sin ^2(c+d x) a+2 \left (20 a^2-3 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^2-12 \left (5 a^2-b^2\right ) \sin ^2(c+d x) a+2 \left (20 a^2-3 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^2-12 \left (5 a^2-b^2\right ) \sin (c+d x)^2 a+2 \left (20 a^2-3 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {2 \left (-10 b \sin (c+d x) a^3-\left (60 a^2-17 b^2\right ) \sin ^2(c+d x) a^2+6 \left (5 a^2-b^2\right ) a^2\right )}{a+b \sin (c+d x)}dx}{2 b}+\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\int \frac {-10 b \sin (c+d x) a^3-\left (60 a^2-17 b^2\right ) \sin ^2(c+d x) a^2+6 \left (5 a^2-b^2\right ) a^2}{a+b \sin (c+d x)}dx}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\int \frac {-10 b \sin (c+d x) a^3-\left (60 a^2-17 b^2\right ) \sin (c+d x)^2 a^2+6 \left (5 a^2-b^2\right ) a^2}{a+b \sin (c+d x)}dx}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\int \frac {3 \left (\left (20 a^2-9 b^2\right ) \sin (c+d x) a^3+2 b \left (5 a^2-b^2\right ) a^2\right )}{a+b \sin (c+d x)}dx}{b}+\frac {a^2 \left (60 a^2-17 b^2\right ) \cos (c+d x)}{b d}}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {3 \int \frac {\left (20 a^2-9 b^2\right ) \sin (c+d x) a^3+2 b \left (5 a^2-b^2\right ) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {a^2 \left (60 a^2-17 b^2\right ) \cos (c+d x)}{b d}}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {3 \int \frac {\left (20 a^2-9 b^2\right ) \sin (c+d x) a^3+2 b \left (5 a^2-b^2\right ) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {a^2 \left (60 a^2-17 b^2\right ) \cos (c+d x)}{b d}}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {3 \left (\frac {a^3 x \left (20 a^2-9 b^2\right )}{b}-\frac {a^2 \left (20 a^4-19 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {a^2 \left (60 a^2-17 b^2\right ) \cos (c+d x)}{b d}}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {3 \left (\frac {a^3 x \left (20 a^2-9 b^2\right )}{b}-\frac {a^2 \left (20 a^4-19 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {a^2 \left (60 a^2-17 b^2\right ) \cos (c+d x)}{b d}}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {3 \left (\frac {a^3 x \left (20 a^2-9 b^2\right )}{b}-\frac {2 a^2 \left (20 a^4-19 a^2 b^2+2 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {a^2 \left (60 a^2-17 b^2\right ) \cos (c+d x)}{b d}}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {3 \left (\frac {4 a^2 \left (20 a^4-19 a^2 b^2+2 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a^3 x \left (20 a^2-9 b^2\right )}{b}\right )}{b}+\frac {a^2 \left (60 a^2-17 b^2\right ) \cos (c+d x)}{b d}}{b}}{3 b}}{2 a^2 b^2}+\frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (6 a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac {\frac {\left (20 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {6 a \left (5 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {a^2 \left (60 a^2-17 b^2\right ) \cos (c+d x)}{b d}+\frac {3 \left (\frac {a^3 x \left (20 a^2-9 b^2\right )}{b}-\frac {2 a^2 \left (20 a^4-19 a^2 b^2+2 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d \sqrt {a^2-b^2}}\right )}{b}}{b}}{3 b}}{2 a^2 b^2}\) |
-1/2*((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(a*b^2*d*(a + b*Sin[c + d*x ])^2) + ((6*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a^2*b^2*d*(a + b*Si n[c + d*x])) - (((20*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d) - ( -(((3*((a^3*(20*a^2 - 9*b^2)*x)/b - (2*a^2*(20*a^4 - 19*a^2*b^2 + 2*b^4)*A rcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2 ]*d)))/b + (a^2*(60*a^2 - 17*b^2)*Cos[c + d*x])/(b*d))/b) + (6*a*(5*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(b*d))/(3*b))/(2*a^2*b^2)
3.12.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1))), x] + (Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b*S in[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2) )), x] - Simp[1/(a^2*b^2*(m + 1)*(m + 2)) Int[(a + b*Sin[e + f*x])^(m + 2 )*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a ^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && !LtQ[n, -1] && (LtQ[m , -2] || EqQ[m + n + 4, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Time = 2.34 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {4 \left (-\frac {a \,b^{2} \left (7 a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {b \left (8 a^{4}+13 a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {5 b^{2} a \left (5 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-2 a^{4} b +\frac {3 a^{2} b^{3}}{4}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (20 a^{4}-19 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b^{6}}-\frac {4 \left (\frac {\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{4}+\left (3 a^{2} b -b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{2} b -b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{4}+3 a^{2} b -\frac {2 b^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {a \left (20 a^{2}-9 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}\right )}{b^{6}}}{d}\) | \(344\) |
