Integrand size = 29, antiderivative size = 525 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {a \left (64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right ) x}{8 b^9}-\frac {2 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^9 d}+\frac {\left (840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}-\frac {a \left (32 a^4-52 a^2 b^2+19 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b^7 d}+\frac {\left (280 a^4-441 a^2 b^2+150 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}-\frac {\left (24 a^4-37 a^2 b^2+12 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a b^5 d}+\frac {\left (224 a^4-340 a^2 b^2+105 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^4 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {3 b \cos (c+d x) \sin ^5(c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {\left (20 a^4-30 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) \sin ^7(c+d x)}{7 b d (a+b \sin (c+d x))} \] Output:
1/8*a*(64*a^6-120*a^4*b^2+60*a^2*b^4-5*b^6)*x/b^9-2*a^2*(8*a^2-3*b^2)*(a^2 -b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^9/d+1/105*( 840*a^6-1435*a^4*b^2+588*a^2*b^4-15*b^6)*cos(d*x+c)/b^8/d-1/8*a*(32*a^4-52 *a^2*b^2+19*b^4)*cos(d*x+c)*sin(d*x+c)/b^7/d+1/105*(280*a^4-441*a^2*b^2+15 0*b^4)*cos(d*x+c)*sin(d*x+c)^2/b^6/d-1/12*(24*a^4-37*a^2*b^2+12*b^4)*cos(d *x+c)*sin(d*x+c)^3/a/b^5/d+1/140*(224*a^4-340*a^2*b^2+105*b^4)*cos(d*x+c)* sin(d*x+c)^4/a^2/b^4/d+1/4*cos(d*x+c)*sin(d*x+c)^4/a/d/(a+b*sin(d*x+c))-3/ 20*b*cos(d*x+c)*sin(d*x+c)^5/a^2/d/(a+b*sin(d*x+c))-1/15*(20*a^4-30*a^2*b^ 2+9*b^4)*cos(d*x+c)*sin(d*x+c)^5/a^2/b^3/d/(a+b*sin(d*x+c))-4/21*a*cos(d*x +c)*sin(d*x+c)^6/b^2/d/(a+b*sin(d*x+c))+1/7*cos(d*x+c)*sin(d*x+c)^7/b/d/(a +b*sin(d*x+c))
Time = 9.56 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-26880 a^2 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+\frac {107520 a^8 c-201600 a^6 b^2 c+100800 a^4 b^4 c-8400 a^2 b^6 c+107520 a^8 d x-201600 a^6 b^2 d x+100800 a^4 b^4 d x-8400 a^2 b^6 d x+840 a b \left (128 a^6-224 a^4 b^2+98 a^2 b^4-5 b^6\right ) \cos (c+d x)+70 \left (64 a^5 b^3-96 a^3 b^5+27 a b^7\right ) \cos (3 (c+d x))-336 a^3 b^5 \cos (5 (c+d x))+350 a b^7 \cos (5 (c+d x))+40 a b^7 \cos (7 (c+d x))+107520 a^7 b c \sin (c+d x)-201600 a^5 b^3 c \sin (c+d x)+100800 a^3 b^5 c \sin (c+d x)-8400 a b^7 c \sin (c+d x)+107520 a^7 b d x \sin (c+d x)-201600 a^5 b^3 d x \sin (c+d x)+100800 a^3 b^5 d x \sin (c+d x)-8400 a b^7 d x \sin (c+d x)+26880 a^6 b^2 \sin (2 (c+d x))-45920 a^4 b^4 \sin (2 (c+d x))+18480 a^2 b^6 \sin (2 (c+d x))-210 b^8 \sin (2 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))+1428 a^2 b^6 \sin (4 (c+d x))-210 b^8 \sin (4 (c+d x))+112 a^2 b^6 \sin (6 (c+d x))-90 b^8 \sin (6 (c+d x))-15 b^8 \sin (8 (c+d x))}{a+b \sin (c+d x)}}{13440 b^9 d} \] Input:
Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]
Output:
(-26880*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/ 2])/Sqrt[a^2 - b^2]] + (107520*a^8*c - 201600*a^6*b^2*c + 100800*a^4*b^4*c - 8400*a^2*b^6*c + 107520*a^8*d*x - 201600*a^6*b^2*d*x + 100800*a^4*b^4*d *x - 8400*a^2*b^6*d*x + 840*a*b*(128*a^6 - 224*a^4*b^2 + 98*a^2*b^4 - 5*b^ 6)*Cos[c + d*x] + 70*(64*a^5*b^3 - 96*a^3*b^5 + 27*a*b^7)*Cos[3*(c + d*x)] - 336*a^3*b^5*Cos[5*(c + d*x)] + 350*a*b^7*Cos[5*(c + d*x)] + 40*a*b^7*Co s[7*(c + d*x)] + 107520*a^7*b*c*Sin[c + d*x] - 201600*a^5*b^3*c*Sin[c + d* x] + 100800*a^3*b^5*c*Sin[c + d*x] - 8400*a*b^7*c*Sin[c + d*x] + 107520*a^ 7*b*d*x*Sin[c + d*x] - 201600*a^5*b^3*d*x*Sin[c + d*x] + 100800*a^3*b^5*d* x*Sin[c + d*x] - 8400*a*b^7*d*x*Sin[c + d*x] + 26880*a^6*b^2*Sin[2*(c + d* x)] - 45920*a^4*b^4*Sin[2*(c + d*x)] + 18480*a^2*b^6*Sin[2*(c + d*x)] - 21 0*b^8*Sin[2*(c + d*x)] - 1120*a^4*b^4*Sin[4*(c + d*x)] + 1428*a^2*b^6*Sin[ 4*(c + d*x)] - 210*b^8*Sin[4*(c + d*x)] + 112*a^2*b^6*Sin[6*(c + d*x)] - 9 0*b^8*Sin[6*(c + d*x)] - 15*b^8*Sin[8*(c + d*x)])/(a + b*Sin[c + d*x]))/(1 3440*b^9*d)
