Integrand size = 21, antiderivative size = 422 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {a \left (8 a^4+20 a^2 b^2+5 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{13/2} d}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))} \] Output:
1/8*a*(8*a^4+20*a^2*b^2+5*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^( 1/2))/(a^2-b^2)^(13/2)/d-1/7*cos(d*x+c)/b/d/(a+b*sin(d*x+c))^7+1/42*a*cos( d*x+c)/b/(a^2-b^2)/d/(a+b*sin(d*x+c))^6+1/210*(5*a^2+6*b^2)*cos(d*x+c)/b/( a^2-b^2)^2/d/(a+b*sin(d*x+c))^5+1/840*a*(20*a^2+79*b^2)*cos(d*x+c)/b/(a^2- b^2)^3/d/(a+b*sin(d*x+c))^4+1/840*(20*a^4+179*a^2*b^2+32*b^4)*cos(d*x+c)/b /(a^2-b^2)^4/d/(a+b*sin(d*x+c))^3+1/1680*a*(40*a^4+718*a^2*b^2+397*b^4)*co s(d*x+c)/b/(a^2-b^2)^5/d/(a+b*sin(d*x+c))^2+1/1680*(40*a^6+1518*a^4*b^2+17 79*a^2*b^4+128*b^6)*cos(d*x+c)/b/(a^2-b^2)^6/d/(a+b*sin(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(1896\) vs. \(2(422)=844\).
Time = 6.36 (sec) , antiderivative size = 1896, normalized size of antiderivative = 4.49 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]
Output:
Cos[c + d*x]^3/(3*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/ 7*(b*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)* (a + b*Sin[c + d*x])^7) - (-1/6*((3*a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^ (3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (-1/5*((2*a*(10*a - b)*b + b*(42*a^2 - 16*a*b + 19*b^2))*(1 - Sin[c + d* x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x]) ^5) - (-1/4*((a*b*(62*a^2 - 18*a*b + 19*b^2) + b*(210*a^3 - 142*a^2*b + 21 3*a*b^2 - 29*b^3))*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^4) - (105*(8*a^4 - 8*a^3*b + 12*a^2*b^2 - 4*a*b^3 + b^4)*(-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/ ((-a + b)*(a + b*Sin[c + d*x])^3) - (-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin [c + d*x])^(3/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt [a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/(( -a - b)^(3/2)*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x] ])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b)))/(3*(-a + b))))/(4*(-a + b)*(a + b)))/(5*(-a + b)*(a + b)))/(6*(-a + b)*(a + b)))/(7*(-a + b)*(a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]]) + (4*b*(Co s[c + d*x]^5/(5*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/7* (b*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-1/6*((a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^(...
Time = 2.08 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.17, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.095, Rules used = {3042, 3172, 3042, 3233, 3042, 3233, 25, 3042, 3233, 27, 3042, 3233, 25, 3042, 3233, 25, 3042, 3233, 27, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{(a+b \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {\int \frac {\sin (c+d x)}{(a+b \sin (c+d x))^7}dx}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin (c+d x)}{(a+b \sin (c+d x))^7}dx}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {-\frac {\int \frac {6 b-5 a \sin (c+d x)}{(a+b \sin (c+d x))^6}dx}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {6 b-5 a \sin (c+d x)}{(a+b \sin (c+d x))^6}dx}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {-\frac {\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}-\frac {\int -\frac {55 a b-4 \left (5 a^2+6 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5}dx}{5 \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {55 a b-4 \left (5 a^2+6 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5}dx}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {55 a b-4 \left (5 a^2+6 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5}dx}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}-\frac {\int -\frac {3 \left (4 b \left (25 a^2+8 b^2\right )-a \left (20 a^2+79 