Integrand size = 29, antiderivative size = 485 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}-\frac {\sqrt {a^2-b^2} \left (42 a^4-29 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^8 d}+\frac {\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac {\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac {\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac {\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))} \] Output:
1/8*a*(168*a^4-200*a^2*b^2+45*b^4)*x/b^8-(a^2-b^2)^(1/2)*(42*a^4-29*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^8/d+1/30*(630* a^4-645*a^2*b^2+91*b^4)*cos(d*x+c)/b^7/d-1/8*(84*a^4-79*a^2*b^2+8*b^4)*cos (d*x+c)*sin(d*x+c)/a/b^6/d+1/30*(210*a^4-187*a^2*b^2+15*b^4)*cos(d*x+c)*si n(d*x+c)^2/a^2/b^5/d+1/3*cos(d*x+c)*sin(d*x+c)^3/a/d/(a+b*sin(d*x+c))^2-1/ 12*b*cos(d*x+c)*sin(d*x+c)^4/a^2/d/(a+b*sin(d*x+c))^2-1/60*(63*a^4-60*a^2* b^2+5*b^4)*cos(d*x+c)*sin(d*x+c)^4/a^2/b^3/d/(a+b*sin(d*x+c))^2-7/20*a*cos (d*x+c)*sin(d*x+c)^5/b^2/d/(a+b*sin(d*x+c))^2+1/5*cos(d*x+c)*sin(d*x+c)^6/ b/d/(a+b*sin(d*x+c))^2-1/12*(63*a^4-54*a^2*b^2+4*b^4)*cos(d*x+c)*sin(d*x+c )^3/a^2/b^4/d/(a+b*sin(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(1883\) vs. \(2(485)=970\).
Time = 17.08 (sec) , antiderivative size = 1883, normalized size of antiderivative = 3.88 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]
Output:
-1/64*(-48*a*(c + d*x) + (6*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*Arc Tan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 16*b*Co s[c + d*x] + (b*(8*a^4 - 8*a^2*b^2 + b^4)*Cos[c + d*x])/((a - b)*(a + b)*( a + b*Sin[c + d*x])^2) + (a*b*(-40*a^4 + 72*a^2*b^2 - 29*b^4)*Cos[c + d*x] )/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/(b^4*d) + (5*((2*(2*a^2 + b^ 2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + ( b*Cos[c + d*x]*(4*a^2 - b^2 + 3*a*b*Sin[c + d*x]))/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^2)))/(256*d) + ((-6*b^2*ArcTan[(b + a*Tan[(c + d*x)/2]) /Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(-(b*(2*a^2 + b^2)) + a *(2*a^2 - 5*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2)/(64*(a - b)^2*(a + b)^2*d) - ((-12*(640*a^8 - 1792*a^6*b^2 + 1680*a^4*b^4 - 560*a^2*b^6 + 35 *b^8)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (3840*a^9*(c + d*x) - 6912*a^7*b^2*(c + d*x) + 1728*a^5*b^4*(c + d*x) + 1920*a^3*b^6*(c + d*x) - 576*a*b^8*(c + d*x) + 3840*a^8*b*Cos[c + d*x] - 7 872*a^6*b^3*Cos[c + d*x] + 4256*a^4*b^5*Cos[c + d*x] - 172*a^2*b^7*Cos[c + d*x] - 70*b^9*Cos[c + d*x] - 1920*a^7*b^2*(c + d*x)*Cos[2*(c + d*x)] + 44 16*a^5*b^4*(c + d*x)*Cos[2*(c + d*x)] - 3072*a^3*b^6*(c + d*x)*Cos[2*(c + d*x)] + 576*a*b^8*(c + d*x)*Cos[2*(c + d*x)] - 320*a^6*b^3*Cos[3*(c + d*x) ] + 696*a^4*b^5*Cos[3*(c + d*x)] - 432*a^2*b^7*Cos[3*(c + d*x)] + 56*b^9*C os[3*(c + d*x)] + 8*a^4*b^5*Cos[5*(c + d*x)] - 16*a^2*b^7*Cos[5*(c + d*...
