Integrand size = 29, antiderivative size = 331 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 \left (40 a^4-24 a^2 b^2+b^4\right ) x}{8 b^7}-\frac {3 a \left (10 a^4-11 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 \sqrt {a^2-b^2} d}+\frac {a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac {3 \left (20 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^5 d}+\frac {\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac {\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))} \] Output:
3/8*(40*a^4-24*a^2*b^2+b^4)*x/b^7-3*a*(10*a^4-11*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^7/(a^2-b^2)^(1/2)/d+1/2*a*(30*a^2 -13*b^2)*cos(d*x+c)/b^6/d-3/8*(20*a^2-7*b^2)*cos(d*x+c)*sin(d*x+c)/b^5/d+1 /2*(10*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)^2/a/b^4/d-1/4*(15*a^2-4*b^2)*cos(d *x+c)*sin(d*x+c)^3/a^2/b^3/d-1/2*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^4/a/b^2/d /(a+b*sin(d*x+c))^2+1/2*(7*a^2-2*b^2)*cos(d*x+c)*sin(d*x+c)^4/a^2/b^2/d/(a +b*sin(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(1250\) vs. \(2(331)=662\).
Time = 12.18 (sec) , antiderivative size = 1250, normalized size of antiderivative = 3.78 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]
Output:
-1/256*((-6*(-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*b*(4*a^2 - 3 *b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d* x]))))/b^3 + (6*((6*a*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/ Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) + (2*(-24*(-8*a^2 + b^2)*(c + d*x) - (6*a*(64*a^6 - 168*a^4*b^2 + 140*a^2*b^4 - 35*b^6)*ArcTa n[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 96*a*b*Co s[c + d*x] + (a*b*(-16*a^4 + 20*a^2*b^2 - 5*b^4)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) + (b*(112*a^6 - 220*a^4*b^2 + 115*a^2*b^4 - 10*b^6)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])) - 8*b^2*S in[2*(c + d*x)]))/b^5 + ((12*a*(640*a^8 - 1920*a^6*b^2 + 2016*a^4*b^4 - 84 0*a^2*b^6 + 105*b^8)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^ 2 - b^2)^(5/2) + (-3840*a^10*(c + d*x) + 7680*a^8*b^2*(c + d*x) - 2976*a^6 *b^4*(c + d*x) - 1776*a^4*b^6*(c + d*x) + 960*a^2*b^8*(c + d*x) - 48*b^10* (c + d*x) - 3840*a^9*b*Cos[c + d*x] + 8640*a^7*b^3*Cos[c + d*x] - 5696*a^5 *b^5*Cos[c + d*x] + 788*a^3*b^7*Cos[c + d*x] + 114*a*b^9*Cos[c + d*x] + 19 20*a^8*b^2*(c + d*x)*Cos[2*(c + d*x)] - 4800*a^6*b^4*(c + d*x)*Cos[2*(c + d*x)] + 3888*a^4*b^6*(c + d*x)*Cos[2*(c + d*x)] - 1056*a^2*b^8*(c + d*x...