Time = 4.40 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.22, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {3042, 3375, 27, 3042, 3526, 27, 3042, 3528, 27, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^6}{(a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {2 \sin ^5(c+d x) \left (-5 \left (112 a^4-180 b^2 a^2+63 b^4\right ) \sin ^2(c+d x)-2 a b \left (10 a^2-21 b^2\right ) \sin (c+d x)+3 \left (160 a^4-245 b^2 a^2+84 b^4\right )\right )}{(a+b \sin (c+d x))^2}dx}{840 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin ^5(c+d x) \left (-5 \left (112 a^4-180 b^2 a^2+63 b^4\right ) \sin ^2(c+d x)-2 a b \left (10 a^2-21 b^2\right ) \sin (c+d x)+3 \left (160 a^4-245 b^2 a^2+84 b^4\right )\right )}{(a+b \sin (c+d x))^2}dx}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^5 \left (-5 \left (112 a^4-180 b^2 a^2+63 b^4\right ) \sin (c+d x)^2-2 a b \left (10 a^2-21 b^2\right ) \sin (c+d x)+3 \left (160 a^4-245 b^2 a^2+84 b^4\right )\right )}{(a+b \sin (c+d x))^2}dx}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {-\frac {\int -\frac {5 \sin ^4(c+d x) \left (-3 \left (224 a^6-564 b^2 a^4+445 b^4 a^2-105 b^6\right ) \sin ^2(c+d x)-a b \left (16 a^4-37 b^2 a^2+21 b^4\right ) \sin (c+d x)+28 \left (20 a^6-50 b^2 a^4+39 b^4 a^2-9 b^6\right )\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \int \frac {\sin ^4(c+d x) \left (-3 \left (224 a^6-564 b^2 a^4+445 b^4 a^2-105 b^6\right ) \sin ^2(c+d x)-a b \left (16 a^4-37 b^2 a^2+21 b^4\right ) \sin (c+d x)+28 \left (20 a^6-50 b^2 a^4+39 b^4 a^2-9 b^6\right )\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \int \frac {\sin (c+d x)^4 \left (-3 \left (224 a^6-564 b^2 a^4+445 b^4 a^2-105 b^6\right ) \sin (c+d x)^2-a b \left (16 a^4-37 b^2 a^2+21 b^4\right ) \sin (c+d x)+28 \left (20 a^6-50 b^2 a^4+39 b^4 a^2-9 b^6\right )\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {5 \left (\frac {\int -\frac {4 \sin ^3(c+d x) \left (-2 b \left (14 a^4-29 b^2 a^2+15 b^4\right ) \sin (c+d x) a^2-35 \left (24 a^6-61 b^2 a^4+49 b^4 a^2-12 b^6\right ) \sin ^2(c+d x) a+3 \left (224 a^6-564 b^2 a^4+445 b^4 a^2-105 b^6\right ) a\right )}{a+b \sin (c+d x)}dx}{5 b}+\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \int \frac {\sin ^3(c+d x) \left (-2 b \left (14 a^4-29 b^2 a^2+15 b^4\right ) \sin (c+d x) a^2-35 \left (24 a^6-61 b^2 a^4+49 b^4 a^2-12 b^6\right ) \sin ^2(c+d x) a+3 \left (224 a^6-564 b^2 a^4+445 b^4 a^2-105 b^6\right ) a\right )}{a+b \sin (c+d x)}dx}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \int \frac {\sin (c+d x)^3 \left (-2 b \left (14 a^4-29 b^2 a^2+15 b^4\right ) \sin (c+d x) a^2-35 \left (24 a^6-61 b^2 a^4+49 b^4 a^2-12 b^6\right ) \sin (c+d x)^2 a+3 \left (224 a^6-564 b^2 a^4+445 b^4 a^2-105 b^6\right ) a\right )}{a+b \sin (c+d x)}dx}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {\int -\frac {3 \sin ^2(c+d x) \left (-b \left (56 a^4-121 b^2 a^2+65 b^4\right ) \sin (c+d x) a^3-4 \left (280 a^6-721 b^2 a^4+591 b^4 a^2-150 b^6\right ) \sin ^2(c+d x) a^2+35 \left (24 a^6-61 b^2 a^4+49 b^4 a^2-12 b^6\right ) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}+\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \int \frac {\sin ^2(c+d x) \left (-b \left (56 a^4-121 b^2 a^2+65 b^4\right ) \sin (c+d x) a^3-4 \left (280 a^6-721 b^2 a^4+591 b^4 a^2-150 b^6\right ) \sin ^2(c+d x) a^2+35 \left (24 a^6-61 b^2 a^4+49 b^4 a^2-12 b^6\right ) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \int \frac {\sin (c+d x)^2 \left (-b \left (56 a^4-121 b^2 a^2+65 b^4\right ) \sin (c+d x) a^3-4 \left (280 a^6-721 b^2 a^4+591 b^4 a^2-150 b^6\right ) \sin (c+d x)^2 a^2+35 \left (24 a^6-61 b^2 a^4+49 b^4 a^2-12 b^6\right ) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {\int -\frac {\sin (c+d x) \left (-105 \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) \sin ^2(c+d x) a^3+8 \left (280 a^6-721 b^2 a^4+591 b^4 a^2-150 b^6\right ) a^3-b \left (280 a^6-637 b^2 a^4+417 b^4 a^2-60 b^6\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-105 \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) \sin ^2(c+d x) a^3+8 \left (280 a^6-721 b^2 a^4+591 b^4 a^2-150 b^6\right ) a^3-b \left (280 a^6-637 b^2 a^4+417 b^4 a^2-60 b^6\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-105 \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) \sin (c+d x)^2 a^3+8 \left (280 a^6-721 b^2 a^4+591 b^4 a^2-150 b^6\right ) a^3-b \left (280 a^6-637 b^2 a^4+417 b^4 a^2-60 b^6\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {105 \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) a^4-b \left (1120 a^6-2716 b^2 a^4+2001 b^4 a^2-405 b^6\right ) \sin (c+d x) a^3-8 \left (840 a^8-2275 b^2 a^6+2023 b^4 a^4-603 b^6 a^2+15 b^8\right ) \sin ^2(c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}+\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {105 \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) a^4-b \left (1120 a^6-2716 b^2 a^4+2001 b^4 a^2-405 b^6\right ) \sin (c+d x) a^3-8 \left (840 a^8-2275 b^2 a^6+2023 b^4 a^4-603 b^6 a^2+15 b^8\right ) \sin ^2(c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {105 \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) a^4-b \left (1120 a^6-2716 b^2 a^4+2001 b^4 a^2-405 b^6\right ) \sin (c+d x) a^3-8 \left (840 a^8-2275 b^2 a^6+2023 b^4 a^4-603 b^6 a^2+15 b^8\right ) \sin (c+d x)^2 a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {\int \frac {105 \left (b \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) a^4+\left (64 a^8-184 b^2 a^6+180 b^4 a^4-65 b^6 a^2+5 b^8\right ) \sin (c+d x) a^3\right )}{a+b \sin (c+d x)}dx}{b}+\frac {8 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \int \frac {b \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) a^4+\left (64 a^8-184 b^2 a^6+180 b^4 a^4-65 b^6 a^2+5 b^8\right ) \sin (c+d x) a^3}{a+b \sin (c+d x)}dx}{b}+\frac {8 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \int \frac {b \left (32 a^6-84 b^2 a^4+71 b^4 a^2-19 b^6\right ) a^4+\left (64 a^8-184 b^2 a^6+180 b^4 a^4-65 b^6 a^2+5 b^8\right ) \sin (c+d x) a^3}{a+b \sin (c+d x)}dx}{b}+\frac {8 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^3 x \left (64 a^8-184 a^6 b^2+180 a^4 b^4-65 a^2 b^6+5 b^8\right )}{b}-\frac {8 a^4 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {8 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^3 x \left (64 a^8-184 a^6 b^2+180 a^4 b^4-65 a^2 b^6+5 b^8\right )}{b}-\frac {8 a^4 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {8 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^3 x \left (64 a^8-184 a^6 b^2+180 a^4 b^4-65 a^2 b^6+5 b^8\right )}{b}-\frac {16 a^4 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {8 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {32 a^4 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a^3 x \left (64 a^8-184 a^6 b^2+180 a^4 b^4-65 a^2 b^6+5 b^8\right )}{b}\right )}{b}+\frac {8 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}+\frac {\frac {5 \left (\frac {3 \left (224 a^6-564 a^4 b^2+445 a^2 b^4-105 b^6\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {4 \left (\frac {35 a \left (24 a^6-61 a^4 b^2+49 a^2 b^4-12 b^6\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a^2 \left (280 a^6-721 a^4 b^2+591 a^2 b^4-150 b^6\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (32 a^6-84 a^4 b^2+71 a^2 b^4-19 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {8 a^2 \left (840 a^8-2275 a^6 b^2+2023 a^4 b^4-603 a^2 b^6+15 b^8\right ) \cos (c+d x)}{b d}+\frac {105 \left (\frac {a^3 x \left (64 a^8-184 a^6 b^2+180 a^4 b^4-65 a^2 b^6+5 b^8\right )}{b}-\frac {16 a^4 \left (8 a^2-3 b^2\right ) \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d}\right )}{b}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}\right )}{b \left (a^2-b^2\right )}-\frac {28 \left (20 a^4-30 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{420 a^2 b^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac {\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}\) |