Time = 3.85 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.23, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {3042, 3375, 27, 3042, 3526, 27, 3042, 3526, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^6}{(a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {4 \sin ^4(c+d x) \left (-3 \left (42 a^4-44 b^2 a^2+5 b^4\right ) \sin ^2(c+d x)-3 a b \left (3 a^2-5 b^2\right ) \sin (c+d x)+5 \left (21 a^4-20 b^2 a^2+2 b^4\right )\right )}{(a+b \sin (c+d x))^3}dx}{240 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin ^4(c+d x) \left (-3 \left (42 a^4-44 b^2 a^2+5 b^4\right ) \sin ^2(c+d x)-3 a b \left (3 a^2-5 b^2\right ) \sin (c+d x)+5 \left (21 a^4-20 b^2 a^2+2 b^4\right )\right )}{(a+b \sin (c+d x))^3}dx}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^4 \left (-3 \left (42 a^4-44 b^2 a^2+5 b^4\right ) \sin (c+d x)^2-3 a b \left (3 a^2-5 b^2\right ) \sin (c+d x)+5 \left (21 a^4-20 b^2 a^2+2 b^4\right )\right )}{(a+b \sin (c+d x))^3}dx}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \sin ^3(c+d x) \left (-3 \left (105 a^6-209 b^2 a^4+114 b^4 a^2-10 b^6\right ) \sin ^2(c+d x)-a b \left (21 a^4-41 b^2 a^2+20 b^4\right ) \sin (c+d x)+4 \left (63 a^6-123 b^2 a^4+65 b^4 a^2-5 b^6\right )\right )}{(a+b \sin (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sin ^3(c+d x) \left (-3 \left (105 a^6-209 b^2 a^4+114 b^4 a^2-10 b^6\right ) \sin ^2(c+d x)-a b \left (21 a^4-41 b^2 a^2+20 b^4\right ) \sin (c+d x)+4 \left (63 a^6-123 b^2 a^4+65 b^4 a^2-5 b^6\right )\right )}{(a+b \sin (c+d x))^2}dx}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin (c+d x)^3 \left (-3 \left (105 a^6-209 b^2 a^4+114 b^4 a^2-10 b^6\right ) \sin (c+d x)^2-a b \left (21 a^4-41 b^2 a^2+20 b^4\right ) \sin (c+d x)+4 \left (63 a^6-123 b^2 a^4+65 b^4 a^2-5 b^6\right )\right )}{(a+b \sin (c+d x))^2}dx}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \sin ^2(c+d x) \left (-2 \left (210 a^4-187 b^2 a^2+15 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+5 \left (63 a^4-54 b^2 a^2+4 b^4\right ) \left (a^2-b^2\right )^2-a b \left (21 a^2-10 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\sin ^2(c+d x) \left (-2 \left (210 a^4-187 b^2 a^2+15 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+5 \left (63 a^4-54 b^2 a^2+4 b^4\right ) \left (a^2-b^2\right )^2-a b \left (21 a^2-10 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\sin (c+d x)^2 \left (-2 \left (210 a^4-187 b^2 a^2+15 b^4\right ) \sin (c+d x)^2 \left (a^2-b^2\right )^2+5 \left (63 a^4-54 b^2 a^2+4 b^4\right ) \left (a^2-b^2\right )^2-a b \left (21 a^2-10 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int -\frac {\sin (c+d x) \left (-15 a \left (84 a^4-79 b^2 a^2+8 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+4 a \left (210 a^4-187 b^2 a^2+15 b^4\right ) \left (a^2-b^2\right )^2-a^2 b \left (105 a^2-62 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-15 a \left (84 a^4-79 b^2 a^2+8 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+4 a \left (210 a^4-187 b^2 a^2+15 b^4\right ) \left (a^2-b^2\right )^2-a^2 b \left (105 a^2-62 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-15 a \left (84 a^4-79 b^2 a^2+8 b^4\right ) \sin (c+d x)^2 \left (a^2-b^2\right )^2+4 a \left (210 a^4-187 b^2 a^2+15 b^4\right ) \left (a^2-b^2\right )^2-a^2 b \left (105 a^2-62 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {-b \left (420 a^2-311 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) a^3-4 \left (a^2-b^2\right )^2 \left (630 a^4-645 b^2 a^2+91 b^4\right ) \sin ^2(c+d x) a^2+15 \left (a^2-b^2\right )^2 \left (84 a^4-79 b^2 a^2+8 b^4\right ) a^2}{a+b \sin (c+d x)}dx}{2 b}+\frac {15 a \left (84 a^4-79 a^2 b^2+8 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {-b \left (420 a^2-311 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) a^3-4 \left (a^2-b^2\right )^2 \left (630 a^4-645 b^2 a^2+91 