Time = 2.22 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.14, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, Rules used = {3042, 3370, 3042, 3528, 27, 3042, 3528, 25, 3042, 3528, 25, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^4}{(a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3370 |
| \(\displaystyle -\frac {\int \frac {\sin ^3(c+d x) \left (-2 \left (15 a^2-4 b^2\right ) \sin ^2(c+d x)-a b \sin (c+d x)+6 \left (4 a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin (c+d x)^3 \left (-2 \left (15 a^2-4 b^2\right ) \sin (c+d x)^2-a b \sin (c+d x)+6 \left (4 a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {\int -\frac {6 \sin ^2(c+d x) \left (-b \sin (c+d x) a^2-2 \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) a+\left (15 a^2-4 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{4 b}+\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \int \frac {\sin ^2(c+d x) \left (-b \sin (c+d x) a^2-2 \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) a+\left (15 a^2-4 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \int \frac {\sin (c+d x)^2 \left (-b \sin (c+d x) a^2-2 \left (10 a^2-3 b^2\right ) \sin (c+d x)^2 a+\left (15 a^2-4 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {\int -\frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^3-3 \left (20 a^2-7 b^2\right ) \sin ^2(c+d x) a^2+4 \left (10 a^2-3 b^2\right ) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^3-3 \left (20 a^2-7 b^2\right ) \sin ^2(c+d x) a^2+4 \left (10 a^2-3 b^2\right ) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^3-3 \left (20 a^2-7 b^2\right ) \sin (c+d x)^2 a^2+4 \left (10 a^2-3 b^2\right ) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {-4 \left (30 a^2-13 b^2\right ) \sin ^2(c+d x) a^3+3 \left (20 a^2-7 b^2\right ) a^3-b \left (20 a^2-3 b^2\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}+\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {-4 \left (30 a^2-13 b^2\right ) \sin ^2(c+d x) a^3+3 \left (20 a^2-7 b^2\right ) a^3-b \left (20 a^2-3 b^2\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {-4 \left (30 a^2-13 b^2\right ) \sin (c+d x)^2 a^3+3 \left (20 a^2-7 b^2\right ) a^3-b \left (20 a^2-3 b^2\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {\int \frac {3 \left (b \left (20 a^2-7 b^2\right ) a^3+\left (40 a^4-24 b^2 a^2+b^4\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{b}+\frac {4 a^3 \left (30 a^2-13 b^2\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {3 \int \frac {b \left (20 a^2-7 b^2\right ) a^3+\left (40 a^4-24 b^2 a^2+b^4\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {4 a^3 \left (30 a^2-13 b^2\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {3 \int \frac {b \left (20 a^2-7 b^2\right ) a^3+\left (40 a^4-24 b^2 a^2+b^4\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {4 a^3 \left (30 a^2-13 b^2\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {3 \left (\frac {a^2 x \left (40 a^4-24 a^2 b^2+b^4\right )}{b}-\frac {4 a^3 \left (10 a^4-11 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {4 a^3 \left (30 a^2-13 b^2\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {3 \left (\frac {a^2 x \left (40 a^4-24 a^2 b^2+b^4\right )}{b}-\frac {4 a^3 \left (10 a^4-11 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {4 a^3 \left (30 a^2-13 b^2\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {3 \left (\frac {a^2 x \left (40 a^4-24 a^2 b^2+b^4\right )}{b}-\frac {8 a^3 \left (10 a^4-11 a^2 b^2+2 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {4 a^3 \left (30 a^2-13 b^2\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {3 \left (\frac {16 a^3 \left (10 a^4-11 a^2 b^2+2 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a^2 x \left (40 a^4-24 a^2 b^2+b^4\right )}{b}\right )}{b}+\frac {4 a^3 \left (30 a^2-13 b^2\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}+\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac {\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {3 a^2 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {4 a^3 \left (30 a^2-13 b^2\right ) \cos (c+d x)}{b d}+\frac {3 \left (\frac {a^2 x \left (40 a^4-24 a^2 b^2+b^4\right )}{b}-\frac {8 a^3 \left (10 a^4-11 a^2 b^2+2 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d \sqrt {a^2-b^2}}\right )}{b}}{2 b}}{3 b}\right )}{2 b}}{2 a^2 b^2}\) |