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \int \frac {4 b \left (25 a^2+8 b^2\right )-a \left (20 a^2+79 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \int \frac {4 b \left (25 a^2+8 b^2\right )-a \left (20 a^2+79 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}-\frac {\int -\frac {9 a b \left (40 a^2+37 b^2\right )-2 \left (20 a^4+179 b^2 a^2+32 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\int \frac {9 a b \left (40 a^2+37 b^2\right )-2 \left (20 a^4+179 b^2 a^2+32 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\int \frac {9 a b \left (40 a^2+37 b^2\right )-2 \left (20 a^4+179 b^2 a^2+32 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {\int -\frac {2 b \left (400 a^4+691 b^2 a^2+64 b^4\right )-a \left (40 a^4+718 b^2 a^2+397 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\frac {\int \frac {2 b \left (400 a^4+691 b^2 a^2+64 b^4\right )-a \left (40 a^4+718 b^2 a^2+397 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\frac {\int \frac {2 b \left (400 a^4+691 b^2 a^2+64 b^4\right )-a \left (40 a^4+718 b^2 a^2+397 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\frac {\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\int -\frac {105 a b \left (8 a^4+20 b^2 a^2+5 b^4\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}}{2 \left (a^2-b^2\right )}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\frac {\frac {105 a b \left (8 a^4+20 a^2 b^2+5 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\frac {\frac {105 a b \left (8 a^4+20 a^2 b^2+5 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\frac {\frac {210 a b \left (8 a^4+20 a^2 b^2+5 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{d \left (a^2-b^2\right )}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 \left (\frac {\frac {\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {420 a b \left (8 a^4+20 a^2 b^2+5 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{5 \left (a^2-b^2\right )}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {a \cos (c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}-\frac {\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}+\frac {\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}+\frac {3 \left (\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}+\frac {\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {\frac {210 a b \left (8 a^4+20 a^2 b^2+5 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{3/2}}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}\right )}{4 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}}{7 b}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
Input:
Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]
Output:
-1/7*Cos[c + d*x]/(b*d*(a + b*Sin[c + d*x])^7) - (-1/6*(a*Cos[c + d*x])/(( a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) - (((5*a^2 + 6*b^2)*Cos[c + d*x])/(5* (a^2 - b^2)*d*(a + b*Sin[c + d*x])^5) + ((a*(20*a^2 + 79*b^2)*Cos[c + d*x] )/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^4) + (3*(((20*a^4 + 179*a^2*b^2 + 32*b^4)*Cos[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^3) + ((a*(40*a ^4 + 718*a^2*b^2 + 397*b^4)*Cos[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((210*a*b*(8*a^4 + 20*a^2*b^2 + 5*b^4)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + ((40*a^6 + 1518* a^4*b^2 + 1779*a^2*b^4 + 128*b^6)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[ c + d*x])))/(2*(a^2 - b^2)))/(3*(a^2 - b^2))))/(4*(a^2 - b^2)))/(5*(a^2 - b^2)))/(6*(a^2 - b^2)))/(7*b)
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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + 1))), x] + Simp[g^2*((p - 1)/(b*(m + 1))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; Fre eQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && I ntegersQ[2*m, 2*p]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Result contains complex when optimal does not.
