Integrand size = 31, antiderivative size = 463 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}-\frac {8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac {8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac {10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac {8 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{45045 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {16 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{45045 b^6 d \sqrt {a+b \sin (c+d x)}} \]
-8/45045*(480*a^4-937*a^2*b^2+231*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/b ^5/d+8/3003*a*(40*a^2-81*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(3/2) /b^4/d-10/1287*(16*a^2-33*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(3 /2)/b^3/d+20/143*a*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(3/2)/b^2/d-2/ 13*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^(3/2)/b/d+16/45045*a*(160*a^4- 279*a^2*b^2+27*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+8/45045*(320*a ^6-798*a^4*b^2+435*a^2*b^4-693*b^6)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/si n(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^6/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-16/ 45045*a*(160*a^6-439*a^4*b^2+306*a^2*b^4-27*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^6/d/(a+b*sin(d*x+ c))^(1/2)
Time = 3.96 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.71 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {\sqrt {a+b \sin (c+d x)} \left (128 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-256 a \left (160 a^5-160 a^4 b-279 a^3 b^2+279 a^2 b^3+27 a b^4-27 b^5\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )-2 b \cos (c+d x) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \left (10240 a^5-21056 a^3 b^2+5898 a b^4-1600 \left (2 a^3 b^2-3 a b^4\right ) \cos (2 (c+d x))+630 a b^4 \cos (4 (c+d x))-7680 a^4 b \sin (c+d x)+13592 a^2 b^3 \sin (c+d x)-19866 b^5 \sin (c+d x)+1400 a^2 b^3 \sin (3 (c+d x))+5775 b^5 \sin (3 (c+d x))+3465 b^5 \sin (5 (c+d x))\right )\right )}{720720 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
(Sqrt[a + b*Sin[c + d*x]]*(128*(320*a^6 - 798*a^4*b^2 + 435*a^2*b^4 - 693* b^6)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 256*a*(160*a^5 - 16 0*a^4*b - 279*a^3*b^2 + 279*a^2*b^3 + 27*a*b^4 - 27*b^5)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 2*b*Cos[c + d*x]*Sqrt[(a + b*Sin[c + d*x] )/(a + b)]*(10240*a^5 - 21056*a^3*b^2 + 5898*a*b^4 - 1600*(2*a^3*b^2 - 3*a *b^4)*Cos[2*(c + d*x)] + 630*a*b^4*Cos[4*(c + d*x)] - 7680*a^4*b*Sin[c + d *x] + 13592*a^2*b^3*Sin[c + d*x] - 19866*b^5*Sin[c + d*x] + 1400*a^2*b^3*S in[3*(c + d*x)] + 5775*b^5*Sin[3*(c + d*x)] + 3465*b^5*Sin[5*(c + d*x)]))) /(720720*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])
Time = 2.75 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.06, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.774, Rules used = {3042, 3374, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 27, 3042, 3232, 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.
| \(\displaystyle \int \sin ^2(c+d x) \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^4 \sqrt {a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3374 |
| \(\displaystyle -\frac {4 \int \frac {1}{4} \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (60 a^2+2 b \sin (c+d x) a-143 b^2-5 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x)\right )dx}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (60 a^2+2 b \sin (c+d x) a-143 b^2-5 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x)\right )dx}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sin (c+d x)^2 \sqrt {a+b \sin (c+d x)} \left (60 a^2+2 b \sin (c+d x) a-143 b^2-5 \left (16 a^2-33 b^2\right ) \sin (c+d x)^2\right )dx}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2 \int -2 \sin (c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 a \left (40 a^2-81 b^2\right ) \sin ^2(c+d x)+b \left (5 a^2+33 b^2\right ) \sin (c+d x)+5 a \left (16 a^2-33 b^2\right )\right )dx}{9 b}+\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \int \sin (c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 a \left (40 a^2-81 b^2\right ) \sin ^2(c+d x)+b \left (5 a^2+33 b^2\right ) \sin (c+d x)+5 a \left (16 a^2-33 b^2\right )\right )dx}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \int \sin (c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 a \left (40 a^2-81 b^2\right ) \sin (c+d x)^2+b \left (5 a^2+33 b^2\right ) \sin (c+d x)+5 a \left (16 a^2-33 b^2\right )\right )dx}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {2 \int -\frac {1}{2} \sqrt {a+b \sin (c+d x)} \left (6 \left (40 a^2-81 b^2\right ) a^2+20 b \left (2 a^2-3 b^2\right ) \sin (c+d x) a-\left (480 a^4-937 b^2 a^2+231 b^4\right ) \sin ^2(c+d x)\right )dx}{7 b}+\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\int \sqrt {a+b \sin (c+d x)} \left (6 \left (40 a^2-81 b^2\right ) a^2+20 b \left (2 a^2-3 b^2\right ) \sin (c+d x) a-\left (480 a^4-937 b^2 a^2+231 b^4\right ) \sin ^2(c+d x)\right )dx}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\int \sqrt {a+b \sin (c+d x)} \left (6 \left (40 a^2-81 b^2\right ) a^2+20 b \left (2 a^2-3 b^2\right ) \sin (c+d x) a-\left (480 a^4-937 b^2 a^2+231 b^4\right ) \sin (c+d x)^2\right )dx}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \int -\frac {3}{2} \sqrt {a+b \sin (c+d x)} \left (b \left (80 a^4-127 b^2 a^2+231 b^4\right )-2 a \left (160 a^4-279 b^2 a^2+27 b^4\right ) \sin (c+d x)\right )dx}{5 b}+\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \int \sqrt {a+b \sin (c+d x)} \left (b \left (80 a^4-127 b^2 a^2+231 b^4\right )-2 a \left (160 a^4-279 b^2 a^2+27 b^4\right ) \sin (c+d x)\right )dx}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \int \sqrt {a+b \sin (c+d x)} \left (b \left (80 a^4-127 b^2 a^2+231 b^4\right )-2 a \left (160 a^4-279 b^2 a^2+27 b^4\right ) \sin (c+d x)\right )dx}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {2}{3} \int -\frac {a b \left (80 a^4-177 b^2 a^2-639 b^4\right )+\left (320 a^6-798 b^2 a^4+435 b^4 a^2-693 b^6\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx+\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}-\frac {1}{3} \int \frac {a b \left (80 a^4-177 b^2 a^2-639 b^4\right )+\left (320 a^6-798 b^2 a^4+435 b^4 a^2-693 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}-\frac {1}{3} \int \frac {a b \left (80 a^4-177 b^2 a^2-639 b^4\right )+\left (320 a^6-798 b^2 a^4+435 b^4 a^2-693 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {1}{3} \left (\frac {2 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {\left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {1}{3} \left (\frac {2 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {\left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {1}{3} \left (\frac {2 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {\left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {1}{3} \left (\frac {2 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {\left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {1}{3} \left (\frac {2 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {1}{3} \left (\frac {2 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {1}{3} \left (\frac {2 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac {4 \left (\frac {6 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {4 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {4 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )\right )}{5 b}}{7 b}\right )}{9 b}}{143 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}\) |
(20*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(143*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2))/(13*b*d) - (( 10*(16*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2 ))/(9*b*d) - (4*((6*a*(40*a^2 - 81*b^2)*Cos[c + d*x]*Sin[c + d*x]*(a + b*S in[c + d*x])^(3/2))/(7*b*d) - ((2*(480*a^4 - 937*a^2*b^2 + 231*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*b*d) - (3*((4*a*(160*a^4 - 279*a^2*b ^2 + 27*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*d) + ((-2*(320*a^6 - 798*a^4*b^2 + 435*a^2*b^4 - 693*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) ]) + (4*a*(160*a^6 - 439*a^4*b^2 + 306*a^2*b^4 - 27*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]]))/3))/(5*b))/(7*b)))/(9*b))/(143*b^2)
3.12.43.3.1 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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a*(n + 3)*Cos[e + f* x]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d*f*(m + n + 3)*(m + n + 4))), x] + (-Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x])^(m + 1)/(b*d^2*f*(m + n + 4))), x] - Simp[1/(b^2*(m + n + 3 )*(m + n + 4)) Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x]) /; F reeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Integ ersQ[2*m, 2*n]) && !m < -1 && !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(1618\) vs. \(2(493)=986\).
Time = 2.57 (sec) , antiderivative size = 1619, normalized size of antiderivative = 3.50
-2/45045*(2772*((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^8-1280*((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^8-2772*((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 lipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^8+1280*((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-960*((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^2-3512*((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*s in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^3+2484*((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 +2448*((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^5-4296*((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*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.37 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} {\left (640 \, a^{7} - 1836 \, a^{5} b^{2} + 1401 \, a^{3} b^{4} + 531 \, 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 (640 \, a^{7} - 1836 \, a^{5} b^{2} + 1401 \, a^{3} b^{4} + 531 \, 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 (-320 i \, a^{6} b + 798 i \, a^{4} b^{3} - 435 i \, a^{2} b^{5} + 693 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 (320 i \, a^{6} b - 798 i \, a^{4} b^{3} + 435 i \, a^{2} b^{5} - 693 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 (315 \, a b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (80 \, a^{3} b^{4} - 57 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (160 \, a^{5} b^{2} - 279 \, a^{3} b^{4} + 27 \, a b^{6}\right )} \cos \left (d x + c\right ) + {\left (3465 \, b^{7} \cos \left (d x + c\right )^{5} + 35 \, {\left (10 \, a^{2} b^{5} - 33 \, b^{7}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (80 \, a^{4} b^{3} - 127 \, a^{2} b^{5} + 231 \, 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 )}}{135135 \, b^{7} d} \]
2/135135*(2*sqrt(2)*(640*a^7 - 1836*a^5*b^2 + 1401*a^3*b^4 + 531*a*b^6)*sq rt(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*sq rt(2)*(640*a^7 - 1836*a^5*b^2 + 1401*a^3*b^4 + 531*a*b^6)*sqrt(-I*b)*weier strassPInverse(-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)*(-320* I*a^6*b + 798*I*a^4*b^3 - 435*I*a^2*b^5 + 693*I*b^7)*sqrt(I*b)*weierstrass Zeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstras sPInverse(-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)*(320*I*a^6* b - 798*I*a^4*b^3 + 435*I*a^2*b^5 - 693*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, weierstrassPIn verse(-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*(315*a*b^6*cos(d*x + c )^5 - 5*(80*a^3*b^4 - 57*a*b^6)*cos(d*x + c)^3 + 4*(160*a^5*b^2 - 279*a^3* b^4 + 27*a*b^6)*cos(d*x + c) + (3465*b^7*cos(d*x + c)^5 + 35*(10*a^2*b^5 - 33*b^7)*cos(d*x + c)^3 - 6*(80*a^4*b^3 - 127*a^2*b^5 + 231*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^7*d)
\[ \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}\, dx \]
\[ \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2} \,d x } \]
\[ \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \]