Integrand size = 23, antiderivative size = 398 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {8 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{15015 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {32 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{15015 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+59 b^2\right )+7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15015 b^3 d} \]
-2/13*b*cos(d*x+c)^5*(a+b*sin(d*x+c))^(3/2)/d-32/143*a*b*cos(d*x+c)^5*(a+b *sin(d*x+c))^(1/2)/d+2/3003*cos(d*x+c)^3*(a*(5*a^2+59*b^2)+7*b*(53*a^2+11* b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b/d-4/15015*cos(d*x+c)*(4*a*(5*a^4 -40*a^2*b^2-93*b^4)-3*b*(5*a^4+430*a^2*b^2+77*b^4)*sin(d*x+c))*(a+b*sin(d* x+c))^(1/2)/b^3/d+8/15015*(20*a^6-175*a^4*b^2-1662*a^2*b^4-231*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)*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^4/d/(( a+b*sin(d*x+c))/(a+b))^(1/2)-32/15015*a*(5*a^6-45*a^4*b^2-53*a^2*b^4+93*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^4/d/(a+b*sin(d*x+c))^(1/2)
Time = 0.91 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.81 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {128 \left (b \left (5 a^5 b-1450 a^3 b^3-603 a b^5\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )+\left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right )\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b (a+b \sin (c+d x)) \left (4 a \left (320 a^4-2710 a^2 b^2+6453 b^4\right ) \cos (c+d x)-10 a b^2 \left (20 a^2-2599 b^2\right ) \cos (3 (c+d x))+5670 a b^4 \cos (5 (c+d x))-b \left (480 a^4+56120 a^2 b^2+4697 b^4\right ) \sin (2 (c+d x))+140 b^3 \left (-53 a^2+22 b^2\right ) \sin (4 (c+d x))+1155 b^5 \sin (6 (c+d x))\right )}{240240 b^4 d \sqrt {a+b \sin (c+d x)}} \]
(128*(b*(5*a^5*b - 1450*a^3*b^3 - 603*a*b^5)*EllipticF[(-2*c + Pi - 2*d*x) /4, (2*b)/(a + b)] + (20*a^6 - 175*a^4*b^2 - 1662*a^2*b^4 - 231*b^6)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*(a + b*Sin[c + d*x])*(4*a*(320*a^4 - 2710*a^2*b^2 + 6453*b^4)*Cos[c + d*x] - 10*a*b^2*(20*a^2 - 2599*b^2)*Cos[3*(c + d*x)] + 5670*a*b^4*Cos[5*(c + d*x) ] - b*(480*a^4 + 56120*a^2*b^2 + 4697*b^4)*Sin[2*(c + d*x)] + 140*b^3*(-53 *a^2 + 22*b^2)*Sin[4*(c + d*x)] + 1155*b^5*Sin[6*(c + d*x)]))/(240240*b^4* d*Sqrt[a + b*Sin[c + d*x]])
Time = 2.21 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.04, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 3171, 27, 3042, 3341, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^4 (a+b \sin (c+d x))^{5/2}dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{13} \int \frac {1}{2} \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (13 a^2+16 b \sin (c+d x) a+3 b^2\right )dx-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (13 a^2+16 b \sin (c+d x) a+3 b^2\right )dx-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \int \cos (c+d x)^4 \sqrt {a+b \sin (c+d x)} \left (13 a^2+16 b \sin (c+d x) a+3 b^2\right )dx-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{13} \left (\frac {2}{11} \int \frac {\cos ^4(c+d x) \left (a \left (143 a^2+49 b^2\right )+3 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \frac {\cos ^4(c+d x) \left (a \left (143 a^2+49 b^2\right )+3 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \frac {\cos (c+d x)^4 \left (a \left (143 a^2+49 b^2\right )+3 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {4 \int \frac {3 \cos ^2(c+d x) \left (8 a \left (47 a^2+17 b^2\right ) b^2+\left (5 a^4+430 b^2 a^2+77 b^4\right ) \sin (c+d x) b\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{21 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \int \frac {\cos ^2(c+d x) \left (8 a \left (47 a^2+17 b^2\right ) b^2+\left (5 a^4+430 b^2 a^2+77 b^4\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \int \frac {\cos (c+d x)^2 \left (8 a \left (47 a^2+17 b^2\right ) b^2+\left (5 a^4+430 b^2 a^2+77 b^4\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (\frac {4 \int -\frac {a \left (5 a^4-1450 b^2 a^2-603 b^4\right ) b^2+\left (20 a^6-175 b^2 a^4-1662 b^4 a^2-231 b^6\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \int \frac {a \left (5 a^4-1450 b^2 a^2-603 b^4\right ) b^2+\left (20 a^6-175 b^2 a^4-1662 b^4 a^2-231 b^6\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \int \frac {a \left (5 a^4-1450 b^2 a^2-603 b^4\right ) b^2+\left (20 a^6-175 b^2 a^4-1662 b^4 a^2-231 b^6\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx-4 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx-4 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {\left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-4 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {\left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-4 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {2 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-4 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {2 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {2 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{21 b d}+\frac {2 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15 b d}-\frac {2 \left (\frac {2 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{7 b^2}\right )-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
(-2*b*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2))/(13*d) + ((-32*a*b*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) + ((2*Cos[c + d*x]^3*Sqrt[a + b* Sin[c + d*x]]*(a*(5*a^2 + 59*b^2) + 7*b*(53*a^2 + 11*b^2)*Sin[c + d*x]))/( 21*b*d) + (2*((-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a*(5*a^4 - 40*a ^2*b^2 - 93*b^4) - 3*b*(5*a^4 + 430*a^2*b^2 + 77*b^4)*Sin[c + d*x]))/(15*b *d) - (2*((2*(20*a^6 - 175*a^4*b^2 - 1662*a^2*b^4 - 231*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*S in[c + d*x])/(a + b)]) - (8*a*(5*a^6 - 45*a^4*b^2 - 53*a^2*b^4 + 93*b^6)*E llipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])))/(15*b^2)))/(7*b^2))/11)/13
3.5.100.3.1 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[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
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[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
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. \(1618\) vs. \(2(436)=872\).