Input:
Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]
Output:
(Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])) - (3*b*Cos[c + d*x]*Sin[c + d*x]^5)/(20*a^2*d*(a + b*Sin[c + d*x])) - (4*a*Cos[c + d*x]*S in[c + d*x]^6)/(21*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x ]^7)/(7*b*d*(a + b*Sin[c + d*x])) + ((-28*(20*a^4 - 30*a^2*b^2 + 9*b^4)*Co s[c + d*x]*Sin[c + d*x]^5)/(b*d*(a + b*Sin[c + d*x])) + (5*((3*(224*a^6 - 564*a^4*b^2 + 445*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x]^4)/(5*b*d) - (4*((35*a*(24*a^6 - 61*a^4*b^2 + 49*a^2*b^4 - 12*b^6)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d) - (3*((4*a^2*(280*a^6 - 721*a^4*b^2 + 591*a^2*b^4 - 150 *b^6)*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d) - (-1/2*((105*((a^3*(64*a^8 - 1 84*a^6*b^2 + 180*a^4*b^4 - 65*a^2*b^6 + 5*b^8)*x)/b - (16*a^4*(8*a^2 - 3*b ^2)*(a^2 - b^2)^(5/2)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^ 2])])/(b*d)))/b + (8*a^2*(840*a^8 - 2275*a^6*b^2 + 2023*a^4*b^4 - 603*a^2* b^6 + 15*b^8)*Cos[c + d*x])/(b*d))/b + (105*a^3*(32*a^6 - 84*a^4*b^2 + 71* a^2*b^4 - 19*b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*b*d))/(3*b)))/(4*b)))/(5*b )))/(b*(a^2 - b^2)))/(420*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_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[ e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x] )^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[a*(n + 5)*Cos[e + f*x]*(d *Sin[e + f*x])^(n + 3)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3*f*(m + n + 5) *(m + n + 6))), x] + Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 4)*((a + b*Sin [e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6))), x] + Simp[1/(a^2*b^2*d^2*(n + 1) *(n + 2)*(m + n + 5)*(m + n + 6)) Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin [e + f*x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2* n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4 )*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && Ne Q[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0] && NeQ[m + n + 6, 0] && !IGtQ[m, 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^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -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 = 4.59 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {-\frac {4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2} \left (\frac {-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a b}{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 {\left (8 a^{2}-3 b^{2}\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^{9}}+\frac {\frac {4 \left (\left (\frac {3}{2} a^{5} b^{2}-\frac {9}{4} a^{3} b^{4}+\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\left (\frac {7}{2} a^{6} b -\frac {15}{2} a^{4} b^{3}+\frac {9}{2} a^{2} b^{5}-\frac {1}{2} b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (6 a^{5} b^{2}-7 a^{3} b^{4}+\frac {7}{12} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (21 a^{6} b -40 a^{4} b^{3}+18 a^{2} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {15}{2} a^{5} b^{2}-\frac {29}{4} a^{3} b^{4}+\frac {85}{48} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {105}{2} a^{6} b -\frac {545}{6} a^{4} b^{3}+\frac {73}{2} a^{2} b^{5}-\frac {5}{2} b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (70 a^{6} b -\frac {340}{3} a^{4} b^{3}+44 a^{2} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {15}{2} a^{5} b^{2}+\frac {29}{4} a^{3} b^{4}-\frac {85}{48} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (\frac {105}{2} a^{6} b -\frac {165}{2} a^{4} b^{3}+\frac {303}{10} a^{2} b^{5}-\frac {3}{2} b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-6 a^{5} b^{2}+7 a^{3} b^{4}-\frac {7}{12} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (21 a^{6} b -\frac {100}{3} a^{4} b^{3}+\frac {58}{5} a^{2} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {3}{2} a^{5} b^{2}+\frac {9}{4} a^{3} b^{4}-\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {7 a^{6} b}{2}-\frac {35 a^{4} b^{3}}{6}+\frac {23 a^{2} b^{5}}{10}-\frac {b^{7}}{14}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {a \left (64 a^{6}-120 a^{4} b^{2}+60 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{9}}}{d}\) | \(656\) |
| default | \(\frac {-\frac {4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2} \left (\frac {-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a b}{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 {\left (8 a^{2}-3 b^{2}\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^{9}}+\frac {\frac {4 \left (\left (\frac {3}{2} a^{5} b^{2}-\frac {9}{4} a^{3} b^{4}+\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\left (\frac {7}{2} a^{6} b -\frac {15}{2} a^{4} b^{3}+\frac {9}{2} a^{2} b^{5}-\frac {1}{2} b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (6 a^{5} b^{2}-7 a^{3} b^{4}+\frac {7}{12} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (21 a^{6} b -40 a^{4} b^{3}+18 a^{2} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {15}{2} a^{5} b^{2}-\frac {29}{4} a^{3} b^{4}+\frac {85}{48} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {105}{2} a^{6} b -\frac {545}{6} a^{4} b^{3}+\frac {73}{2} a^{2} b^{5}-\frac {5}{2} b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (70 a^{6} b -\frac {340}{3} a^{4} b^{3}+44 a^{2} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {15}{2} a^{5} b^{2}+\frac {29}{4} a^{3} b^{4}-\frac {85}{48} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (\frac {105}{2} a^{6} b -\frac {165}{2} a^{4} b^{3}+\frac {303}{10} a^{2} b^{5}-\frac {3}{2} b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-6 a^{5} b^{2}+7 a^{3} b^{4}-\frac {7}{12} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (21 a^{6} b -\frac {100}{3} a^{4} b^{3}+\frac {58}{5} a^{2} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {3}{2} a^{5} b^{2}+\frac {9}{4} a^{3} b^{4}-\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {7 a^{6} b}{2}-\frac {35 a^{4} b^{3}}{6}+\frac {23 a^{2} b^{5}}{10}-\frac {b^{7}}{14}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {a \left (64 a^{6}-120 a^{4} b^{2}+60 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{9}}}{d}\) | \(656\) |
| risch | \(-\frac {5 a x}{8 b^{3}}+\frac {3 \cos \left (5 d x +5 c \right ) a^{2}}{80 d \,b^{4}}+\frac {a^{3} \sin \left (4 d x +4 c \right )}{8 b^{5} d}-\frac {3 a \sin \left (4 d x +4 c \right )}{32 b^{3} d}-\frac {5 \cos \left (3 d x +3 c \right ) a^{4}}{12 d \,b^{6}}+\frac {7 \cos \left (3 d x +3 c \right ) a^{2}}{16 d \,b^{4}}-\frac {a \sin \left (6 d x +6 c \right )}{96 b^{3} d}+\frac {7 \,{\mathrm e}^{i \left (d x +c \right )} a^{6}}{2 d \,b^{8}}-\frac {45 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{8 d \,b^{6}}+\frac {33 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 d \,b^{4}}+\frac {7 \,{\mathrm e}^{-i \left (d x +c \right )} a^{6}}{2 d \,b^{8}}-\frac {45 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{8 d \,b^{6}}+\frac {33 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 d \,b^{4}}+\frac {3 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{7} d}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{b^{5} d}+\frac {15 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{64 