b^4\right ) \sin ^2(c+d x) a^2+15 \left (a^2-b^2\right )^2 \left (84 a^4-79 b^2 a^2+8 b^4\right ) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {-b \left (420 a^2-311 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) a^3-4 \left (a^2-b^2\right )^2 \left (630 a^4-645 b^2 a^2+91 b^4\right ) \sin (c+d x)^2 a^2+15 \left (a^2-b^2\right )^2 \left (84 a^4-79 b^2 a^2+8 b^4\right ) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {\int \frac {15 \left (\left (a^2-b^2\right )^2 \left (168 a^4-200 b^2 a^2+45 b^4\right ) \sin (c+d x) a^3+b \left (a^2-b^2\right )^2 \left (84 a^4-79 b^2 a^2+8 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{b}+\frac {4 a^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \int \frac {\left (a^2-b^2\right )^2 \left (168 a^4-200 b^2 a^2+45 b^4\right ) \sin (c+d x) a^3+b \left (a^2-b^2\right )^2 \left (84 a^4-79 b^2 a^2+8 b^4\right ) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {4 a^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \int \frac {\left (a^2-b^2\right )^2 \left (168 a^4-200 b^2 a^2+45 b^4\right ) \sin (c+d x) a^3+b \left (a^2-b^2\right )^2 \left (84 a^4-79 b^2 a^2+8 b^4\right ) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {4 a^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^3 x \left (a^2-b^2\right )^2 \left (168 a^4-200 a^2 b^2+45 b^4\right )}{b}-\frac {4 a^2 \left (a^2-b^2\right )^3 \left (42 a^4-29 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {4 a^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^3 x \left (a^2-b^2\right )^2 \left (168 a^4-200 a^2 b^2+45 b^4\right )}{b}-\frac {4 a^2 \left (a^2-b^2\right )^3 \left (42 a^4-29 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {4 a^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^3 x \left (a^2-b^2\right )^2 \left (168 a^4-200 a^2 b^2+45 b^4\right )}{b}-\frac {8 a^2 \left (a^2-b^2\right )^3 \left (42 a^4-29 a^2 b^2+2 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {4 a^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {16 a^2 \left (a^2-b^2\right )^3 \left (42 a^4-29 a^2 b^2+2 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a^3 x \left (a^2-b^2\right )^2 \left (168 a^4-200 a^2 b^2+45 b^4\right )}{b}\right )}{b}+\frac {4 a^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}+\frac {\frac {\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {4 a^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{b d}+\frac {15 \left (\frac {a^3 x \left (a^2-b^2\right )^2 \left (168 a^4-200 a^2 b^2+45 b^4\right )}{b}-\frac {8 a^2 \left (a^2-b^2\right )^{5/2} \left (42 a^4-29 a^2 b^2+2 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d}\right )}{b}}{2 b}}{3 b}\right )}{b \left (a^2-b^2\right )}-\frac {5 \left (a^2-b^2\right ) \left (63 a^4-54 a^2 b^2+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (63 a^4-60 a^2 b^2+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
Input:
Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]
Output:
(Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(12*a^2*d*(a + b*Sin[c + d*x])^2) - (7*a*Cos[c + d*x] *Sin[c + d*x]^5)/(20*b^2*d*(a + b*Sin[c + d*x])^2) + (Cos[c + d*x]*Sin[c + d*x]^6)/(5*b*d*(a + b*Sin[c + d*x])^2) + (-(((63*a^4 - 60*a^2*b^2 + 5*b^4 )*Cos[c + d*x]*Sin[c + d*x]^4)/(b*d*(a + b*Sin[c + d*x])^2)) + ((-5*(a^2 - b^2)*(63*a^4 - 54*a^2*b^2 + 4*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(b*d*(a + b*Sin[c + d*x])) + (3*((2*(a^2 - b^2)^2*(210*a^4 - 187*a^2*b^2 + 15*b^4)* Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d) - (-1/2*((15*((a^3*(a^2 - b^2)^2*(168 *a^4 - 200*a^2*b^2 + 45*b^4)*x)/b - (8*a^2*(a^2 - b^2)^(5/2)*(42*a^4 - 29* a^2*b^2 + 2*b^4)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])]) /(b*d)))/b + (4*a^2*(a^2 - b^2)^2*(630*a^4 - 645*a^2*b^2 + 91*b^4)*Cos[c + d*x])/(b*d))/b + (15*a*(a^2 - b^2)^2*(84*a^4 - 79*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b*d))/(3*b)))/(b*(a^2 - b^2)))/(b*(a^2 - b^2)))/(6 0*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 = 4.77 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {-\frac {a \,b^{2} \left (11 a^{4}-13 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-\frac {3 b \left (4 a^{6}+3 a^{4} b^{2}-9 a^{2} b^{4}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-\frac {b^{2} a \left (37 a^{4}-47 a^{2} b^{2}+10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-6 a^{6} b +\frac {15 a^{4} b^{3}}{2}-\frac {3 a^{2} b^{5}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (42 a^{6}-71 a^{4} b^{2}+31 a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{8}}+\frac {\frac {2 \left (\left (5 a^{3} b^{2}-\frac {27}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (15 a^{4} b -18 a^{2} b^{3}+3 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (10 a^{3} b^{2}-\frac {15}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (60 a^{4} b -60 a^{2} b^{3}+6 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (90 a^{4} b -80 a^{2} b^{3}+\frac {28}{3} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-10 a^{3} b^{2}+\frac {15}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (60 a^{4} b -52 a^{2} b^{3}+\frac {14}{3} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-5 a^{3} b^{2}+\frac {27}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+15 a^{4} b -14 a^{2} b^{3}+\frac {23 b^{5}}{15}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {a \left (168 a^{4}-200 a^{2} b^{2}+45 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{8}}}{d}\) | \(554\) |
| default | \(\frac {-\frac {2 \left (\frac {-\frac {a \,b^{2} \left (11 a^{4}-13 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-\frac {3 b \left (4 a^{6}+3 a^{4} b^{2}-9 a^{2} b^{4}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-\frac {b^{2} a \left (37 a^{4}-47 a^{2} b^{2}+10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-6 a^{6} b +\frac {15 a^{4} b^{3}}{2}-\frac {3 a^{2} b^{5}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (42 a^{6}-71 a^{4} b^{2}+31 a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{8}}+\frac {\frac {2 \left (\left (5 a^{3} b^{2}-\frac {27}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (15 a^{4} b -18 a^{2} b^{3}+3 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (10 a^{3} b^{2}-\frac {15}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (60 a^{4} b -60 a^{2} b^{3}+6 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (90 a^{4} b -80 a^{2} b^{3}+\frac {28}{3} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-10 a^{3} b^{2}+\frac {15}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (60 a^{4} b -52 a^{2} b^{3}+\frac {14}{3} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-5 a^{3} b^{2}+\frac {27}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+15 a^{4} b -14 a^{2} b^{3}+\frac {23 b^{5}}{15}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {a \left (168 a^{4}-200 a^{2} b^{2}+45 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{8}}}{d}\) | \(554\) |
| risch | \(\frac {45 a x}{8 b^{4}}+\frac {3 a \sin \left (4 d x +4 c \right )}{32 b^{4} d}-\frac {i a \left (-14 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+19 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-5 i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+38 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-49 i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+11 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+26 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-21 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+4 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-13 a^{4} b^{2}+17 a^{2} b^{4}-4 b^{6}\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^{8}}+\frac {\cos \left (5 d x +5 c \right )}{80 b^{3} d}-\frac {5 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{6} d}+\frac {3 i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{4} d}-\frac {21 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{4}}{d \,b^{8}}+\frac {29 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{2}}{2 d \,b^{6}}+\frac {21 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{4}}{d \,b^{8}}-\frac {29 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{2}}{2 d \,b^{6}}+\frac {5 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{6} d}-\frac {3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{4} d}+\frac {21 a^{5} x}{b^{8}}-\frac {25 a^{3} x}{b^{6}}+\frac {15 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 b^{7} d}-\frac {27 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{4 b^{5} d}+\frac {15 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 b^{7} d}-\frac {27 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{4 b^{5} d}-\frac {{\mathrm e}^{-3 i \left (d x +c \right )} a^{2}}{4 b^{5} d}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}-\frac {{\mathrm e}^{3 i \left (d x +c \right )} a^{2}}{4 b^{5} d}+\frac {7 \,{\mathrm e}^{3 i \left (d x +c \right )}}{96 b^{3} d}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b^{3} d}+\frac {7 \,{\mathrm e}^{-3 i \left (d x +c \right )}}{96 b^{3} d}+\frac {11 \,{\mathrm e}^{i \left (d x +c \right )}}{16 b^{3} d}\) | \(870\) |
Input:
int(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-2/b^8*((-1/2*a*b^2*(11*a^4-13*a^2*b^2+2*b^4)*tan(1/2*d*x+1/2*c)^3-3/ 2*b*(4*a^6+3*a^4*b^2-9*a^2*b^4+2*b^6)*tan(1/2*d*x+1/2*c)^2-1/2*b^2*a*(37*a ^4-47*a^2*b^2+10*b^4)*tan(1/2*d*x+1/2*c)-6*a^6*b+15/2*a^4*b^3-3/2*a^2*b^5) /(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*(42*a^6-71*a^4*b^ 2+31*a^2*b^4-2*b^6)/(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^8*(((5*a^3*b^2-27/8*a*b^4)*tan(1/2*d*x+1/2*c)^9+(1 5*a^4*b-18*a^2*b^3+3*b^5)*tan(1/2*d*x+1/2*c)^8+(10*a^3*b^2-15/4*a*b^4)*tan (1/2*d*x+1/2*c)^7+(60*a^4*b-60*a^2*b^3+6*b^5)*tan(1/2*d*x+1/2*c)^6+(90*a^4 *b-80*a^2*b^3+28/3*b^5)*tan(1/2*d*x+1/2*c)^4+(-10*a^3*b^2+15/4*a*b^4)*tan( 1/2*d*x+1/2*c)^3+(60*a^4*b-52*a^2*b^3+14/3*b^5)*tan(1/2*d*x+1/2*c)^2+(-5*a ^3*b^2+27/8*a*b^4)*tan(1/2*d*x+1/2*c)+15*a^4*b-14*a^2*b^3+23/15*b^5)/(1+ta n(1/2*d*x+1/2*c)^2)^5+1/8*a*(168*a^4-200*a^2*b^2+45*b^4)*arctan(tan(1/2*d* x+1/2*c))))
Time = 0.16 (sec) , antiderivative size = 995, normalized size of antiderivative = 2.05 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="frica s")
Output:
[1/120*(24*b^7*cos(d*x + c)^7 - 4*(21*a^2*b^5 - 4*b^7)*cos(d*x + c)^5 + 15 *(168*a^5*b^2 - 200*a^3*b^4 + 45*a*b^6)*d*x*cos(d*x + c)^2 + 10*(84*a^4*b^ 3 - 79*a^2*b^5 + 8*b^7)*cos(d*x + c)^3 - 15*(168*a^7 - 32*a^5*b^2 - 155*a^ 3*b^4 + 45*a*b^6)*d*x - 30*(42*a^6 + 13*a^4*b^2 - 27*a^2*b^4 + 2*b^6 - (42 *a^4*b^2 - 29*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(42*a^5*b - 29*a^3*b^3 + 2*a*b^5)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos (d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^ 2 - b^2)) - 30*(84*a^6*b - 58*a^4*b^3 - 17*a^2*b^5 + 4*b^7)*cos(d*x + c) + (42*a*b^6*cos(d*x + c)^5 - 5*(42*a^3*b^4 - 29*a*b^6)*cos(d*x + c)^3 - 30* (168*a^6*b - 200*a^4*b^3 + 45*a^2*b^5)*d*x - 15*(252*a^5*b^2 - 279*a^3*b^4 + 53*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^10*d*cos(d*x + c)^2 - 2*a*b^9* d*sin(d*x + c) - (a^2*b^8 + b^10)*d), 1/120*(24*b^7*cos(d*x + c)^7 - 4*(21 *a^2*b^5 - 4*b^7)*cos(d*x + c)^5 + 15*(168*a^5*b^2 - 200*a^3*b^4 + 45*a*b^ 6)*d*x*cos(d*x + c)^2 + 10*(84*a^4*b^3 - 79*a^2*b^5 + 8*b^7)*cos(d*x + c)^ 3 - 15*(168*a^7 - 32*a^5*b^2 - 155*a^3*b^4 + 45*a*b^6)*d*x - 60*(42*a^6 + 13*a^4*b^2 - 27*a^2*b^4 + 2*b^6 - (42*a^4*b^2 - 29*a^2*b^4 + 2*b^6)*cos(d* x + c)^2 + 2*(42*a^5*b - 29*a^3*b^3 + 2*a*b^5)*sin(d*x + c))*sqrt(a^2 - b^ 2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 30*(84*a ^6*b - 58*a^4*b^3 - 17*a^2*b^5 + 4*b^7)*cos(d*x + c) + (42*a*b^6*cos(d*...
