Integrand size = 29, antiderivative size = 307 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}+\frac {6 a^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 \sqrt {a^2-b^2} d}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))} \] Output:
-3/4*a*(8*a^4-8*a^2*b^2+b^4)*x/b^7+6*a^2*(2*a^4-3*a^2*b^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-1/5*(30*a^4-25 *a^2*b^2+b^4)*cos(d*x+c)/b^6/d+3/4*a*(4*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)/b ^5/d-1/5*(10*a^2-7*b^2)*cos(d*x+c)*sin(d*x+c)^2/b^4/d+1/2*(3*a^2-2*b^2)*co s(d*x+c)*sin(d*x+c)^3/a/b^3/d-1/5*cos(d*x+c)*sin(d*x+c)^4/b^2/d-(a^2-b^2)* cos(d*x+c)*sin(d*x+c)^4/a/b^2/d/(a+b*sin(d*x+c))
Time = 4.98 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.23 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {960 a^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {960 a^6 c-960 a^4 b^2 c+120 a^2 b^4 c+960 a^6 d x-960 a^4 b^2 d x+120 a^2 b^4 d x+60 a b \left (16 a^4-14 a^2 b^2+b^4\right ) \cos (c+d x)+5 \left (8 a^3 b^3-5 a b^5\right ) \cos (3 (c+d x))-3 a b^5 \cos (5 (c+d x))+960 a^5 b c \sin (c+d x)-960 a^3 b^3 c \sin (c+d x)+120 a b^5 c \sin (c+d x)+960 a^5 b d x \sin (c+d x)-960 a^3 b^3 d x \sin (c+d x)+120 a b^5 d x \sin (c+d x)+240 a^4 b^2 \sin (2 (c+d x))-200 a^2 b^4 \sin (2 (c+d x))+5 b^6 \sin (2 (c+d x))-10 a^2 b^4 \sin (4 (c+d x))+4 b^6 \sin (4 (c+d x))+b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}}{160 b^7 d} \] Input:
Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]
Output:
((960*a^2*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a ^2 - b^2]])/Sqrt[a^2 - b^2] - (960*a^6*c - 960*a^4*b^2*c + 120*a^2*b^4*c + 960*a^6*d*x - 960*a^4*b^2*d*x + 120*a^2*b^4*d*x + 60*a*b*(16*a^4 - 14*a^2 *b^2 + b^4)*Cos[c + d*x] + 5*(8*a^3*b^3 - 5*a*b^5)*Cos[3*(c + d*x)] - 3*a* b^5*Cos[5*(c + d*x)] + 960*a^5*b*c*Sin[c + d*x] - 960*a^3*b^3*c*Sin[c + d* x] + 120*a*b^5*c*Sin[c + d*x] + 960*a^5*b*d*x*Sin[c + d*x] - 960*a^3*b^3*d *x*Sin[c + d*x] + 120*a*b^5*d*x*Sin[c + d*x] + 240*a^4*b^2*Sin[2*(c + d*x) ] - 200*a^2*b^4*Sin[2*(c + d*x)] + 5*b^6*Sin[2*(c + d*x)] - 10*a^2*b^4*Sin [4*(c + d*x)] + 4*b^6*Sin[4*(c + d*x)] + b^6*Sin[6*(c + d*x)])/(a + b*Sin[ c + d*x]))/(160*b^7*d)
Time = 2.18 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.15, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, Rules used = {3042, 3371, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^4}{(a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3371 |
| \(\displaystyle \frac {\int \frac {\sin ^3(c+d x) \left (-10 \left (3 a^2-2 b^2\right ) \sin ^2(c+d x)-a b \sin (c+d x)+3 \left (8 a^2-5 b^2\right )\right )}{a+b \sin (c+d x)}dx}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (-10 \left (3 a^2-2 b^2\right ) \sin (c+d x)^2-a b \sin (c+d x)+3 \left (8 a^2-5 b^2\right )\right )}{a+b \sin (c+d x)}dx}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\int -\frac {6 \sin ^2(c+d x) \left (-b \sin (c+d x) a^2-2 \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) a+5 \left (3 a^2-2 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{4 b}+\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \int \frac {\sin ^2(c+d x) \left (-b \sin (c+d x) a^2-2 \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) a+5 \left (3 a^2-2 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \int \frac {\sin (c+d x)^2 \left (-b \sin (c+d x) a^2-2 \left (10 a^2-7 b^2\right ) \sin (c+d x)^2 a+5 \left (3 a^2-2 b^2\right ) a\right )}{a+b \sin (c+d x)}dx}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {\int -\frac {\sin (c+d x) \left (-15 \left (4 a^2-3 b^2\right ) \sin ^2(c+d x) a^2+4 \left (10 a^2-7 b^2\right ) a^2-b \left (5 a^2-2 b^2\right ) \sin (c+d x) a\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-15 \left (4 a^2-3 b^2\right ) \sin ^2(c+d x) a^2+4 \left (10 a^2-7 b^2\right ) a^2-b \left (5 a^2-2 b^2\right ) \sin (c+d x) a\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-15 \left (4 a^2-3 b^2\right ) \sin (c+d x)^2 a^2+4 \left (10 a^2-7 b^2\right ) a^2-b \left (5 a^2-2 b^2\right ) \sin (c+d x) a\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {15 \left (4 a^2-3 b^2\right ) a^3-b \left (20 a^2-11 b^2\right ) \sin (c+d x) a^2-4 \left (30 a^4-25 b^2 a^2+b^4\right ) \sin ^2(c+d x) a}{a+b \sin (c+d x)}dx}{2 b}+\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {15 \left (4 a^2-3 b^2\right ) a^3-b \left (20 a^2-11 b^2\right ) \sin (c+d x) a^2-4 \left (30 a^4-25 b^2 a^2+b^4\right ) \sin ^2(c+d x) a}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {15 \left (4 a^2-3 b^2\right ) a^3-b \left (20 a^2-11 b^2\right ) \sin (c+d x) a^2-4 \left (30 a^4-25 b^2 a^2+b^4\right ) \sin (c+d x)^2 a}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {\int \frac {15 \left (b \left (4 a^2-3 b^2\right ) a^3+\left (8 a^4-8 b^2 a^2+b^4\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{b}+\frac {4 a \left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \int \frac {b \left (4 a^2-3 b^2\right ) a^3+\left (8 a^4-8 b^2 a^2+b^4\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {4 a \left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \int \frac {b \left (4 a^2-3 b^2\right ) a^3+\left (8 a^4-8 b^2 a^2+b^4\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {4 a \left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^2 x \left (8 a^4-8 a^2 b^2+b^4\right )}{b}-\frac {4 a^3 \left (2 a^4-3 a^2 b^2+b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {4 a \left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^2 x \left (8 a^4-8 a^2 b^2+b^4\right )}{b}-\frac {4 a^3 \left (2 a^4-3 a^2 b^2+b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {4 a \left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^2 x \left (8 a^4-8 a^2 b^2+b^4\right )}{b}-\frac {8 a^3 \left (2 a^4-3 a^2 b^2+b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {4 a \left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {16 a^3 \left (2 a^4-3 a^2 b^2+b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a^2 x \left (8 a^4-8 a^2 b^2+b^4\right )}{b}\right )}{b}+\frac {4 a \left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\frac {5 \left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 b d}-\frac {3 \left (\frac {2 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {4 a \left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{b d}+\frac {15 \left (\frac {a^2 x \left (8 a^4-8 a^2 b^2+b^4\right )}{b}-\frac {8 a^3 \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d \sqrt {a^2-b^2}}\right )}{b}}{2 b}}{3 b}\right )}{2 b}}{5 a b^2}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}\) |
Input:
Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]
Output:
-1/5*(Cos[c + d*x]*Sin[c + d*x]^4)/(b^2*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin [c + d*x]^4)/(a*b^2*d*(a + b*Sin[c + d*x])) + ((5*(3*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*b*d) - (3*((2*a*(10*a^2 - 7*b^2)*Cos[c + d*x]*Sin[ c + d*x]^2)/(3*b*d) - (-1/2*((15*((a^2*(8*a^4 - 8*a^2*b^2 + b^4)*x)/b - (8 *a^3*(2*a^4 - 3*a^2*b^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*(30*a^4 - 25*a^2*b^2 + b^4 )*Cos[c + d*x])/(b*d))/b + (15*a^2*(4*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d* x])/(2*b*d))/(3*b)))/(2*b))/(5*a*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[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*((d*Sin[e + f* x])^(n + 1)/(b^2*d*f*(m + n + 4))), x] - Simp[1/(a*b^2*(m + 1)*(m + n + 4)) Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 2 )*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; Fre eQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && Lt Q[m, -1] && !LtQ[n, -1] && NeQ[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 = 2.90 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.41