b^{3} d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{b^{5} d}-\frac {3 i a^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{7} d}-\frac {15 i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{64 b^{3} d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{128 b^{2} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 b^{2} d}-\frac {\cos \left (7 d x +7 c \right )}{448 d \,b^{2}}-\frac {\cos \left (5 d x +5 c \right )}{64 d \,b^{2}}-\frac {3 \cos \left (3 d x +3 c \right )}{64 d \,b^{2}}+\frac {8 a^{7} x}{b^{9}}-\frac {15 a^{5} x}{b^{7}}+\frac {15 a^{3} x}{2 b^{5}}-\frac {8 i \sqrt {a^{2}-b^{2}}\, a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{9}}+\frac {11 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 {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 {8 i \sqrt {a^{2}-b^{2}}\, a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{9}}+\frac {2 i a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{9} 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 {11 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 {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}}\) | \(875\) |
Input:
int(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-4*(a^4-2*a^2*b^2+b^4)*a^2/b^9*((-1/2*b^2*tan(1/2*d*x+1/2*c)-1/2*a*b) /(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+1/2*(8*a^2-3*b^2)/(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^9 *(((3/2*a^5*b^2-9/4*a^3*b^4+11/16*a*b^6)*tan(1/2*d*x+1/2*c)^13+(7/2*a^6*b- 15/2*a^4*b^3+9/2*a^2*b^5-1/2*b^7)*tan(1/2*d*x+1/2*c)^12+(6*a^5*b^2-7*a^3*b ^4+7/12*a*b^6)*tan(1/2*d*x+1/2*c)^11+(21*a^6*b-40*a^4*b^3+18*a^2*b^5)*tan( 1/2*d*x+1/2*c)^10+(15/2*a^5*b^2-29/4*a^3*b^4+85/48*a*b^6)*tan(1/2*d*x+1/2* c)^9+(105/2*a^6*b-545/6*a^4*b^3+73/2*a^2*b^5-5/2*b^7)*tan(1/2*d*x+1/2*c)^8 +(70*a^6*b-340/3*a^4*b^3+44*a^2*b^5)*tan(1/2*d*x+1/2*c)^6+(-15/2*a^5*b^2+2 9/4*a^3*b^4-85/48*a*b^6)*tan(1/2*d*x+1/2*c)^5+(105/2*a^6*b-165/2*a^4*b^3+3 03/10*a^2*b^5-3/2*b^7)*tan(1/2*d*x+1/2*c)^4+(-6*a^5*b^2+7*a^3*b^4-7/12*a*b ^6)*tan(1/2*d*x+1/2*c)^3+(21*a^6*b-100/3*a^4*b^3+58/5*a^2*b^5)*tan(1/2*d*x +1/2*c)^2+(-3/2*a^5*b^2+9/4*a^3*b^4-11/16*a*b^6)*tan(1/2*d*x+1/2*c)+7/2*a^ 6*b-35/6*a^4*b^3+23/10*a^2*b^5-1/14*b^7)/(1+tan(1/2*d*x+1/2*c)^2)^7+1/16*a *(64*a^6-120*a^4*b^2+60*a^2*b^4-5*b^6)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.23 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.66 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="frica s")
Output:
[1/840*(160*a*b^7*cos(d*x + c)^7 - 14*(24*a^3*b^5 - 5*a*b^7)*cos(d*x + c)^ 5 + 35*(32*a^5*b^3 - 36*a^3*b^5 + 5*a*b^7)*cos(d*x + c)^3 + 105*(64*a^8 - 120*a^6*b^2 + 60*a^4*b^4 - 5*a^2*b^6)*d*x + 420*(8*a^7 - 11*a^5*b^2 + 3*a^ 3*b^4 + (8*a^6*b - 11*a^4*b^3 + 3*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)) + 105*(64*a^7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*cos(d*x + c) - (120*b^8*cos(d*x + c)^7 - 224*a^2*b^ 6*cos(d*x + c)^5 + 70*(8*a^4*b^4 - 7*a^2*b^6)*cos(d*x + c)^3 - 105*(64*a^7 *b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*d*x - 105*(32*a^6*b^2 - 52*a^4*b^ 4 + 19*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/(b^10*d*sin(d*x + c) + a*b^9*d ), 1/840*(160*a*b^7*cos(d*x + c)^7 - 14*(24*a^3*b^5 - 5*a*b^7)*cos(d*x + c )^5 + 35*(32*a^5*b^3 - 36*a^3*b^5 + 5*a*b^7)*cos(d*x + c)^3 + 105*(64*a^8 - 120*a^6*b^2 + 60*a^4*b^4 - 5*a^2*b^6)*d*x + 840*(8*a^7 - 11*a^5*b^2 + 3* a^3*b^4 + (8*a^6*b - 11*a^4*b^3 + 3*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))) + 105*(64*a^ 7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*cos(d*x + c) - (120*b^8*cos(d*x + c)^7 - 224*a^2*b^6*cos(d*x + c)^5 + 70*(8*a^4*b^4 - 7*a^2*b^6)*cos(d*x + c)^3 - 105*(64*a^7*b - 120*a^5*b^3 + 60*a^3*b^5 - 5*a*b^7)*d*x - 105*(32* a^6*b^2 - 52*a^4*b^4 + 19*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/(b^10*d*...