| default | \(\frac {\frac {\frac {4 \left (-\frac {a \,b^{2} \left (7 a^{2}-2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {b \left (8 a^{4}+13 a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {5 b^{2} a \left (5 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-2 a^{4} b +\frac {3 a^{2} b^{3}}{4}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (20 a^{4}-19 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b^{6}}-\frac {4 \left (\frac {\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{4}+\left (3 a^{2} b -b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{2} b -b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{4}+3 a^{2} b -\frac {2 b^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {a \left (20 a^{2}-9 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}\right )}{b^{6}}}{d}\) | \(344\) |
| risch | \(-\frac {10 x \,a^{3}}{b^{6}}+\frac {9 a x}{2 b^{4}}+\frac {{\mathrm e}^{3 i \left (d x +c \right )}}{24 b^{3} d}+\frac {i a \left (-10 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+5 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+26 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-11 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+18 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{2}+4 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \,b^{6}}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{b^{5} d}+\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{3} d}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{b^{5} d}+\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{3} d}-\frac {3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{4} d}+\frac {{\mathrm e}^{-3 i \left (d x +c \right )}}{24 b^{3} d}+\frac {3 i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{4} d}+\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{4}}{\sqrt {-a^{2}+b^{2}}\, d \,b^{6}}-\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{2}}{2 \sqrt {-a^{2}+b^{2}}\, d \,b^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}-\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{4}}{\sqrt {-a^{2}+b^{2}}\, d \,b^{6}}+\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{2}}{2 \sqrt {-a^{2}+b^{2}}\, d \,b^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}\) | \(753\) |
1/d*(4/b^6*((-1/4*a*b^2*(7*a^2-2*b^2)*tan(1/2*d*x+1/2*c)^3-1/4*b*(8*a^4+13 *a^2*b^2-6*b^4)*tan(1/2*d*x+1/2*c)^2-5/4*b^2*a*(5*a^2-2*b^2)*tan(1/2*d*x+1 /2*c)-2*a^4*b+3/4*a^2*b^3)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+ a)^2+1/4*(20*a^4-19*a^2*b^2+2*b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2 *d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))-4/b^6*((3/4*tan(1/2*d*x+1/2*c)^5*a*b^2+ (3*a^2*b-b^3)*tan(1/2*d*x+1/2*c)^4+(6*a^2*b-b^3)*tan(1/2*d*x+1/2*c)^2-3/4* tan(1/2*d*x+1/2*c)*a*b^2+3*a^2*b-2/3*b^3)/(1+tan(1/2*d*x+1/2*c)^2)^3+1/4*a *(20*a^2-9*b^2)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.37 (sec) , antiderivative size = 976, normalized size of antiderivative = 3.44 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
[1/12*(4*(a^2*b^5 - b^7)*cos(d*x + c)^5 - 6*(20*a^5*b^2 - 29*a^3*b^4 + 9*a *b^6)*d*x*cos(d*x + c)^2 - 8*(5*a^4*b^3 - 6*a^2*b^5 + b^7)*cos(d*x + c)^3 + 6*(20*a^7 - 9*a^5*b^2 - 20*a^3*b^4 + 9*a*b^6)*d*x + 3*(20*a^6 + a^4*b^2 - 17*a^2*b^4 + 2*b^6 - (20*a^4*b^2 - 19*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(20*a^5*b - 19*a^3*b^3 + 2*a*b^5)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(((2 *a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 6*(20*a^6*b - 19*a^4*b^3 - 3*a^2*b^5 + 2*b^7)*cos(d*x + c) + 2*(5*(a^3*b^4 - a*b^6)*cos(d*x + c)^3 + 6*(20*a^6 *b - 29*a^4*b^3 + 9*a^2*b^5)*d*x + 3*(30*a^5*b^2 - 41*a^3*b^4 + 11*a*b^6)* cos(d*x + c))*sin(d*x + c))/((a^2*b^8 - b^10)*d*cos(d*x + c)^2 - 2*(a^3*b^ 7 - a*b^9)*d*sin(d*x + c) - (a^4*b^6 - b^10)*d), 1/6*(2*(a^2*b^5 - b^7)*co s(d*x + c)^5 - 3*(20*a^5*b^2 - 29*a^3*b^4 + 9*a*b^6)*d*x*cos(d*x + c)^2 - 4*(5*a^4*b^3 - 6*a^2*b^5 + b^7)*cos(d*x + c)^3 + 3*(20*a^7 - 9*a^5*b^2 - 2 0*a^3*b^4 + 9*a*b^6)*d*x + 3*(20*a^6 + a^4*b^2 - 17*a^2*b^4 + 2*b^6 - (20* a^4*b^2 - 19*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(20*a^5*b - 19*a^3*b^3 + 2*a*b^5)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt( a^2 - b^2)*cos(d*x + c))) + 3*(20*a^6*b - 19*a^4*b^3 - 3*a^2*b^5 + 2*b^7)* cos(d*x + c) + (5*(a^3*b^4 - a*b^6)*cos(d*x + c)^3 + 6*(20*a^6*b - 29*a^4* b^3 + 9*a^2*b^5)*d*x + 3*(30*a^5*b^2 - 41*a^3*b^4 + 11*a*b^6)*cos(d*x +...
