Integrand size = 29, antiderivative size = 394 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {8 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{15015 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d} \]
-2/13*cos(d*x+c)^5*(a+b*sin(d*x+c))^(3/2)/d-6/143*a*cos(d*x+c)^5*(a+b*sin( d*x+c))^(1/2)/d-2/3003*cos(d*x+c)^3*(4*a*(2*a^2-5*b^2)-7*b*(a^2+11*b^2)*si n(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^2/d+4/15015*cos(d*x+c)*(a*(32*a^4-113*a ^2*b^2+177*b^4)-3*b*(8*a^4-27*a^2*b^2-77*b^4)*sin(d*x+c))*(a+b*sin(d*x+c)) ^(1/2)/b^4/d-8/15015*(32*a^6-137*a^4*b^2+258*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^5/d/((a+b*si n(d*x+c))/(a+b))^(1/2)+8/15015*a*(32*a^6-145*a^4*b^2+290*a^2*b^4-177*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(co s(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^5/d/(a+b*sin(d*x+c))^(1/2)
Time = 8.60 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.97 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {-384 \left (32 a^7+32 a^6 b-137 a^5 b^2-137 a^4 b^3+258 a^3 b^4+258 a^2 b^5+231 a b^6+231 b^7\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+384 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-3 b \cos (c+d x) \left (-2048 a^6+8640 a^4 b^2+1980 a^2 b^4-6622 b^6+\left (-128 a^4 b^2+24512 a^2 b^4+8547 b^6\right ) \cos (2 (c+d x))+70 \left (86 a^2 b^4-11 b^6\right ) \cos (4 (c+d x))-1155 b^6 \cos (6 (c+d x))-512 a^5 b \sin (c+d x)+2088 a^3 b^3 \sin (c+d x)-19492 a b^5 \sin (c+d x)+40 a^3 b^3 \sin (3 (c+d x))+11870 a b^5 \sin (3 (c+d x))+5250 a b^5 \sin (5 (c+d x))\right )}{720720 b^5 d \sqrt {a+b \sin (c+d x)}} \]
(-384*(32*a^7 + 32*a^6*b - 137*a^5*b^2 - 137*a^4*b^3 + 258*a^3*b^4 + 258*a ^2*b^5 + 231*a*b^6 + 231*b^7)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 384*a*(32*a^6 - 145*a^4*b^2 + 290 *a^2*b^4 - 177*b^6)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[( a + b*Sin[c + d*x])/(a + b)] - 3*b*Cos[c + d*x]*(-2048*a^6 + 8640*a^4*b^2 + 1980*a^2*b^4 - 6622*b^6 + (-128*a^4*b^2 + 24512*a^2*b^4 + 8547*b^6)*Cos[ 2*(c + d*x)] + 70*(86*a^2*b^4 - 11*b^6)*Cos[4*(c + d*x)] - 1155*b^6*Cos[6* (c + d*x)] - 512*a^5*b*Sin[c + d*x] + 2088*a^3*b^3*Sin[c + d*x] - 19492*a* b^5*Sin[c + d*x] + 40*a^3*b^3*Sin[3*(c + d*x)] + 11870*a*b^5*Sin[3*(c + d* x)] + 5250*a*b^5*Sin[5*(c + d*x)]))/(720720*b^5*d*Sqrt[a + b*Sin[c + d*x]] )
Time = 2.04 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {3042, 3341, 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.