Timed out. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**3/(a+b*sin(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^6*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 = 0.24 (sec) , antiderivative size = 965, normalized size of antiderivative = 1.84 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/840*(105*(64*a^7 - 120*a^5*b^2 + 60*a^3*b^4 - 5*a*b^6)*(d*x + c)/b^9 - 1 680*(8*a^8 - 19*a^6*b^2 + 14*a^4*b^4 - 3*a^2*b^6)*(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^9) + 1680*(a^6*b*tan(1/2*d*x + 1/2*c) - 2*a^4*b^3*tan(1/ 2*d*x + 1/2*c) + a^2*b^5*tan(1/2*d*x + 1/2*c) + a^7 - 2*a^5*b^2 + a^3*b^4) /((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*b^8) + 2*(2520 *a^5*b*tan(1/2*d*x + 1/2*c)^13 - 3780*a^3*b^3*tan(1/2*d*x + 1/2*c)^13 + 11 55*a*b^5*tan(1/2*d*x + 1/2*c)^13 + 5880*a^6*tan(1/2*d*x + 1/2*c)^12 - 1260 0*a^4*b^2*tan(1/2*d*x + 1/2*c)^12 + 7560*a^2*b^4*tan(1/2*d*x + 1/2*c)^12 - 840*b^6*tan(1/2*d*x + 1/2*c)^12 + 10080*a^5*b*tan(1/2*d*x + 1/2*c)^11 - 1 1760*a^3*b^3*tan(1/2*d*x + 1/2*c)^11 + 980*a*b^5*tan(1/2*d*x + 1/2*c)^11 + 35280*a^6*tan(1/2*d*x + 1/2*c)^10 - 67200*a^4*b^2*tan(1/2*d*x + 1/2*c)^10 + 30240*a^2*b^4*tan(1/2*d*x + 1/2*c)^10 + 12600*a^5*b*tan(1/2*d*x + 1/2*c )^9 - 12180*a^3*b^3*tan(1/2*d*x + 1/2*c)^9 + 2975*a*b^5*tan(1/2*d*x + 1/2* c)^9 + 88200*a^6*tan(1/2*d*x + 1/2*c)^8 - 152600*a^4*b^2*tan(1/2*d*x + 1/2 *c)^8 + 61320*a^2*b^4*tan(1/2*d*x + 1/2*c)^8 - 4200*b^6*tan(1/2*d*x + 1/2* c)^8 + 117600*a^6*tan(1/2*d*x + 1/2*c)^6 - 190400*a^4*b^2*tan(1/2*d*x + 1/ 2*c)^6 + 73920*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 - 12600*a^5*b*tan(1/2*d*x + 1/2*c)^5 + 12180*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 - 2975*a*b^5*tan(1/2*d*x + 1/2*c)^5 + 88200*a^6*tan(1/2*d*x + 1/2*c)^4 - 138600*a^4*b^2*tan(1/2*d...