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.37 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.38 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {3 \, {\left (20 \, a^{3} - 9 \, a b^{2}\right )} {\left (d x + c\right )}}{b^{6}} - \frac {6 \, {\left (20 \, a^{4} - 19 \, a^{2} b^{2} + 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{6}} + \frac {6 \, {\left (7 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 13 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 25 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 10 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a^{4} - 3 \, a^{2} b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} b^{5}} + \frac {2 \, {\left (9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 72 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} - 8 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{5}}}{6 \, d} \]
-1/6*(3*(20*a^3 - 9*a*b^2)*(d*x + c)/b^6 - 6*(20*a^4 - 19*a^2*b^2 + 2*b^4) *(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^6) + 6*(7*a^3*b*tan(1/2*d*x + 1 /2*c)^3 - 2*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 8*a^4*tan(1/2*d*x + 1/2*c)^2 + 13*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 6*b^4*tan(1/2*d*x + 1/2*c)^2 + 25*a^3* b*tan(1/2*d*x + 1/2*c) - 10*a*b^3*tan(1/2*d*x + 1/2*c) + 8*a^4 - 3*a^2*b^2 )/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*b^5) + 2*(9 *a*b*tan(1/2*d*x + 1/2*c)^5 + 36*a^2*tan(1/2*d*x + 1/2*c)^4 - 12*b^2*tan(1 /2*d*x + 1/2*c)^4 + 72*a^2*tan(1/2*d*x + 1/2*c)^2 - 12*b^2*tan(1/2*d*x + 1 /2*c)^2 - 9*a*b*tan(1/2*d*x + 1/2*c) + 36*a^2 - 8*b^2)/((tan(1/2*d*x + 1/2 *c)^2 + 1)^3*b^5))/d
Time = 16.86 (sec) , antiderivative size = 2034, normalized size of antiderivative = 7.16 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
- ((60*a^4 - 17*a^2*b^2)/(3*b^5) - (20*tan(c/2 + (d*x)/2)^7*(a*b^2 - 5*a^3 ))/b^4 - (2*tan(c/2 + (d*x)/2)^9*(a*b^2 - 5*a^3))/b^4 - (4*tan(c/2 + (d*x) /2)^5*(17*a*b^2 - 60*a^3))/b^4 - (4*tan(c/2 + (d*x)/2)^3*(53*a*b^2 - 165*a ^3))/(3*b^4) + (tan(c/2 + (d*x)/2)^8*(20*a^4 - 6*b^4 + 21*a^2*b^2))/b^5 + (2*tan(c/2 + (d*x)/2)^6*(40*a^4 - 17*b^4 + 42*a^2*b^2))/b^5 + (2*tan(c/2 + (d*x)/2)^2*(120*a^4 - 25*b^4 + 46*a^2*b^2))/(3*b^5) + (2*tan(c/2 + (d*x)/ 2)^4*(180*a^4 - 51*b^4 + 149*a^2*b^2))/(3*b^5) - (2*tan(c/2 + (d*x)/2)*(31 *a*b^2 - 105*a^3))/(3*b^4))/(d*(tan(c/2 + (d*x)/2)^2*(5*a^2 + 4*b^2) + tan (c/2 + (d*x)/2)^8*(5*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^4*(10*a^2 + 12*b^2) + tan(c/2 + (d*x)/2)^6*(10*a^2 + 12*b^2) + a^2*tan(c/2 + (d*x)/2)^10 + a^ 2 + 16*a*b*tan(c/2 + (d*x)/2)^3 + 24*a*b*tan(c/2 + (d*x)/2)^5 + 16*a*b*tan (c/2 + (d*x)/2)^7 + 4*a*b*tan(c/2 + (d*x)/2)^9 + 4*a*b*tan(c/2 + (d*x)/2)) ) - (a*atan((280*a^4*tan(c/2 + (d*x)/2))/(280*a^4 - 288*a^2*b^2 + (800*a^6 )/b^2) - (288*a^2*tan(c/2 + (d*x)/2))/((280*a^4)/b^2 - 288*a^2 + (800*a^6) /b^4) + (800*a^6*tan(c/2 + (d*x)/2))/(800*a^6 - 288*a^2*b^4 + 280*a^4*b^2) )*(20*a^2 - 9*b^2))/(b^6*d) - (atan((((-(a + b)*(a - b))^(1/2)*(10*a^4 + b ^4 - (19*a^2*b^2)/2)*((8*(81*a^4*b^9 - 360*a^6*b^7 + 400*a^8*b^5))/b^14 - (8*tan(c/2 + (d*x)/2)*(4*a*b^13 - 238*a^3*b^11 + 1242*a^5*b^9 - 1920*a^7*b ^7 + 800*a^9*b^5))/b^15 + ((-(a + b)*(a - b))^(1/2)*(10*a^4 + b^4 - (19*a^ 2*b^2)/2)*((8*tan(c/2 + (d*x)/2)*(8*a*b^16 - 76*a^3*b^14 + 80*a^5*b^12)...