Time = 8.81 (sec) , antiderivative size = 1703, normalized size of antiderivative = 4.04
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1703\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1872\) |
| default | \(\text {Expression too large to display}\) | \(1872\) |
Input:
int(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
-1/840*I*(-128*I*b^13-840*b^8*a^5*exp(13*I*(d*x+c))-2100*b^10*a^3*exp(13*I *(d*x+c))-525*b^12*a*exp(13*I*(d*x+c))-2688*I*b^13*exp(4*I*(d*x+c))-40*I*a ^6*b^7-1779*I*a^2*b^11+59920*b^6*a^7*exp(11*I*(d*x+c))+155400*b^8*a^5*exp( 11*I*(d*x+c))+51450*b^10*a^3*exp(11*I*(d*x+c))+3500*b^12*a*exp(11*I*(d*x+c ))-899388*b^8*a^5*exp(9*I*(d*x+c))-212310*b^10*a^3*exp(9*I*(d*x+c))-9905*b ^12*a*exp(9*I*(d*x+c))+124032*b^2*a^11*exp(7*I*(d*x+c))-26880*a^11*b^2*exp (5*I*(d*x+c))-756448*a^9*b^4*exp(5*I*(d*x+c))-1856624*a^7*b^6*exp(5*I*(d*x +c))-1497888*a^5*b^8*exp(5*I*(d*x+c))-304640*a^3*b^10*exp(5*I*(d*x+c))-896 0*I*b*a^12*exp(8*I*(d*x+c))-245952*I*b^3*a^10*exp(8*I*(d*x+c))-1522416*I*b ^5*a^8*exp(8*I*(d*x+c))-890190*I*b^9*a^4*exp(8*I*(d*x+c))-1518*I*a^4*b^9+1 16872*I*a^6*b^7*exp(2*I*(d*x+c))-22400*I*a^10*b^3*exp(4*I*(d*x+c))-585088* I*a^4*b^9*exp(4*I*(d*x+c))-1747592*I*a^6*b^7*exp(8*I*(d*x+c))+2560*a^13*ex p(7*I*(d*x+c))-308448*b^4*a^9*exp(9*I*(d*x+c))-1047424*b^6*a^7*exp(9*I*(d* x+c))-10920*I*b^7*a^6*exp(12*I*(d*x+c))-27300*I*b^9*a^4*exp(12*I*(d*x+c))- 6825*I*b^11*a^2*exp(12*I*(d*x+c))+8960*I*b*a^12*exp(6*I*(d*x+c))+178640*I* b^5*a^8*exp(10*I*(d*x+c))+265650*I*b^9*a^4*exp(10*I*(d*x+c))+38500*I*b^11* a^2*exp(10*I*(d*x+c))-8960*I*b^13*exp(8*I*(d*x+c))-4480*I*b^13*exp(6*I*(d* x+c))+896*I*b^13*exp(2*I*(d*x+c))+132762*I*a^4*b^9*exp(2*I*(d*x+c))+16380* I*a^2*b^11*exp(2*I*(d*x+c))+501312*I*a^10*b^3*exp(6*I*(d*x+c))+1858416*I*a ^8*b^5*exp(6*I*(d*x+c))+2574992*I*a^6*b^7*exp(6*I*(d*x+c))+872340*I*a^4...
Leaf count of result is larger than twice the leaf count of optimal. 1444 vs. \(2 (401) = 802\).
Time = 0.33 (sec) , antiderivative size = 2972, normalized size of antiderivative = 7.04 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="fricas")
Output:
[1/3360*(2*(40*a^8*b^5 + 1478*a^6*b^7 + 261*a^4*b^9 - 1651*a^2*b^11 - 128* b^13)*cos(d*x + c)^7 - 28*(60*a^10*b^3 + 1837*a^8*b^5 + 176*a^6*b^7 - 1680 *a^4*b^9 - 361*a^2*b^11 - 32*b^13)*cos(d*x + c)^5 + 70*(40*a^12*b + 900*a^ 10*b^3 + 1111*a^8*b^5 - 501*a^6*b^7 - 1395*a^4*b^9 - 139*a^2*b^11 - 16*b^1 3)*cos(d*x + c)^3 + 105*(8*a^12 + 188*a^10*b^2 + 705*a^8*b^4 + 861*a^6*b^6 + 315*a^4*b^8 + 35*a^2*b^10 - 7*(8*a^6*b^6 + 20*a^4*b^8 + 5*a^2*b^10)*cos (d*x + c)^6 + 7*(40*a^8*b^4 + 124*a^6*b^6 + 85*a^4*b^8 + 15*a^2*b^10)*cos( d*x + c)^4 - 7*(24*a^10*b^2 + 140*a^8*b^4 + 