Integrand size = 23, antiderivative size = 384 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {128 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{99 b^8 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {32 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{99 b^8 d \sqrt {a+b \sin (c+d x)}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d} \]
-2/3*cos(d*x+c)^7/b/d/(a+b*sin(d*x+c))^(3/2)-28/33*cos(d*x+c)^5*(12*a+b*si n(d*x+c))/b^3/d/(a+b*sin(d*x+c))^(1/2)+40/99*cos(d*x+c)^3*(32*a^2-3*b^2-28 *a*b*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^5/d-16/99*cos(d*x+c)*(128*a^4-14 4*a^2*b^2+15*b^4-3*a*b*(32*a^2-31*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/ b^7/d+128/99*a*(8*a^2-9*b^2)*(4*a^2-3*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^( 1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2) *(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/b^8/d/((a+b*sin(d*x+c))/(a+b))^(1 /2)-32/99*(128*a^6-272*a^4*b^2+159*a^2*b^4-15*b^6)*(sin(1/2*c+1/4*Pi+1/2*d *x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x) ,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/b^8/d/(a+b*sin(d* x+c))^(1/2)
Time = 1.21 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {256 (a+b) \left (b \left (32 a^4 b-51 a^2 b^3+15 b^5\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )+4 \left (32 a^5-60 a^3 b^2+27 a b^4\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}+\frac {1}{2} b \cos (c+d x) \left (-32768 a^6+55296 a^4 b^2-18144 a^2 b^4-2574 b^6+\left (2048 a^4 b^2-3648 a^2 b^4+1383 b^6\right ) \cos (2 (c+d x))+\left (-96 a^2 b^4+126 b^6\right ) \cos (4 (c+d x))+9 b^6 \cos (6 (c+d x))-40960 a^5 b \sin (c+d x)+74112 a^3 b^3 \sin (c+d x)-30920 a b^5 \sin (c+d x)-384 a^3 b^3 \sin (3 (c+d x))+596 a b^5 \sin (3 (c+d x))+28 a b^5 \sin (5 (c+d x))\right )}{792 b^8 d (a+b \sin (c+d x))^{3/2}} \]
(256*(a + b)*(b*(32*a^4*b - 51*a^2*b^3 + 15*b^5)*EllipticF[(-2*c + Pi - 2* d*x)/4, (2*b)/(a + b)] + 4*(32*a^5 - 60*a^3*b^2 + 27*a*b^4)*((a + b)*Ellip ticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d* x)/4, (2*b)/(a + b)]))*((a + b*Sin[c + d*x])/(a + b))^(3/2) + (b*Cos[c + d *x]*(-32768*a^6 + 55296*a^4*b^2 - 18144*a^2*b^4 - 2574*b^6 + (2048*a^4*b^2 - 3648*a^2*b^4 + 1383*b^6)*Cos[2*(c + d*x)] + (-96*a^2*b^4 + 126*b^6)*Cos [4*(c + d*x)] + 9*b^6*Cos[6*(c + d*x)] - 40960*a^5*b*Sin[c + d*x] + 74112* a^3*b^3*Sin[c + d*x] - 30920*a*b^5*Sin[c + d*x] - 384*a^3*b^3*Sin[3*(c + d *x)] + 596*a*b^5*Sin[3*(c + d*x)] + 28*a*b^5*Sin[5*(c + d*x)]))/2)/(792*b^ 8*d*(a + b*Sin[c + d*x])^(3/2))
Time = 2.12 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 3172, 3042, 3342, 27, 3042, 3344, 27, 3042, 3344, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{(a+b \sin (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {14 \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}}dx}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {14 \int \frac {\cos (c+d x)^6 \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}}dx}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {14 \left (\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {20 \int -\frac {\cos ^4(c+d x) (b+12 a \sin (c+d x))}{2 \sqrt {a+b \sin (c+d x)}}dx}{11 b^2}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {14 \left (\frac {10 \int \frac {\cos ^4(c+d x) (b+12 a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {14 \left (\frac {10 \int \frac {\cos (c+d x)^4 (b+12 a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (\frac {4 \int -\frac {3 \cos ^2(c+d x) \left (b \left (4 a^2-3 b^2\right )+a \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \int \frac {\cos ^2(c+d x) \left (b \left (4 a^2-3 b^2\right )+a \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \int \frac {\cos (c+d x)^2 \left (b \left (4 a^2-3 b^2\right )+a \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (\frac {4 \int -\frac {b \left (32 a^4-51 b^2 a^2+15 b^4\right )+4 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \int \frac {b \left (32 a^4-51 b^2 a^2+15 b^4\right )+4 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \int \frac {b \left (32 a^4-51 b^2 a^2+15 b^4\right )+4 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \left (\frac {4 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \left (\frac {4 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \left (\frac {4 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \left (\frac {4 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \left (\frac {8 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \left (\frac {8 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \left (-\frac {2 \left (\frac {8 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {14 \left (\frac {10 \left (-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{21 b^2 d}-\frac {2 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{15 b^2 d}-\frac {2 \left (\frac {8 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{7 b^2}\right )}{11 b^2}+\frac {2 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{11 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
(-2*Cos[c + d*x]^7)/(3*b*d*(a + b*Sin[c + d*x])^(3/2)) - (14*((2*Cos[c + d *x]^5*(12*a + b*Sin[c + d*x]))/(11*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (10*( (-2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 3*b^2 - 28*a*b*Sin[c + d*x]))/(21*b^2*d) - (2*((-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(128* a^4 - 144*a^2*b^2 + 15*b^4 - 3*a*b*(32*a^2 - 31*b^2)*Sin[c + d*x]))/(15*b^ 2*d) - (2*((8*a*(8*a^2 - 9*b^2)*(4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x) /2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b*d*Sqrt[(a + b*Sin[c + d*x] )/(a + b)]) - (2*(128*a^6 - 272*a^4*b^2 + 159*a^2*b^4 - 15*b^6)*EllipticF[ (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b* d*Sqrt[a + b*Sin[c + d*x]])))/(15*b^2)))/(7*b^2)))/(11*b^2)))/(3*b)
3.6.33.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 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[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 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[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(2252\) vs. \(2(422)=844\).