| \(\displaystyle \int \sin (c+d x) \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x) \cos (c+d x)^4 (a+b \sin (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {2}{13} \int \frac {3}{2} \cos ^4(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}dx-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{13} \int \cos ^4(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}dx-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{13} \int \cos (c+d x)^4 (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}dx-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {3}{13} \left (\frac {2}{11} \int \frac {\cos ^4(c+d x) \left (12 a b+\left (a^2+11 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \int \frac {\cos ^4(c+d x) \left (12 a b+\left (a^2+11 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \int \frac {\cos (c+d x)^4 \left (12 a b+\left (a^2+11 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (\frac {4 \int -\frac {\cos ^2(c+d x) \left (a b \left (a^2-97 b^2\right )+\left (8 a^4-27 b^2 a^2-77 b^4\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \int \frac {\cos ^2(c+d x) \left (a b \left (a^2-97 b^2\right )+\left (8 a^4-27 b^2 a^2-77 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \int \frac {\cos (c+d x)^2 \left (a b \left (a^2-97 b^2\right )+\left (8 a^4-27 b^2 a^2-77 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (\frac {4 \int -\frac {8 a b \left (a^4-4 b^2 a^2+51 b^4\right )+\left (32 a^6-137 b^2 a^4+258 b^4 a^2+231 b^6\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \int \frac {8 a b \left (a^4-4 b^2 a^2+51 b^4\right )+\left (32 a^6-137 b^2 a^4+258 b^4 a^2+231 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \int \frac {8 a b \left (a^4-4 b^2 a^2+51 b^4\right )+\left (32 a^6-137 b^2 a^4+258 b^4 a^2+231 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {\left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {\left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {\left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {\left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {3}{13} \left (\frac {1}{11} \left (-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{63 b^2 d}-\frac {2 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15 b^2 d}-\frac {2 \left (\frac {2 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{21 b^2}\right )-\frac {2 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\) |
(-2*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2))/(13*d) + (3*((-2*a*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) + ((-2*Cos[c + d*x]^3*Sqrt[a + b*S in[c + d*x]]*(4*a*(2*a^2 - 5*b^2) - 7*b*(a^2 + 11*b^2)*Sin[c + d*x]))/(63* b^2*d) - (2*((-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^4 - 113*a^ 2*b^2 + 177*b^4) - 3*b*(8*a^4 - 27*a^2*b^2 - 77*b^4)*Sin[c + d*x]))/(15*b^ 2*d) - (2*((2*(32*a^6 - 137*a^4*b^2 + 258*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]])/(b*d*Sqrt[(a + b *Sin[c + d*x])/(a + b)]) - (2*a*(32*a^6 - 145*a^4*b^2 + 290*a^2*b^4 - 177* 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)))/(21*b^2))/11))/13
3.12.52.3.1 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[((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. \(2(432)=864\).
Time = 2.51 (sec) , antiderivative size = 1619, normalized size of antiderivative = 4.11
2/15015*(-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)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2 ),((a-b)/(a+b))^(1/2))*b^8-128*((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)*Ellip ticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^8+128*((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-96*((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^2-580*((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+420*((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)*El lipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^4+1160*( (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^3*b^5+600*((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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.60 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (4 \, \sqrt {2} {\left (32 \, a^{7} - 149 \, a^{5} b^{2} + 306 \, a^{3} b^{4} - 381 \, 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 ) + 4 \, \sqrt {2} {\left (32 \, a^{7} - 149 \, a^{5} b^{2} + 306 \, a^{3} b^{4} - 381 \, 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 (32 i \, a^{6} b - 137 i \, a^{4} b^{3} + 258 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 (-32 i \, a^{6} b + 137 i \, a^{4} b^{3} - 258 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 (1470 \, a b^{6} \cos \left (d x + c\right )^{5} + 20 \, {\left (2 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (32 \, a^{5} b^{2} - 113 \, a^{3} b^{4} + 177 \, a b^{6}\right )} \cos \left (d x + c\right ) + {\left (1155 \, b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (a^{2} b^{5} + 11 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (8 \, a^{4} b^{3} - 27 \, 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^{6} d} \]
-2/45045*(4*sqrt(2)*(32*a^7 - 149*a^5*b^2 + 306*a^3*b^4 - 381*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) + 4*sqrt( 2)*(32*a^7 - 149*a^5*b^2 + 306*a^3*b^4 - 381*a*b^6)*sqrt(-I*b)*weierstrass PInverse(-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)*(32*I*a^6*b - 137*I*a^4*b^3 + 258*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, weierstrassPInvers e(-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)*(-32*I*a^6*b + 137* I*a^4*b^3 - 258*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, 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)) + 3*(1470*a*b^6*cos(d*x + c)^5 + 2 0*(2*a^3*b^4 - 5*a*b^6)*cos(d*x + c)^3 - 2*(32*a^5*b^2 - 113*a^3*b^4 + 177 *a*b^6)*cos(d*x + c) + (1155*b^7*cos(d*x + c)^5 - 35*(a^2*b^5 + 11*b^7)*co s(d*x + c)^3 + 6*(8*a^4*b^3 - 27*a^2*b^5 - 77*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^6*d)
Timed out. \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) \,d x } \]
\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]