| method | result | size |
| derivativedivides | \(\frac {\frac {4 a^{2} \left (\frac {-\frac {b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3} b}{2}+\frac {a \,b^{3}}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {3 \left (2 a^{4}-3 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}-\frac {4 \left (\frac {\left (a^{3} b^{2}-\frac {5}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {5}{2} a^{4} b -3 a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (2 a^{3} b^{2}-\frac {1}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (10 a^{4} b -9 a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (15 a^{4} b -11 a^{2} b^{3}+b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-2 a^{3} b^{2}+\frac {1}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (10 a^{4} b -7 a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-a^{3} b^{2}+\frac {5}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 a^{4} b}{2}-2 a^{2} b^{3}+\frac {b^{5}}{10}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {3 a \left (8 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}\right )}{b^{7}}}{d}\) | \(434\) |
| default | \(\frac {\frac {4 a^{2} \left (\frac {-\frac {b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3} b}{2}+\frac {a \,b^{3}}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {3 \left (2 a^{4}-3 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}-\frac {4 \left (\frac {\left (a^{3} b^{2}-\frac {5}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {5}{2} a^{4} b -3 a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (2 a^{3} b^{2}-\frac {1}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (10 a^{4} b -9 a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (15 a^{4} b -11 a^{2} b^{3}+b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-2 a^{3} b^{2}+\frac {1}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (10 a^{4} b -7 a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-a^{3} b^{2}+\frac {5}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 a^{4} b}{2}-2 a^{2} b^{3}+\frac {b^{5}}{10}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {3 a \left (8 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}\right )}{b^{7}}}{d}\) | \(434\) |
| risch | \(-\frac {6 a^{5} x}{b^{7}}+\frac {6 a^{3} x}{b^{5}}-\frac {3 a x}{4 b^{3}}+\frac {2 i a^{3} \left (-a^{2}+b^{2}\right ) \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{7} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}-\frac {6 i \sqrt {a^{2}-b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{7}}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 b^{6} d}+\frac {15 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{8 b^{4} d}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{16 b^{2} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 b^{6} d}+\frac {15 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{8 b^{4} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{16 b^{2} d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 b^{5} d}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{3} d}+\frac {3 i \sqrt {a^{2}-b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}+\frac {6 i \sqrt {a^{2}-b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{7}}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{3} d}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{2 b^{5} d}-\frac {3 i \sqrt {a^{2}-b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}-\frac {\cos \left (5 d x +5 c \right )}{80 d \,b^{2}}-\frac {a \sin \left (4 d x +4 c \right )}{16 b^{3} d}+\frac {\cos \left (3 d x +3 c \right ) a^{2}}{4 d \,b^{4}}-\frac {\cos \left (3 d x +3 c \right )}{16 d \,b^{2}}\) | \(575\) |
Input:
int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(4*a^2/b^7*((-1/2*b^2*(a^2-b^2)*tan(1/2*d*x+1/2*c)-1/2*a^3*b+1/2*a*b^3 )/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+3/2*(2*a^4-3*a^2*b^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^7*(((a^3*b^2-5/8*a*b^4)*tan(1/2*d*x+1/2*c)^9+(5/2*a^4*b-3*a^2*b^3+ 1/2*b^5)*tan(1/2*d*x+1/2*c)^8+(2*a^3*b^2-1/4*a*b^4)*tan(1/2*d*x+1/2*c)^7+( 10*a^4*b-9*a^2*b^3)*tan(1/2*d*x+1/2*c)^6+(15*a^4*b-11*a^2*b^3+b^5)*tan(1/2 *d*x+1/2*c)^4+(-2*a^3*b^2+1/4*a*b^4)*tan(1/2*d*x+1/2*c)^3+(10*a^4*b-7*a^2* b^3)*tan(1/2*d*x+1/2*c)^2+(-a^3*b^2+5/8*a*b^4)*tan(1/2*d*x+1/2*c)+5/2*a^4* b-2*a^2*b^3+1/10*b^5)/(1+tan(1/2*d*x+1/2*c)^2)^5+3/8*a*(8*a^4-8*a^2*b^2+b^ 4)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.13 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.11 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="frica s")
Output:
[1/20*(6*a*b^5*cos(d*x + c)^5 - 5*(4*a^3*b^3 - a*b^5)*cos(d*x + c)^3 - 15* (8*a^6 - 8*a^4*b^2 + a^2*b^4)*d*x - 30*(2*a^5 - a^3*b^2 + (2*a^4*b - a^2*b ^3)*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)) - 15*(8*a^5*b - 8*a^3*b^3 + a*b^5)*cos(d*x + c) - (4*b^6*cos(d*x + c)^ 5 - 10*a^2*b^4*cos(d*x + c)^3 + 15*(8*a^5*b - 8*a^3*b^3 + a*b^5)*d*x + 15* (4*a^4*b^2 - 3*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/(b^8*d*sin(d*x + c) + a*b^7*d), 1/20*(6*a*b^5*cos(d*x + c)^5 - 5*(4*a^3*b^3 - a*b^5)*cos(d*x + c )^3 - 15*(8*a^6 - 8*a^4*b^2 + a^2*b^4)*d*x - 60*(2*a^5 - a^3*b^2 + (2*a^4* b - a^2*b^3)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(s qrt(a^2 - b^2)*cos(d*x + c))) - 15*(8*a^5*b - 8*a^3*b^3 + a*b^5)*cos(d*x + c) - (4*b^6*cos(d*x + c)^5 - 10*a^2*b^4*cos(d*x + c)^3 + 15*(8*a^5*b - 8* a^3*b^3 + a*b^5)*d*x + 15*(4*a^4*b^2 - 3*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/(b^8*d*sin(d*x + c) + a*b^7*d)]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**4*sin(d*x+c)**3/(a+b*sin(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,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 = 0.14 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.75 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="giac" )
Output:
-1/20*(15*(8*a^5 - 8*a^3*b^2 + a*b^4)*(d*x + c)/b^7 - 120*(2*a^6 - 3*a^4*b ^2 + a^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^7) + 40*(a^4*b*tan (1/2*d*x + 1/2*c) - a^2*b^3*tan(1/2*d*x + 1/2*c) + a^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)*b^6) + 2*(40*a^3*b*tan( 1/2*d*x + 1/2*c)^9 - 25*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 100*a^4*tan(1/2*d*x + 1/2*c)^8 - 120*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 + 20*b^4*tan(1/2*d*x + 1/ 2*c)^8 + 80*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 10*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 400*a^4*tan(1/2*d*x + 1/2*c)^6 - 360*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 + 6 00*a^4*tan(1/2*d*x + 1/2*c)^4 - 440*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 40*b^ 4*tan(1/2*d*x + 1/2*c)^4 - 80*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 10*a*b^3*tan( 1/2*d*x + 1/2*c)^3 + 400*a^4*tan(1/2*d*x + 1/2*c)^2 - 280*a^2*b^2*tan(1/2* d*x + 1/2*c)^2 - 40*a^3*b*tan(1/2*d*x + 1/2*c) + 25*a*b^3*tan(1/2*d*x + 1/ 2*c) + 100*a^4 - 80*a^2*b^2 + 4*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*b^6)) /d
Time = 36.96 (sec) , antiderivative size = 2390, normalized size of antiderivative = 7.79 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^2,x)
Output:
