Integrand size = 21, antiderivative size = 491 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {x}{b^8}-\frac {a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{8 b^8 \left (a^2-b^2\right )^{7/2} d}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {\cos ^3(c+d x) \left (8 \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{24 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{16 b^7 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \]
x/b^8-1/8*a*(16*a^6-56*a^4*b^2+70*a^2*b^4-35*b^6)*arctan((b+a*tan(1/2*d*x+ 1/2*c))/(a^2-b^2)^(1/2))/b^8/(a^2-b^2)^(7/2)/d-1/7*cos(d*x+c)^7/b/d/(a+b*s in(d*x+c))^7+1/6*a*cos(d*x+c)^7/b/(a^2-b^2)/d/(a+b*sin(d*x+c))^6-1/24*a*(6 *a^2-11*b^2)*cos(d*x+c)^5/b^3/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^4+1/16*a*(8*a ^4-22*a^2*b^2+19*b^4)*cos(d*x+c)^3/b^5/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^2+1/ 30*cos(d*x+c)^5*(6*a^2-6*b^2+5*a*b*sin(d*x+c))/b^3/(a^2-b^2)/d/(a+b*sin(d* x+c))^5-1/24*cos(d*x+c)^3*(8*(a^2-b^2)^2+a*b*(6*a^2-11*b^2)*sin(d*x+c))/b^ 5/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^3+1/16*cos(d*x+c)*(16*(a^2-b^2)^3+a*b*(8* a^4-22*a^2*b^2+19*b^4)*sin(d*x+c))/b^7/(a^2-b^2)^3/d/(a+b*sin(d*x+c))
Time = 17.17 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {c+d x}{b^8 d}-\frac {a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{8 b^8 \left (a^2-b^2\right )^{7/2} d}+\frac {a^6 \cos (c+d x)-3 a^4 b^2 \cos (c+d x)+3 a^2 b^4 \cos (c+d x)-b^6 \cos (c+d x)}{7 b^7 d (a+b \sin (c+d x))^7}-\frac {43 \left (a^5 \cos (c+d x)-2 a^3 b^2 \cos (c+d x)+a b^4 \cos (c+d x)\right )}{42 b^7 d (a+b \sin (c+d x))^6}+\frac {667 a^4 \cos (c+d x)-799 a^2 b^2 \cos (c+d x)+132 b^4 \cos (c+d x)}{210 b^7 d (a+b \sin (c+d x))^5}+\frac {-4682 a^3 \cos (c+d x)+2777 a b^2 \cos (c+d x)}{840 b^7 d (a+b \sin (c+d x))^4}+\frac {-5118 a^4 \cos (c+d x)+6059 a^2 b^2 \cos (c+d x)-976 b^4 \cos (c+d x)}{840 b^7 \left (-a^2+b^2\right ) d (a+b \sin (c+d x))^3}+\frac {-7404 a^5 \cos (c+d x)+14528 a^3 b^2 \cos (c+d x)-6949 a b^4 \cos (c+d x)}{1680 b^7 \left (-a^2+b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {-4356 a^6 \cos (c+d x)+12508 a^4 b^2 \cos (c+d x)-11493 a^2 b^4 \cos (c+d x)+2816 b^6 \cos (c+d x)}{1680 b^7 \left (-a^2+b^2\right )^3 d (a+b \sin (c+d x))} \]
(c + d*x)/(b^8*d) - (a*(16*a^6 - 56*a^4*b^2 + 70*a^2*b^4 - 35*b^6)*ArcTan[ (Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^ 2]])/(8*b^8*(a^2 - b^2)^(7/2)*d) + (a^6*Cos[c + d*x] - 3*a^4*b^2*Cos[c + d *x] + 3*a^2*b^4*Cos[c + d*x] - b^6*Cos[c + d*x])/(7*b^7*d*(a + b*Sin[c + d *x])^7) - (43*(a^5*Cos[c + d*x] - 2*a^3*b^2*Cos[c + d*x] + a*b^4*Cos[c + d *x]))/(42*b^7*d*(a + b*Sin[c + d*x])^6) + (667*a^4*Cos[c + d*x] - 799*a^2* b^2*Cos[c + d*x] + 132*b^4*Cos[c + d*x])/(210*b^7*d*(a + b*Sin[c + d*x])^5 ) + (-4682*a^3*Cos[c + d*x] + 2777*a*b^2*Cos[c + d*x])/(840*b^7*d*(a + b*S in[c + d*x])^4) + (-5118*a^4*Cos[c + d*x] + 6059*a^2*b^2*Cos[c + d*x] - 97 6*b^4*Cos[c + d*x])/(840*b^7*(-a^2 + b^2)*d*(a + b*Sin[c + d*x])^3) + (-74 04*a^5*Cos[c + d*x] + 14528*a^3*b^2*Cos[c + d*x] - 6949*a*b^4*Cos[c + d*x] )/(1680*b^7*(-a^2 + b^2)^2*d*(a + b*Sin[c + d*x])^2) + (-4356*a^6*Cos[c + d*x] + 12508*a^4*b^2*Cos[c + d*x] - 11493*a^2*b^4*Cos[c + d*x] + 2816*b^6* Cos[c + d*x])/(1680*b^7*(-a^2 + b^2)^3*d*(a + b*Sin[c + d*x]))
Time = 2.71 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.08, 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, 3343, 3042, 3342, 25, 3042, 3343, 3042, 3342, 27, 3042, 3343, 3042, 3342, 25, 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.