Time = 4.44 (sec) , antiderivative size = 2253, normalized size of antiderivative = 5.87
-2/99*(-9*b^8*cos(d*x+c)^8-14*cos(d*x+c)^6*sin(d*x+c)*a*b^7+(24*a^2*b^6-18 *b^8)*cos(d*x+c)^6+(48*a^3*b^5-64*a*b^7)*cos(d*x+c)^4*sin(d*x+c)+(-128*a^4 *b^4+204*a^2*b^6-60*b^8)*cos(d*x+c)^4+(1280*a^5*b^3-2328*a^3*b^5+984*a*b^7 )*cos(d*x+c)^2*sin(d*x+c)+(1024*a^6*b^2-1664*a^4*b^4+456*a^2*b^6+120*b^8)* cos(d*x+c)^2-16*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/ (a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*b*(128*EllipticE((b/(a-b) *sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7-368*EllipticE((b/(a-b) *sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2+348*EllipticE((b/( a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4-108*EllipticE( (b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6-128*Elliptic F((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b+96*Ellipti cF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2+272*Ell ipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^3-189 *EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4 -159*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2 *b^5+93*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))* a*b^6+15*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2)) *b^7)*sin(d*x+c)+2048*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x +c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)* sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7*b-1536*(b/(a-b)*sin(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.28 (sec) , antiderivative size = 1043, normalized size of antiderivative = 2.72 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
2/297*(8*(sqrt(2)*(256*a^6*b^2 - 576*a^4*b^4 + 369*a^2*b^6 - 45*b^8)*cos(d *x + c)^2 - 2*sqrt(2)*(256*a^7*b - 576*a^5*b^3 + 369*a^3*b^5 - 45*a*b^7)*s in(d*x + c) - sqrt(2)*(256*a^8 - 320*a^6*b^2 - 207*a^4*b^4 + 324*a^2*b^6 - 45*b^8))*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8 *I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I* a)/b) + 8*(sqrt(2)*(256*a^6*b^2 - 576*a^4*b^4 + 369*a^2*b^6 - 45*b^8)*cos( d*x + c)^2 - 2*sqrt(2)*(256*a^7*b - 576*a^5*b^3 + 369*a^3*b^5 - 45*a*b^7)* sin(d*x + c) - sqrt(2)*(256*a^8 - 320*a^6*b^2 - 207*a^4*b^4 + 324*a^2*b^6 - 45*b^8))*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27* (-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2 *I*a)/b) + 96*(sqrt(2)*(32*I*a^5*b^3 - 60*I*a^3*b^5 + 27*I*a*b^7)*cos(d*x + c)^2 + 2*sqrt(2)*(-32*I*a^6*b^2 + 60*I*a^4*b^4 - 27*I*a^2*b^6)*sin(d*x + c) + sqrt(2)*(-32*I*a^7*b + 28*I*a^5*b^3 + 33*I*a^3*b^5 - 27*I*a*b^7))*sq rt(I*b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b ^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9* I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) + 96 *(sqrt(2)*(-32*I*a^5*b^3 + 60*I*a^3*b^5 - 27*I*a*b^7)*cos(d*x + c)^2 + 2*s qrt(2)*(32*I*a^6*b^2 - 60*I*a^4*b^4 + 27*I*a^2*b^6)*sin(d*x + c) + sqrt(2) *(32*I*a^7*b - 28*I*a^5*b^3 - 33*I*a^3*b^5 + 27*I*a*b^7))*sqrt(-I*b)*weier strassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, ...
Timed out. \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{8}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^8}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]