- ((2*(a*b^4 + 30*a^5 - 25*a^3*b^2))/(5*b^6) - (3*tan(c/2 + (d*x)/2)^10*(a *b^4 - 4*a^5 + 2*a^3*b^2))/b^6 + (6*tan(c/2 + (d*x)/2)^4*(a*b^4 + 20*a^5 - 18*a^3*b^2))/b^6 + (4*tan(c/2 + (d*x)/2)^6*(a*b^4 + 30*a^5 - 25*a^3*b^2)) /b^6 + (3*tan(c/2 + (d*x)/2)^2*(9*a*b^4 + 100*a^5 - 90*a^3*b^2))/(5*b^6) + (tan(c/2 + (d*x)/2)*(180*a^4 + 8*b^4 - 155*a^2*b^2))/(10*b^5) + (3*tan(c/ 2 + (d*x)/2)^11*(4*a^4 - 3*a^2*b^2))/(2*b^5) + (6*tan(c/2 + (d*x)/2)^8*(10 *a^5 - 7*a^3*b^2))/b^6 + (3*tan(c/2 + (d*x)/2)^7*(36*a^4 - 31*a^2*b^2))/b^ 5 + (tan(c/2 + (d*x)/2)^3*(156*a^4 - 125*a^2*b^2))/(2*b^5) + (tan(c/2 + (d *x)/2)^9*(84*a^4 + 8*b^4 - 75*a^2*b^2))/(2*b^5) + (tan(c/2 + (d*x)/2)^5*(1 32*a^4 + 8*b^4 - 107*a^2*b^2))/b^5)/(d*(a + 2*b*tan(c/2 + (d*x)/2) + 6*a*t an(c/2 + (d*x)/2)^2 + 15*a*tan(c/2 + (d*x)/2)^4 + 20*a*tan(c/2 + (d*x)/2)^ 6 + 15*a*tan(c/2 + (d*x)/2)^8 + 6*a*tan(c/2 + (d*x)/2)^10 + a*tan(c/2 + (d *x)/2)^12 + 10*b*tan(c/2 + (d*x)/2)^3 + 20*b*tan(c/2 + (d*x)/2)^5 + 20*b*t an(c/2 + (d*x)/2)^7 + 10*b*tan(c/2 + (d*x)/2)^9 + 2*b*tan(c/2 + (d*x)/2)^1 1)) - (2*atanh((108*a^6*(b^2 - a^2)^(1/2))/(108*a^6*b - (324*a^8)/b + (216 *a^10)/b^3 - 648*a^7*tan(c/2 + (d*x)/2) + 216*a^5*b^2*tan(c/2 + (d*x)/2) + (432*a^9*tan(c/2 + (d*x)/2))/b^2) - (216*a^8*(b^2 - a^2)^(1/2))/(108*a^6* b^3 - 324*a^8*b + (216*a^10)/b + 432*a^9*tan(c/2 + (d*x)/2) + 216*a^5*b^4* tan(c/2 + (d*x)/2) - 648*a^7*b^2*tan(c/2 + (d*x)/2)) + (216*a^5*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))/(108*a^6 - (324*a^8)/b^2 + (216*a^10)/b^4 +...
Time = 0.21 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.81 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {240 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right ) a^{4} b -120 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right ) a^{2} b^{3}-120 \sin \left (d x +c \right ) a^{5} b d x +120 \sin \left (d x +c \right ) a^{3} b^{3} d x -15 \sin \left (d x +c \right ) a \,b^{5} d x +240 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{5}-120 \cos \left (d x +c \right ) a^{5} b +100 \cos \left (d x +c \right ) a^{3} b^{3}-60 \sin \left (d x +c \right ) a^{4} b^{2}+55 \sin \left (d x +c \right ) a^{2} b^{4}-120 a^{6} d x -4 \cos \left (d x +c \right ) a \,b^{5}-120 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{3} b^{2}+20 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3} b^{3}-17 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{5}-60 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b^{2}+55 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{4}+120 a^{4} b^{2} d x -15 a^{2} b^{4} d x -4 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{6}-60 a^{5} b +55 a^{3} b^{3}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{6}-10 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{4}-4 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{6}-4 \sin \left (d x +c \right ) b^{6}-4 a \,b^{5}+6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{5}}{20 b^{7} d \left (\sin \left (d x +c \right ) b +a \right )} \] Input:
int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x)
Output:
(240*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*si n(c + d*x)*a**4*b - 120*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sq rt(a**2 - b**2))*sin(c + d*x)*a**2*b**3 + 240*sqrt(a**2 - b**2)*atan((tan( (c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**5 - 120*sqrt(a**2 - b**2)*atan(( tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**3*b**2 - 4*cos(c + d*x)*sin( c + d*x)**5*b**6 + 6*cos(c + d*x)*sin(c + d*x)**4*a*b**5 - 10*cos(c + d*x) *sin(c + d*x)**3*a**2*b**4 + 8*cos(c + d*x)*sin(c + d*x)**3*b**6 + 20*cos( c + d*x)*sin(c + d*x)**2*a**3*b**3 - 17*cos(c + d*x)*sin(c + d*x)**2*a*b** 5 - 60*cos(c + d*x)*sin(c + d*x)*a**4*b**2 + 55*cos(c + d*x)*sin(c + d*x)* a**2*b**4 - 4*cos(c + d*x)*sin(c + d*x)*b**6 - 120*cos(c + d*x)*a**5*b + 1 00*cos(c + d*x)*a**3*b**3 - 4*cos(c + d*x)*a*b**5 - 120*sin(c + d*x)*a**5* b*d*x - 60*sin(c + d*x)*a**4*b**2 + 120*sin(c + d*x)*a**3*b**3*d*x + 55*si n(c + d*x)*a**2*b**4 - 15*sin(c + d*x)*a*b**5*d*x - 4*sin(c + d*x)*b**6 - 120*a**6*d*x - 60*a**5*b + 120*a**4*b**2*d*x + 55*a**3*b**3 - 15*a**2*b**4 *d*x - 4*a*b**5)/(20*b**7*d*(sin(c + d*x)*b + a))