| \(\displaystyle \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{(a+b \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {\int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^7}dx}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x)^6 \sin (c+d x)}{(a+b \sin (c+d x))^7}dx}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos ^6(c+d x) (6 b+a \sin (c+d x))}{(a+b \sin (c+d x))^6}dx}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos (c+d x)^6 (6 b+a \sin (c+d x))}{(a+b \sin (c+d x))^6}dx}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {-\frac {\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}-\frac {\int -\frac {\cos ^4(c+d x) \left (5 a b+6 \left (a^2-b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^5}dx}{b^2}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {\cos ^4(c+d x) \left (5 a b+6 \left (a^2-b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^5}dx}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {\cos (c+d x)^4 \left (5 a b+6 \left (a^2-b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^5}dx}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\int \frac {\cos ^4(c+d x) \left (4 b \left (a^2-6 b^2\right )+a \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\int \frac {\cos (c+d x)^4 \left (4 b \left (a^2-6 b^2\right )+a \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}-\frac {\int -\frac {3 \cos ^2(c+d x) \left (8 \sin (c+d x) \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right )\right )}{(a+b \sin (c+d x))^3}dx}{b^2}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \int \frac {\cos ^2(c+d x) \left (8 \sin (c+d x) \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right )\right )}{(a+b \sin (c+d x))^3}dx}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \int \frac {\cos (c+d x)^2 \left (8 \sin (c+d x) \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right )\right )}{(a+b \sin (c+d x))^3}dx}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\int \frac {\cos ^2(c+d x) \left (2 b \left (2 a^4-5 b^2 a^2+8 b^4\right )+a \left (8 a^4-22 b^2 a^2+19 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\int \frac {\cos (c+d x)^2 \left (2 b \left (2 a^4-5 b^2 a^2+8 b^4\right )+a \left (8 a^4-22 b^2 a^2+19 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{b^2 d (a+b \sin (c+d x))}-\frac {\int -\frac {16 \sin (c+d x) \left (a^2-b^2\right )^3+a b \left (8 a^4-22 b^2 a^2+19 b^4\right )}{a+b \sin (c+d x)}dx}{b^2}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\frac {\int \frac {16 \sin (c+d x) \left (a^2-b^2\right )^3+a b \left (8 a^4-22 b^2 a^2+19 b^4\right )}{a+b \sin (c+d x)}dx}{b^2}+\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{b^2 d (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\frac {\int \frac {16 \sin (c+d x) \left (a^2-b^2\right )^3+a b \left (8 a^4-22 b^2 a^2+19 b^4\right )}{a+b \sin (c+d x)}dx}{b^2}+\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{b^2 d (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\frac {\frac {16 x \left (a^2-b^2\right )^3}{b}-\frac {a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{b^2}+\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{b^2 d (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\frac {\frac {16 x \left (a^2-b^2\right )^3}{b}-\frac {a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{b^2}+\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{b^2 d (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\frac {\frac {16 x \left (a^2-b^2\right )^3}{b}-\frac {2 a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{b^2}+\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{b^2 d (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {3 \left (-\frac {\frac {\frac {4 a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {16 x \left (a^2-b^2\right )^3}{b}}{b^2}+\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{b^2 d (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{b^2}+\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{b^2}+\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}}{6 \left (a^2-b^2\right )}-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {a \cos ^7(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}-\frac {\frac {\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{5 