239*a^6*b^6 + 110*a^4*b^8 + 15 *a^2*b^10)*cos(d*x + c)^2 + (56*a^11*b + 420*a^9*b^3 + 903*a^7*b^5 + 603*a ^5*b^7 + 125*a^3*b^9 + 5*a*b^11 - (8*a^5*b^7 + 20*a^3*b^9 + 5*a*b^11)*cos( d*x + c)^6 + 3*(56*a^7*b^5 + 148*a^5*b^7 + 55*a^3*b^9 + 5*a*b^11)*cos(d*x + c)^4 - (280*a^9*b^3 + 1036*a^7*b^5 + 1039*a^5*b^7 + 270*a^3*b^9 + 15*a*b ^11)*cos(d*x + c)^2)*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)) - 420*(24*a^12*b + 116*a^10*b^3 + 99*a^8*b^5 - 129*a^ 6*b^7 - 95*a^4*b^9 - 15*a^2*b^11)*cos(d*x + c) - 14*((40*a^9*b^4 + 1358*a^ 7*b^6 + 81*a^5*b^8 - 1426*a^3*b^10 - 53*a*b^12)*cos(d*x + c)^5 - 10*(20*a^ 11*b^2 + 535*a^9*b^4 + 147*a^7*b^6 - 407*a^5*b^8 - 283*a^3*b^10 - 12*a*b^1 2)*cos(d*x + c)^3 + 15*(8*a^13 + 132*a^11*b^2 + 285*a^9*b^4 - 42*a^7*b^...
Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2/(a+b*sin(d*x+c))**8,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="maxima")
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
Leaf count of result is larger than twice the leaf count of optimal. 2207 vs. \(2 (401) = 802\).
Time = 0.31 (sec) , antiderivative size = 2207, normalized size of antiderivative = 5.23 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="giac")
Output:
1/840*(105*(8*a^5 + 20*a^3*b^2 + 5*a*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2 )*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b^6 + 15*a^4*b^8 - 6*a^2*b^10 + b^12)*sqr t(a^2 - b^2)) - (840*a^18*tan(1/2*d*x + 1/2*c)^13 - 12180*a^16*b^2*tan(1/2 *d*x + 1/2*c)^13 + 24675*a^14*b^4*tan(1/2*d*x + 1/2*c)^13 - 33600*a^12*b^6 *tan(1/2*d*x + 1/2*c)^13 + 25200*a^10*b^8*tan(1/2*d*x + 1/2*c)^13 - 10080* a^8*b^10*tan(1/2*d*x + 1/2*c)^13 + 1680*a^6*b^12*tan(1/2*d*x + 1/2*c)^13 - 840*a^17*b*tan(1/2*d*x + 1/2*c)^12 - 87780*a^15*b^3*tan(1/2*d*x + 1/2*c)^ 12 + 144375*a^13*b^5*tan(1/2*d*x + 1/2*c)^12 - 201600*a^11*b^7*tan(1/2*d*x + 1/2*c)^12 + 151200*a^9*b^9*tan(1/2*d*x + 1/2*c)^12 - 60480*a^7*b^11*tan (1/2*d*x + 1/2*c)^12 + 10080*a^5*b^13*tan(1/2*d*x + 1/2*c)^12 + 3360*a^18* tan(1/2*d*x + 1/2*c)^11 - 94080*a^16*b^2*tan(1/2*d*x + 1/2*c)^11 - 220500* a^14*b^4*tan(1/2*d*x + 1/2*c)^11 + 287350*a^12*b^6*tan(1/2*d*x + 1/2*c)^11 - 537600*a^10*b^8*tan(1/2*d*x + 1/2*c)^11 + 450240*a^8*b^10*tan(1/2*d*x + 1/2*c)^11 - 192640*a^6*b^12*tan(1/2*d*x + 1/2*c)^11 + 33600*a^4*b^14*tan( 1/2*d*x + 1/2*c)^11 - 13440*a^17*b*tan(1/2*d*x + 1/2*c)^10 - 554400*a^15*b ^3*tan(1/2*d*x + 1/2*c)^10 - 165900*a^13*b^5*tan(1/2*d*x + 1/2*c)^10 - 668 50*a^11*b^7*tan(1/2*d*x + 1/2*c)^10 - 621600*a^9*b^9*tan(1/2*d*x + 1/2*c)^ 10 + 719040*a^7*b^11*tan(1/2*d*x + 1/2*c)^10 - 355040*a^5*b^13*tan(1/2*d*x + 1/2*c)^10 + 67200*a^3*b^15*tan(1/2*d*x + 1/2*c)^10 + 4200*a^18*tan(1...