Input:
Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]
Output:
-1/2*((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(a*b^2*d*(a + b*Sin[c + d*x ])^2) + ((7*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a^2*b^2*d*(a + b* Sin[c + d*x])) - (((15*a^2 - 4*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*b*d) - (3*((2*a*(10*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d) - (-1/2*(( 3*((a^2*(40*a^4 - 24*a^2*b^2 + b^4)*x)/b - (8*a^3*(10*a^4 - 11*a^2*b^2 + 2 *b^4)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*Sqrt[a^ 2 - b^2]*d)))/b + (4*a^3*(30*a^2 - 13*b^2)*Cos[c + d*x])/(b*d))/b + (3*a^2 *(20*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b*d))/(3*b)))/(2*b))/(2*a^ 2*b^2)
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 = 3.11 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a \left (\frac {\left (-\frac {9}{2} a^{3} b^{2}+2 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {5 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-\frac {a \,b^{2} \left (31 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {5 a^{2} b \left (2 a^{2}-b^{2}\right )}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {3 \left (10 a^{4}-11 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}+\frac {\frac {2 \left (\left (3 a^{2} b^{2}-\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (10 a^{3} b -6 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (3 a^{2} b^{2}+\frac {3}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (30 a^{3} b -12 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-3 a^{2} b^{2}-\frac {3}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (30 a^{3} b -10 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-3 a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+10 a^{3} b -4 a \,b^{3}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (40 a^{4}-24 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{7}}}{d}\) | \(450\) |
| default | \(\frac {-\frac {2 a \left (\frac {\left (-\frac {9}{2} a^{3} b^{2}+2 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {5 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-\frac {a \,b^{2} \left (31 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {5 a^{2} b \left (2 a^{2}-b^{2}\right )}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {3 \left (10 a^{4}-11 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}+\frac {\frac {2 \left (\left (3 a^{2} b^{2}-\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (10 a^{3} b -6 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (3 a^{2} b^{2}+\frac {3}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (30 a^{3} b -12 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-3 a^{2} b^{2}-\frac {3}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (30 a^{3} b -10 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-3 a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+10 a^{3} b -4 a \,b^{3}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (40 a^{4}-24 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{7}}}{d}\) | \(450\) |