b^2 d (a+b \sin (c+d x))^5}+\frac {-\frac {a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}-\frac {\frac {\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{b^2 d (a+b \sin (c+d x))^3}+\frac {3 \left (-\frac {a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {\frac {\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{b^2 d (a+b \sin (c+d x))}+\frac {\frac {16 x \left (a^2-b^2\right )^3}{b}-\frac {2 a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d \sqrt {a^2-b^2}}}{b^2}}{2 \left (a^2-b^2\right )}\right )}{b^2}}{4 \left (a^2-b^2\right )}}{b^2}}{6 \left (a^2-b^2\right )}}{b}-\frac {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
-1/7*Cos[c + d*x]^7/(b*d*(a + b*Sin[c + d*x])^7) - (-1/6*(a*Cos[c + d*x]^7 )/((a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) - ((Cos[c + d*x]^5*(6*(a^2 - b^2) + 5*a*b*Sin[c + d*x]))/(5*b^2*d*(a + b*Sin[c + d*x])^5) + (-1/4*(a*(6*a^2 - 11*b^2)*Cos[c + d*x]^5)/((a^2 - b^2)*d*(a + b*Sin[c + d*x])^4) - ((Cos[ c + d*x]^3*(8*(a^2 - b^2)^2 + a*b*(6*a^2 - 11*b^2)*Sin[c + d*x]))/(b^2*d*( a + b*Sin[c + d*x])^3) + (3*(-1/2*(a*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Cos[c + d*x]^3)/((a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (((16*(a^2 - b^2)^3*x)/b - (2*a*(16*a^6 - 56*a^4*b^2 + 70*a^2*b^4 - 35*b^6)*ArcTan[(2*b + 2*a*Tan[ (c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d))/b^2 + (Cos[c + d*x]*(16*(a^2 - b^2)^3 + a*b*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Sin[c + d*x]))/ (b^2*d*(a + b*Sin[c + d*x])))/(2*(a^2 - b^2))))/b^2)/(4*(a^2 - b^2)))/b^2) /(6*(a^2 - b^2)))/b
3.5.67.3.1 Defintions of rubi rules used
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_)])/((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_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g*C os[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d*p + b*d*(m + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Simp[g^2*(( p - 1)/(b^2*(m + 1)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin [e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x ], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && Lt Q[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p *(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ [a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(1651\) vs. \(2(469)=938\).
Time = 14.47 (sec) , antiderivative size = 1652, normalized size of antiderivative = 3.36
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1652\) |
| default | \(\text {Expression too large to display}\) | \(1652\) |
| risch | \(\text {Expression too large to display}\) | \(1998\) |
1/d*(2/b^8*arctan(tan(1/2*d*x+1/2*c))-2/b^8*((-1/16*b^2*(8*a^12-22*a^10*b^ 2+19*a^8*b^4-16*a^6*b^6+48*a^4*b^8-48*a^2*b^10+16*b^12)/a/(a^6-3*a^4*b^2+3 *a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^13-1/16*b*(16*a^14+56*a^12*b^2-238*a^10*b ^4+231*a^8*b^6-96*a^6*b^8+288*a^4*b^10-288*a^2*b^12+96*b^14)/(a^6-3*a^4*b^ 2+3*a^2*b^4-b^6)/a^2*tan(1/2*d*x+1/2*c)^12-1/24/a^3*b^2*(384*a^14-276*a^12 *b^2-1252*a^10*b^4+1697*a^8*b^6-384*a^6*b^8+1344*a^4*b^10-1408*a^2*b^12+48 0*b^14)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^11-1/24/a^4*b*(14 4*a^16+2064*a^14*b^2-4284*a^12*b^4-224*a^10*b^6+3949*a^8*b^8+528*a^6*b^10+ 1392*a^4*b^12-2384*a^2*b^14+960*b^16)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/ 2*d*x+1/2*c)^10-1/240/a^5*b^2*(17400*a^16+34110*a^14*b^2-152515*a^12*b^4+9 3770*a^10*b^6+49308*a^8*b^8+30576*a^6*b^10-12496*a^4*b^12-18048*a^2*b^14+1 1520*b^16)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^9-1/240/a^6*b* (3600*a^18+73480*a^16*b^2-70050*a^14*b^4-198555*a^12*b^6+252090*a^10*b^8+2 8792*a^8*b^10+43584*a^6*b^12-48064*a^4*b^14+3968*a^2*b^16+7680*b^18)/(a^6- 3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^8-1/420/a^7*b^2*(58800*a^18+18 6200*a^16*b^2-565950*a^14*b^4+95655*a^12*b^6+444108*a^10*b^8+51212*a^8*b^1 0-11904*a^6*b^12-64640*a^4*b^14+27904*a^2*b^16+3840*b^18)/(a^6-3*a^4*b^2+3 *a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^7-1/60/a^6*b*(1200*a^18+26080*a^16*b^2-19 080*a^14*b^4-80730*a^12*b^6+87285*a^10*b^8+21208*a^8*b^10+5316*a^6*b^12-12 016*a^4*b^14+992*a^2*b^16+1920*b^18)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(...