Time = 20.00 (sec) , antiderivative size = 2440, normalized size of antiderivative = 5.78 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:
int(cos(c + d*x)^2/(a + b*sin(c + d*x))^8,x)
Output:
((3640*a^10*b - 240*b^11 + 1448*a^2*b^9 - 3646*a^4*b^7 + 4923*a^6*b^5 - 26 60*a^8*b^3)/(840*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15* a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^6*(2800*a^16*b - 1920*b^17 + 4384*a^2*b^15 + 10672*a^4*b^13 - 48276*a^6*b^11 + 74800*a^8*b^9 + 29395*a^ 10*b^7 + 83100*a^12*b^5 + 57400*a^14*b^3))/(30*a^6*(a^12 + b^12 - 6*a^2*b^ 10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x )/2)^8*(7000*a^16*b - 7680*b^17 + 17536*a^2*b^15 + 42688*a^4*b^13 - 194304 *a^6*b^11 + 281800*a^8*b^9 + 49510*a^10*b^7 + 246615*a^12*b^5 + 193900*a^1 4*b^3))/(120*a^6*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15* a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^10*(192*a^14*b - 960*b^15 + 5 072*a^2*b^13 - 10272*a^4*b^11 + 8880*a^6*b^9 + 955*a^8*b^7 + 2370*a^10*b^5 + 7920*a^12*b^3))/(12*a^4*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6 *b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^4*(9000*a^14*b - 96 00*b^15 + 50720*a^2*b^13 - 102800*a^4*b^11 + 90624*a^6*b^9 + 62092*a^8*b^7 + 122669*a^10*b^5 + 131220*a^12*b^3))/(120*a^4*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2 )^12*(8*a^12*b - 96*b^13 + 576*a^2*b^11 - 1440*a^4*b^9 + 1920*a^6*b^7 - 13 75*a^8*b^5 + 836*a^10*b^3))/(8*a^2*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^2*(1760*a^1 2*b - 720*b^13 + 4248*a^2*b^11 - 10352*a^4*b^9 + 13315*a^6*b^7 - 3186*a...
Time = 5.70 (sec) , antiderivative size = 3876, normalized size of antiderivative = 9.18 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x)
Output:
(1680*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*s in(c + d*x)**7*a**10*b**8 + 4200*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)* a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**7*a**8*b**10 + 1050*sqrt(a**2 - b* *2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**7*a**6* b**12 + 11760*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**6*a**11*b**7 + 29400*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**6*a**9*b**9 + 7350*sqrt(a **2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)* *6*a**7*b**11 + 35280*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt (a**2 - b**2))*sin(c + d*x)**5*a**12*b**6 + 88200*sqrt(a**2 - b**2)*atan(( tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*a**10*b**8 + 22 050*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin (c + d*x)**5*a**8*b**10 + 58800*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a**13*b**5 + 147000*sqrt(a**2 - b **2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a**1 1*b**7 + 36750*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a**9*b**9 + 58800*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**3*a**14*b**4 + 147000*sqr t(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d* x)**3*a**12*b**6 + 36750*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b...