| risch | \(\frac {15 x \,a^{4}}{b^{7}}-\frac {9 x \,a^{2}}{b^{5}}+\frac {3 x}{8 b^{3}}-\frac {a \,{\mathrm e}^{3 i \left (d x +c \right )}}{8 b^{4} d}-\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {5 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{b^{6} d}-\frac {15 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{4} d}+\frac {5 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{b^{6} d}-\frac {15 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{4} d}-\frac {33 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,b^{5}}-\frac {15 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{7}}-\frac {a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{8 b^{4} d}+\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {33 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,b^{5}}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{4 b^{5} d}+\frac {15 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{7}}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{4 b^{5} d}-\frac {i a^{2} \left (-12 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+7 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+32 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-17 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+22 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-11 a^{2} b^{2}+6 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^{7}}+\frac {\sin \left (4 d x +4 c \right )}{32 b^{3} d}\) | \(834\) |
Input:
int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-2*a/b^7*(((-9/2*a^3*b^2+2*a*b^4)*tan(1/2*d*x+1/2*c)^3-5/2*b*(2*a^4+3 *a^2*b^2-2*b^4)*tan(1/2*d*x+1/2*c)^2-1/2*a*b^2*(31*a^2-16*b^2)*tan(1/2*d*x +1/2*c)-5/2*a^2*b*(2*a^2-b^2))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2 *c)+a)^2+3/2*(10*a^4-11*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)))+2/b^7*(((3*a^2*b^2-5/8*b^4)*tan(1/2 *d*x+1/2*c)^7+(10*a^3*b-6*a*b^3)*tan(1/2*d*x+1/2*c)^6+(3*a^2*b^2+3/8*b^4)* tan(1/2*d*x+1/2*c)^5+(30*a^3*b-12*a*b^3)*tan(1/2*d*x+1/2*c)^4+(-3*a^2*b^2- 3/8*b^4)*tan(1/2*d*x+1/2*c)^3+(30*a^3*b-10*a*b^3)*tan(1/2*d*x+1/2*c)^2+(-3 *a^2*b^2+5/8*b^4)*tan(1/2*d*x+1/2*c)+10*a^3*b-4*a*b^3)/(1+tan(1/2*d*x+1/2* c)^2)^4+3/8*(40*a^4-24*a^2*b^2+b^4)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.18 (sec) , antiderivative size = 1110, normalized size of antiderivative = 3.35 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="frica s")
Output:
[-1/8*(4*(a^3*b^5 - a*b^7)*cos(d*x + c)^5 - 3*(40*a^6*b^2 - 64*a^4*b^4 + 2 5*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2 - 2*(20*a^5*b^3 - 27*a^3*b^5 + 7*a*b^7 )*cos(d*x + c)^3 + 3*(40*a^8 - 24*a^6*b^2 - 39*a^4*b^4 + 24*a^2*b^6 - b^8) *d*x - 6*(10*a^7 - a^5*b^2 - 9*a^3*b^4 + 2*a*b^6 - (10*a^5*b^2 - 11*a^3*b^ 4 + 2*a*b^6)*cos(d*x + c)^2 + 2*(10*a^6*b - 11*a^4*b^3 + 2*a^2*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^7*b - 22*a^5*b^3 - a^3*b^5 + 3*a*b^7)*cos(d*x + c) - (2*(a^2*b^6 - b^ 8)*cos(d*x + c)^5 - (10*a^4*b^4 - 11*a^2*b^6 + b^8)*cos(d*x + c)^3 - 6*(40 *a^7*b - 64*a^5*b^3 + 25*a^3*b^5 - a*b^7)*d*x - 3*(60*a^6*b^2 - 91*a^4*b^4 + 32*a^2*b^6 - b^8)*cos(d*x + c))*sin(d*x + c))/((a^2*b^9 - b^11)*d*cos(d *x + c)^2 - 2*(a^3*b^8 - a*b^10)*d*sin(d*x + c) - (a^4*b^7 - b^11)*d), -1/ 8*(4*(a^3*b^5 - a*b^7)*cos(d*x + c)^5 - 3*(40*a^6*b^2 - 64*a^4*b^4 + 25*a^ 2*b^6 - b^8)*d*x*cos(d*x + c)^2 - 2*(20*a^5*b^3 - 27*a^3*b^5 + 7*a*b^7)*co s(d*x + c)^3 + 3*(40*a^8 - 24*a^6*b^2 - 39*a^4*b^4 + 24*a^2*b^6 - b^8)*d*x + 12*(10*a^7 - a^5*b^2 - 9*a^3*b^4 + 2*a*b^6 - (10*a^5*b^2 - 11*a^3*b^4 + 2*a*b^6)*cos(d*x + c)^2 + 2*(10*a^6*b - 11*a^4*b^3 + 2*a^2*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))) + 6*(20*a^7*b - 22*a^5*b^3 - a^3*b^5 + 3*a*b^7)*cos(d*x + c) - ...
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**4*sin(d*x+c)**3/(a+b*sin(d*x+c))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
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 = 0.17 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.63 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/8*(3*(40*a^4 - 24*a^2*b^2 + b^4)*(d*x + c)/b^7 - 24*(10*a^5 - 11*a^3*b^2 + 2*a*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^7) + 8*(9*a^4*b*tan( 1/2*d*x + 1/2*c)^3 - 4*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 10*a^5*tan(1/2*d*x + 1/2*c)^2 + 15*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 - 10*a*b^4*tan(1/2*d*x + 1 /2*c)^2 + 31*a^4*b*tan(1/2*d*x + 1/2*c) - 16*a^2*b^3*tan(1/2*d*x + 1/2*c) + 10*a^5 - 5*a^3*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^6) + 