Leaf count of result is larger than twice the leaf count of optimal. 1818 vs. \(2 (469) = 938\).
Time = 0.73 (sec) , antiderivative size = 3721, normalized size of antiderivative = 7.58 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
[1/3360*(23520*(a^9*b^6 - 4*a^7*b^8 + 6*a^5*b^10 - 4*a^3*b^12 + a*b^14)*d* x*cos(d*x + c)^6 + 2*(4356*a^8*b^7 - 16864*a^6*b^9 + 24001*a^4*b^11 - 1430 9*a^2*b^13 + 2816*b^15)*cos(d*x + c)^7 - 23520*(5*a^11*b^4 - 17*a^9*b^6 + 18*a^7*b^8 - 2*a^5*b^10 - 7*a^3*b^12 + 3*a*b^14)*d*x*cos(d*x + c)^4 - 28*( 2754*a^10*b^5 - 9717*a^8*b^7 + 11528*a^6*b^9 - 3782*a^4*b^11 - 1247*a^2*b^ 13 + 464*b^15)*cos(d*x + c)^5 + 23520*(3*a^13*b^2 - 2*a^11*b^4 - 19*a^9*b^ 6 + 36*a^7*b^8 - 19*a^5*b^10 - 2*a^3*b^12 + 3*a*b^14)*d*x*cos(d*x + c)^2 + 70*(856*a^12*b^3 - 1090*a^10*b^5 - 3477*a^8*b^7 + 7907*a^6*b^9 - 4423*a^4 *b^11 + 67*a^2*b^13 + 160*b^15)*cos(d*x + c)^3 - 3360*(a^15 + 17*a^13*b^2 - 43*a^11*b^4 - 11*a^9*b^6 + 99*a^7*b^8 - 77*a^5*b^10 + 7*a^3*b^12 + 7*a*b ^14)*d*x + 105*(16*a^14 + 280*a^12*b^2 - 546*a^10*b^4 - 413*a^8*b^6 + 1323 *a^6*b^8 - 735*a^4*b^10 - 245*a^2*b^12 - 7*(16*a^8*b^6 - 56*a^6*b^8 + 70*a ^4*b^10 - 35*a^2*b^12)*cos(d*x + c)^6 + 7*(80*a^10*b^4 - 232*a^8*b^6 + 182 *a^6*b^8 + 35*a^4*b^10 - 105*a^2*b^12)*cos(d*x + c)^4 - 7*(48*a^12*b^2 - 8 *a^10*b^4 - 302*a^8*b^6 + 427*a^6*b^8 - 140*a^4*b^10 - 105*a^2*b^12)*cos(d *x + c)^2 + (112*a^13*b + 168*a^11*b^3 - 1134*a^9*b^5 + 1045*a^7*b^7 + 189 *a^5*b^9 - 665*a^3*b^11 - 35*a*b^13 - (16*a^7*b^7 - 56*a^5*b^9 + 70*a^3*b^ 11 - 35*a*b^13)*cos(d*x + c)^6 + 3*(112*a^9*b^5 - 376*a^7*b^7 + 434*a^5*b^ 9 - 175*a^3*b^11 - 35*a*b^13)*cos(d*x + c)^4 - (560*a^11*b^3 - 1288*a^9*b^ 5 + 146*a^7*b^7 + 1547*a^5*b^9 - 1260*a^3*b^11 - 105*a*b^13)*cos(d*x + ...
Timed out. \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Exception raised: ValueError} \]
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. 2326 vs. \(2 (469) = 938\).