Time = 27.16 (sec) , antiderivative size = 3724, normalized size of antiderivative = 7.09 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^6*sin(c + d*x)^3)/(a + b*sin(c + d*x))^2,x)
Output:
((tan(c/2 + (d*x)/2)^14*(7*a*b^6 + 32*a^7 + 4*a^3*b^4 - 44*a^5*b^2))/(2*b^ 8) - (2*(15*a*b^6 - 840*a^7 - 588*a^3*b^4 + 1435*a^5*b^2))/(105*b^8) + (2* tan(c/2 + (d*x)/2)^12*(4*a*b^6 + 168*a^7 + 72*a^3*b^4 - 255*a^5*b^2))/(3*b ^8) - (2*tan(c/2 + (d*x)/2)^8*(15*a*b^6 - 840*a^7 - 588*a^3*b^4 + 1435*a^5 *b^2))/(3*b^8) + (tan(c/2 + (d*x)/2)^10*(25*a*b^6 + 2016*a^7 + 1212*a^3*b^ 4 - 3284*a^5*b^2))/(6*b^8) - (2*tan(c/2 + (d*x)/2)^4*(80*a*b^6 - 2520*a^7 - 1992*a^3*b^4 + 4465*a^5*b^2))/(15*b^8) - (tan(c/2 + (d*x)/2)^6*(605*a*b^ 6 - 16800*a^7 - 12756*a^3*b^4 + 29500*a^5*b^2))/(30*b^8) - (tan(c/2 + (d*x )/2)^2*(1215*a*b^6 - 23520*a^7 - 18396*a^3*b^4 + 41300*a^5*b^2))/(210*b^8) + (tan(c/2 + (d*x)/2)*(10080*a^6 - 240*b^6 + 7413*a^2*b^4 - 17500*a^4*b^2 ))/(420*b^7) + (tan(c/2 + (d*x)/2)^15*(32*a^6 + 19*a^2*b^4 - 52*a^4*b^2))/ (4*b^7) + (tan(c/2 + (d*x)/2)^11*(3168*a^6 + 2345*a^2*b^4 - 5532*a^4*b^2)) /(12*b^7) + (tan(c/2 + (d*x)/2)^7*(7200*a^6 + 4979*a^2*b^4 - 12212*a^4*b^2 ))/(12*b^7) + (tan(c/2 + (d*x)/2)^3*(9120*a^6 + 6103*a^2*b^4 - 15460*a^4*b ^2))/(60*b^7) + (tan(c/2 + (d*x)/2)^13*(864*a^6 - 48*b^6 + 661*a^2*b^4 - 1 500*a^4*b^2))/(12*b^7) + (tan(c/2 + (d*x)/2)^9*(6240*a^6 - 240*b^6 + 4429* a^2*b^4 - 10748*a^4*b^2))/(12*b^7) + (tan(c/2 + (d*x)/2)^5*(24480*a^6 - 72 0*b^6 + 16499*a^2*b^4 - 41220*a^4*b^2))/(60*b^7))/(d*(a + 2*b*tan(c/2 + (d *x)/2) + 8*a*tan(c/2 + (d*x)/2)^2 + 28*a*tan(c/2 + (d*x)/2)^4 + 56*a*tan(c /2 + (d*x)/2)^6 + 70*a*tan(c/2 + (d*x)/2)^8 + 56*a*tan(c/2 + (d*x)/2)^1...
Time = 0.74 (sec) , antiderivative size = 860, normalized size of antiderivative = 1.64 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x)
Output:
( - 13440*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2 ))*sin(c + d*x)*a**6*b + 18480*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a**4*b**3 - 5040*sqrt(a**2 - b**2)*at an((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a**2*b**5 - 13 440*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a** 7 + 18480*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2 ))*a**5*b**2 - 5040*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a **2 - b**2))*a**3*b**4 + 120*cos(c + d*x)*sin(c + d*x)**7*b**8 - 160*cos(c + d*x)*sin(c + d*x)**6*a*b**7 + 224*cos(c + d*x)*sin(c + d*x)**5*a**2*b** 6 - 360*cos(c + d*x)*sin(c + d*x)**5*b**8 - 336*cos(c + d*x)*sin(c + d*x)* *4*a**3*b**5 + 550*cos(c + d*x)*sin(c + d*x)**4*a*b**7 + 560*cos(c + d*x)* sin(c + d*x)**3*a**4*b**4 - 938*cos(c + d*x)*sin(c + d*x)**3*a**2*b**6 + 3 60*cos(c + d*x)*sin(c + d*x)**3*b**8 - 1120*cos(c + d*x)*sin(c + d*x)**2*a **5*b**3 + 1932*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 - 795*cos(c + d*x)* sin(c + d*x)**2*a*b**7 + 3360*cos(c + d*x)*sin(c + d*x)*a**6*b**2 - 6020*c os(c + d*x)*sin(c + d*x)*a**4*b**4 + 2709*cos(c + d*x)*sin(c + d*x)*a**2*b **6 - 120*cos(c + d*x)*sin(c + d*x)*b**8 + 6720*cos(c + d*x)*a**7*b - 1148 0*cos(c + d*x)*a**5*b**3 + 4704*cos(c + d*x)*a**3*b**5 - 120*cos(c + d*x)* a*b**7 + 6720*sin(c + d*x)*a**7*b*d*x + 3360*sin(c + d*x)*a**6*b**2 - 1260 0*sin(c + d*x)*a**5*b**3*d*x - 6020*sin(c + d*x)*a**4*b**4 + 6300*sin(c...