2*(24*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 5*b^3*tan(1/2*d*x + 1 /2*c)^7 + 80*a^3*tan(1/2*d*x + 1/2*c)^6 - 48*a*b^2*tan(1/2*d*x + 1/2*c)^6 + 24*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 3*b^3*tan(1/2*d*x + 1/2*c)^5 + 240*a^3 *tan(1/2*d*x + 1/2*c)^4 - 96*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 24*a^2*b*tan(1 /2*d*x + 1/2*c)^3 - 3*b^3*tan(1/2*d*x + 1/2*c)^3 + 240*a^3*tan(1/2*d*x + 1 /2*c)^2 - 80*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 24*a^2*b*tan(1/2*d*x + 1/2*c) + 5*b^3*tan(1/2*d*x + 1/2*c) + 80*a^3 - 32*a*b^2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*b^6))/d
Time = 41.08 (sec) , antiderivative size = 3581, normalized size of antiderivative = 10.82 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^3,x)
Output:
((30*a^5 - 13*a^3*b^2)/b^6 + (15*tan(c/2 + (d*x)/2)^8*(10*a^5 - 6*a*b^4 + 9*a^3*b^2))/b^6 + (3*tan(c/2 + (d*x)/2)^10*(10*a^5 - 5*a*b^4 + 9*a^3*b^2)) /b^6 + (tan(c/2 + (d*x)/2)^2*(150*a^5 - 37*a*b^4 + 15*a^3*b^2))/b^6 + (2*t an(c/2 + (d*x)/2)^4*(150*a^5 - 59*a*b^4 + 75*a^3*b^2))/b^6 + (tan(c/2 + (d *x)/2)*(420*a^4 - 187*a^2*b^2))/(4*b^5) + (3*tan(c/2 + (d*x)/2)^11*(20*a^4 - 7*a^2*b^2))/(4*b^5) + (3*tan(c/2 + (d*x)/2)^7*(340*a^4 + 2*b^4 - 139*a^ 2*b^2))/(2*b^5) - (tan(c/2 + (d*x)/2)^9*(20*b^4 - 660*a^4 + 231*a^2*b^2))/ (4*b^5) - (tan(c/2 + (d*x)/2)^5*(6*b^4 - 1380*a^4 + 623*a^2*b^2))/(2*b^5) + (tan(c/2 + (d*x)/2)^3*(1740*a^4 + 20*b^4 - 809*a^2*b^2))/(4*b^5) - (2*ta n(c/2 + (d*x)/2)^6*(13*a*b^2 - 30*a^3)*(5*a^2 + 6*b^2))/b^6)/(d*(tan(c/2 + (d*x)/2)^2*(6*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^10*(6*a^2 + 4*b^2) + tan( c/2 + (d*x)/2)^4*(15*a^2 + 16*b^2) + tan(c/2 + (d*x)/2)^8*(15*a^2 + 16*b^2 ) + tan(c/2 + (d*x)/2)^6*(20*a^2 + 24*b^2) + a^2*tan(c/2 + (d*x)/2)^12 + a ^2 + 20*a*b*tan(c/2 + (d*x)/2)^3 + 40*a*b*tan(c/2 + (d*x)/2)^5 + 40*a*b*ta n(c/2 + (d*x)/2)^7 + 20*a*b*tan(c/2 + (d*x)/2)^9 + 4*a*b*tan(c/2 + (d*x)/2 )^11 + 4*a*b*tan(c/2 + (d*x)/2))) + (atan((((a^4*40i + b^4*1i - a^2*b^2*24 i)*(((9*a^2*b^14)/2 - 216*a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*a ^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(18*a*b^16 - 1449*a^3*b^14 + 18576*a^5 *b^12 - 63648*a^7*b^10 + 77760*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) - (3*(a ^4*40i + b^4*1i - a^2*b^2*24i)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^1...
Time = 0.21 (sec) , antiderivative size = 1101, normalized size of antiderivative = 3.33 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x)
Output:
( - 480*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2)) *sin(c + d*x)**2*a**5*b**2 + 528*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)* a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**3*b**4 - 96*sqrt(a**2 - b**2) *atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a*b**6 - 960*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*si n(c + d*x)*a**6*b + 1056*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/s qrt(a**2 - b**2))*sin(c + d*x)*a**4*b**3 - 192*sqrt(a**2 - b**2)*atan((tan ((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a**2*b**5 - 480*sqrt( a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**7 + 528*s qrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**5*b** 2 - 96*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* a**3*b**4 - 4*cos(c + d*x)*sin(c + d*x)**5*a**2*b**6 + 4*cos(c + d*x)*sin( c + d*x)**5*b**8 + 8*cos(c + d*x)*sin(c + d*x)**4*a**3*b**5 - 8*cos(c + d* x)*sin(c + d*x)**4*a*b**7 - 20*cos(c + d*x)*sin(c + d*x)**3*a**4*b**4 + 30 *cos(c + d*x)*sin(c + d*x)**3*a**2*b**6 - 10*cos(c + d*x)*sin(c + d*x)**3* b**8 + 80*cos(c + d*x)*sin(c + d*x)**2*a**5*b**3 - 124*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 + 44*cos(c + d*x)*sin(c + d*x)**2*a*b**7 + 360*cos(c + d*x)*sin(c + d*x)*a**6*b**2 - 526*cos(c + d*x)*sin(c + d*x)*a**4*b**4 + 1 66*cos(c + d*x)*sin(c + d*x)*a**2*b**6 + 240*cos(c + d*x)*a**7*b - 344*cos (c + d*x)*a**5*b**3 + 104*cos(c + d*x)*a**3*b**5 + 240*sin(c + d*x)**2*...