Time = 0.58 (sec) , antiderivative size = 2326, normalized size of antiderivative = 4.74 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
-1/840*(105*(16*a^7 - 56*a^5*b^2 + 70*a^3*b^4 - 35*a*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)))/((a^6*b^8 - 3*a^4*b^10 + 3*a^2*b^12 - b^14)*sqrt(a^2 - b^2)) - (840 *a^18*b*tan(1/2*d*x + 1/2*c)^13 - 2310*a^16*b^3*tan(1/2*d*x + 1/2*c)^13 + 1995*a^14*b^5*tan(1/2*d*x + 1/2*c)^13 - 1680*a^12*b^7*tan(1/2*d*x + 1/2*c) ^13 + 5040*a^10*b^9*tan(1/2*d*x + 1/2*c)^13 - 5040*a^8*b^11*tan(1/2*d*x + 1/2*c)^13 + 1680*a^6*b^13*tan(1/2*d*x + 1/2*c)^13 + 1680*a^19*tan(1/2*d*x + 1/2*c)^12 + 5880*a^17*b^2*tan(1/2*d*x + 1/2*c)^12 - 24990*a^15*b^4*tan(1 /2*d*x + 1/2*c)^12 + 24255*a^13*b^6*tan(1/2*d*x + 1/2*c)^12 - 10080*a^11*b ^8*tan(1/2*d*x + 1/2*c)^12 + 30240*a^9*b^10*tan(1/2*d*x + 1/2*c)^12 - 3024 0*a^7*b^12*tan(1/2*d*x + 1/2*c)^12 + 10080*a^5*b^14*tan(1/2*d*x + 1/2*c)^1 2 + 26880*a^18*b*tan(1/2*d*x + 1/2*c)^11 - 19320*a^16*b^3*tan(1/2*d*x + 1/ 2*c)^11 - 87640*a^14*b^5*tan(1/2*d*x + 1/2*c)^11 + 118790*a^12*b^7*tan(1/2 *d*x + 1/2*c)^11 - 26880*a^10*b^9*tan(1/2*d*x + 1/2*c)^11 + 94080*a^8*b^11 *tan(1/2*d*x + 1/2*c)^11 - 98560*a^6*b^13*tan(1/2*d*x + 1/2*c)^11 + 33600* a^4*b^15*tan(1/2*d*x + 1/2*c)^11 + 10080*a^19*tan(1/2*d*x + 1/2*c)^10 + 14 4480*a^17*b^2*tan(1/2*d*x + 1/2*c)^10 - 299880*a^15*b^4*tan(1/2*d*x + 1/2* c)^10 - 15680*a^13*b^6*tan(1/2*d*x + 1/2*c)^10 + 276430*a^11*b^8*tan(1/2*d *x + 1/2*c)^10 + 36960*a^9*b^10*tan(1/2*d*x + 1/2*c)^10 + 97440*a^7*b^12*t an(1/2*d*x + 1/2*c)^10 - 166880*a^5*b^14*tan(1/2*d*x + 1/2*c)^10 + 6720...
Time = 32.39 (sec) , antiderivative size = 9647, normalized size of antiderivative = 19.65 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
(2*atan((((((((32*a^2*b^35 - 192*a^4*b^33 + 480*a^6*b^31 - 640*a^8*b^29 + 480*a^10*b^27 - 192*a^12*b^25 + 32*a^14*b^23)*1i)/(b^32 - 6*a^2*b^30 + 15* a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) + (tan(c/2 + (d*x)/2)*(768*a*b^37 - 5120*a^3*b^35 + 14592*a^5*b^33 - 23040*a^7*b^31 + 21760*a^9*b^29 - 12288*a^11*b^27 + 3840*a^13*b^25 - 512*a^15*b^23)*1i)/( 8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^10*b^ 23 + a^12*b^21)))*1i)/b^8 + ((32*a*b^28 - 154*a^3*b^26 + 322*a^5*b^24 - 37 8*a^7*b^22 + 262*a^9*b^20 - 100*a^11*b^18 + 16*a^13*b^16)*1i)/(b^32 - 6*a^ 2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20 ) + (tan(c/2 + (d*x)/2)*(1120*a^2*b^28 - 5600*a^4*b^26 + 11872*a^6*b^24 - 13728*a^8*b^22 + 9152*a^10*b^20 - 3328*a^12*b^18 + 512*a^14*b^16)*1i)/(8*( b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^10*b^23 + a^12*b^21)))/b^8 + (32*a^2*b^19 - 192*a^4*b^17 + 480*a^6*b^15 - 640*a^8* b^13 + 480*a^10*b^11 - 192*a^12*b^9 + 32*a^14*b^7)/(b^32 - 6*a^2*b^30 + 15 *a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) + (tan(c/ 2 + (d*x)/2)*(512*a*b^21 - 4553*a^3*b^19 + 14116*a^5*b^17 - 22900*a^7*b^15 + 21760*a^9*b^13 - 12288*a^11*b^11 + 3840*a^13*b^9 - 512*a^15*b^7))/(8*(b ^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^10*b^23 + a^12*b^21)))/b^8 + ((32*a^2*b^19 - 192*a^4*b^17 + 480*a^6*b^15 - 640*a^8* b^13 + 480*a^10*b^11 - 192*a^12*b^9 + 32*a^14*b^7)/(b^32 - 6*a^2*b^30 +...