Time = 23.96 (sec) , antiderivative size = 1619, normalized size of antiderivative = 4.07
-2/15015*(-1155*sin(d*x+c)^8*b^8-10*sin(d*x+c)^3*a^5*b^3-40*sin(d*x+c)^2*a ^6*b^2-3990*sin(d*x+c)^7*a*b^7-4690*sin(d*x+c)^6*a^2*b^6-1880*sin(d*x+c)^5 *a^3*b^5+5*sin(d*x+c)^4*a^4*b^4+10*sin(d*x+c)*a^5*b^3-4780*sin(d*x+c)*a^3* b^5+2104*sin(d*x+c)*a*b^7+40*a^6*b^2-345*a^4*b^4+1796*a^2*b^6+924*((a+b*si n(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/ (a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2)) *b^8-80*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-( 1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a- b)/(a+b))^(1/2))*a^8-924*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b /(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/ (a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^8+80*((a+b*sin(d*x+c))/(a-b))^(1/2)*(- (sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF((( a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7*b-60*((a+b*sin(d*x+c ))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^ (1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b^ 2-720*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+ sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b) /(a+b))^(1/2))*a^5*b^3-5100*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1 )*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c ))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^4-848*((a+b*sin(d*x+c))/(a-b...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.59 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} {\left (40 \, a^{7} - 365 \, a^{5} b^{2} + 1026 \, a^{3} b^{4} + 1347 \, a b^{6}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (40 \, a^{7} - 365 \, a^{5} b^{2} + 1026 \, a^{3} b^{4} + 1347 \, a b^{6}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 6 \, \sqrt {2} {\left (-20 i \, a^{6} b + 175 i \, a^{4} b^{3} + 1662 i \, a^{2} b^{5} + 231 i \, b^{7}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 6 \, \sqrt {2} {\left (20 i \, a^{6} b - 175 i \, a^{4} b^{3} - 1662 i \, a^{2} b^{5} - 231 i \, b^{7}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 3 \, {\left (2835 \, a b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (5 \, a^{3} b^{4} + 59 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, a^{5} b^{2} - 40 \, a^{3} b^{4} - 93 \, a b^{6}\right )} \cos \left (d x + c\right ) + {\left (1155 \, b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (53 \, a^{2} b^{5} + 11 \, b^{7}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (5 \, a^{4} b^{3} + 430 \, a^{2} b^{5} + 77 \, b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{45045 \, b^{5} d} \]
2/45045*(2*sqrt(2)*(40*a^7 - 365*a^5*b^2 + 1026*a^3*b^4 + 1347*a*b^6)*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) + 2*sqrt (2)*(40*a^7 - 365*a^5*b^2 + 1026*a^3*b^4 + 1347*a*b^6)*sqrt(-I*b)*weierstr assPInverse(-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) - 6*sqrt(2)*(-20*I*a^ 6*b + 175*I*a^4*b^3 + 1662*I*a^2*b^5 + 231*I*b^7)*sqrt(I*b)*weierstrassZet a(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstrassPI nverse(-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)) - 6*sqrt(2)*(20*I*a^6*b - 175*I*a^4*b^3 - 1662*I*a^2*b^5 - 231*I*b^7)*sqrt(-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, weierstrassPInver se(-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*co s(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) - 3*(2835*a*b^6*cos(d*x + c)^ 5 - 5*(5*a^3*b^4 + 59*a*b^6)*cos(d*x + c)^3 + 8*(5*a^5*b^2 - 40*a^3*b^4 - 93*a*b^6)*cos(d*x + c) + (1155*b^7*cos(d*x + c)^5 - 35*(53*a^2*b^5 + 11*b^ 7)*cos(d*x + c)^3 - 6*(5*a^4*b^3 + 430*a^2*b^5 + 77*b^7)*cos(d*x + c))*sin (d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^5*d)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]