Timed out. \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**2/(a+b*sin(d*x+c))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(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 = 0.21 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.49 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/120*(15*(168*a^5 - 200*a^3*b^2 + 45*a*b^4)*(d*x + c)/b^8 - 120*(42*a^6 - 71*a^4*b^2 + 31*a^2*b^4 - 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^8) + 120*(11*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 13*a^3*b^3*tan(1/2*d*x + 1/2 *c)^3 + 2*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 12*a^6*tan(1/2*d*x + 1/2*c)^2 + 9 *a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 27*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 + 6*b^ 6*tan(1/2*d*x + 1/2*c)^2 + 37*a^5*b*tan(1/2*d*x + 1/2*c) - 47*a^3*b^3*tan( 1/2*d*x + 1/2*c) + 10*a*b^5*tan(1/2*d*x + 1/2*c) + 12*a^6 - 15*a^4*b^2 + 3 *a^2*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*b^7 ) + 2*(600*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 405*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 1800*a^4*tan(1/2*d*x + 1/2*c)^8 - 2160*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 + 360*b^4*tan(1/2*d*x + 1/2*c)^8 + 1200*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 450* a*b^3*tan(1/2*d*x + 1/2*c)^7 + 7200*a^4*tan(1/2*d*x + 1/2*c)^6 - 7200*a^2* b^2*tan(1/2*d*x + 1/2*c)^6 + 720*b^4*tan(1/2*d*x + 1/2*c)^6 + 10800*a^4*ta n(1/2*d*x + 1/2*c)^4 - 9600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 1120*b^4*tan( 1/2*d*x + 1/2*c)^4 - 1200*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 450*a*b^3*tan(1/2 *d*x + 1/2*c)^3 + 7200*a^4*tan(1/2*d*x + 1/2*c)^2 - 6240*a^2*b^2*tan(1/2*d *x + 1/2*c)^2 + 560*b^4*tan(1/2*d*x + 1/2*c)^2 - 600*a^3*b*tan(1/2*d*x + 1 /2*c) + 405*a*b^3*tan(1/2*d*x + 1/2*c) + 1800*a^4 - 1680*a^2*b^2 + 184*b^4 )/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*b^7))/d
Time = 31.18 (sec) , antiderivative size = 3700, normalized size of antiderivative = 7.63 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^6*sin(c + d*x)^2)/(a + b*sin(c + d*x))^3,x)
Output:
((630*a^6 + 91*a^2*b^4 - 645*a^4*b^2)/(15*b^7) + (tan(c/2 + (d*x)/2)^13*(8 *a*b^4 + 84*a^5 - 79*a^3*b^2))/(4*b^6) + (tan(c/2 + (d*x)/2)^11*(17*a*b^4 + 252*a^5 - 237*a^3*b^2))/b^6 + (8*tan(c/2 + (d*x)/2)^7*(91*a*b^4 + 630*a^ 5 - 645*a^3*b^2))/(3*b^6) + (9*tan(c/2 + (d*x)/2)^5*(112*a*b^4 + 700*a^5 - 733*a^3*b^2))/(4*b^6) + (tan(c/2 + (d*x)/2)^9*(448*a*b^4 + 3780*a^5 - 372 3*a^3*b^2))/(4*b^6) + (tan(c/2 + (d*x)/2)^3*(643*a*b^4 + 3780*a^5 - 3975*a ^3*b^2))/(5*b^6) + (tan(c/2 + (d*x)/2)^12*(42*a^6 + 6*b^6 - 48*a^2*b^4 + 1 3*a^4*b^2))/b^7 + (3*tan(c/2 + (d*x)/2)^10*(84*a^6 + 18*b^6 - 103*a^2*b^4 + 26*a^4*b^2))/b^7 + (2*tan(c/2 + (d*x)/2)^6*(1260*a^6 + 202*b^6 - 1187*a^ 2*b^4 + 54*a^4*b^2))/(3*b^7) + (tan(c/2 + (d*x)/2)^8*(1890*a^6 + 324*b^6 - 2149*a^2*b^4 + 417*a^4*b^2))/(3*b^7) + (tan(c/2 + (d*x)/2)^2*(3780*a^6 + 274*b^6 - 1223*a^2*b^4 - 2190*a^4*b^2))/(15*b^7) + (tan(c/2 + (d*x)/2)^4*( 9450*a^6 + 1010*b^6 - 6354*a^2*b^4 - 2115*a^4*b^2))/(15*b^7) + (tan(c/2 + (d*x)/2)*(1336*a*b^4 + 8820*a^5 - 9135*a^3*b^2))/(60*b^6))/(d*(tan(c/2 + ( d*x)/2)^2*(7*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^12*(7*a^2 + 4*b^2) + tan(c/ 2 + (d*x)/2)^4*(21*a^2 + 20*b^2) + tan(c/2 + (d*x)/2)^10*(21*a^2 + 20*b^2) + tan(c/2 + (d*x)/2)^6*(35*a^2 + 40*b^2) + tan(c/2 + (d*x)/2)^8*(35*a^2 + 40*b^2) + a^2*tan(c/2 + (d*x)/2)^14 + a^2 + 24*a*b*tan(c/2 + (d*x)/2)^3 + 60*a*b*tan(c/2 + (d*x)/2)^5 + 80*a*b*tan(c/2 + (d*x)/2)^7 + 60*a*b*tan(c/ 2 + (d*x)/2)^9 + 24*a*b*tan(c/2 + (d*x)/2)^11 + 4*a*b*tan(c/2 + (d*x)/2...
Time = 0.30 (sec) , antiderivative size = 989, normalized size of antiderivative = 2.04 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^2/(a+b*sin(d*x+c))^3,x)
Output:
( - 10080*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2 ))*sin(c + d*x)**2*a**4*b**2 + 6960*sqrt(a**2 - b**2)*atan((tan((c + d*x)/ 2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**2*b**4 - 480*sqrt(a**2 - b **2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*b**6 - 20160*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2) )*sin(c + d*x)*a**5*b + 13920*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a**3*b**3 - 960*sqrt(a**2 - b**2)*atan ((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a*b**5 - 10080*s qrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**6 + 6 960*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a** 4*b**2 - 480*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b **2))*a**2*b**4 + 48*cos(c + d*x)*sin(c + d*x)**6*b**7 - 84*cos(c + d*x)*s in(c + d*x)**5*a*b**6 + 168*cos(c + d*x)*sin(c + d*x)**4*a**2*b**5 - 176*c os(c + d*x)*sin(c + d*x)**4*b**7 - 420*cos(c + d*x)*sin(c + d*x)**3*a**3*b **4 + 458*cos(c + d*x)*sin(c + d*x)**3*a*b**6 + 1680*cos(c + d*x)*sin(c + d*x)**2*a**4*b**3 - 1916*cos(c + d*x)*sin(c + d*x)**2*a**2*b**5 + 368*cos( c + d*x)*sin(c + d*x)**2*b**7 + 7560*cos(c + d*x)*sin(c + d*x)*a**5*b**2 - 7950*cos(c + d*x)*sin(c + d*x)*a**3*b**4 + 1216*cos(c + d*x)*sin(c + d*x) *a*b**6 + 5040*cos(c + d*x)*a**6*b - 5160*cos(c + d*x)*a**4*b**3 + 728*cos (c + d*x)*a**2*b**5 + 5040*sin(c + d*x)**2*a**5*b**2*d*x + 3780*sin(c +...