Integrand size = 31, antiderivative size = 528 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {16 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{45045 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{45045 b^6 d \sqrt {a+b \sin (c+d x)}} \]
16/45045*a*(32*a^4-47*a^2*b^2-27*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/b^ 5/d-8/45045*(160*a^4-375*a^2*b^2+117*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(5/2 )/b^5/d+8/1287*a*(8*a^2-21*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(5/ 2)/b^4/d-2/2145*(80*a^2-221*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^ (5/2)/b^3/d+4/39*a*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(5/2)/b^2/d-2/ 15*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^(5/2)/b/d+8/45045*(64*a^6-174* a^4*b^2+81*a^2*b^4-195*b^6)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+16/450 45*a*(32*a^6-111*a^4*b^2+102*a^2*b^4-471*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^6/d/((a+b*sin(d*x+c))/(a+b)) ^(1/2)-8/45045*(64*a^8-238*a^6*b^2+255*a^4*b^4-276*a^2*b^6+195*b^8)*(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 = 10.64 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.72 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {a+b \sin (c+d x)} \left (512 \left (32 a^7-111 a^5 b^2+102 a^3 b^4-471 a b^6\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-256 \left (64 a^7-64 a^6 b-174 a^5 b^2+174 a^4 b^3+81 a^3 b^4-81 a^2 b^5-195 a b^6+195 b^7\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )-2 b \cos (c+d x) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \left (4096 a^6-12416 a^4 b^2+8100 a^2 b^4+6786 b^6+\left (-1280 a^4 b^2+3168 a^2 b^4+21723 b^6\right ) \cos (2 (c+d x))+42 \left (6 a^2 b^4-13 b^6\right ) \cos (4 (c+d x))-3003 b^6 \cos (6 (c+d x))-3072 a^5 b \sin (c+d x)+8432 a^3 b^3 \sin (c+d x)-41424 a b^5 \sin (c+d x)+560 a^3 b^3 \sin (3 (c+d x))+13776 a b^5 \sin (3 (c+d x))+7392 a b^5 \sin (5 (c+d x))\right )\right )}{1441440 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
(Sqrt[a + b*Sin[c + d*x]]*(512*(32*a^7 - 111*a^5*b^2 + 102*a^3*b^4 - 471*a *b^6)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 256*(64*a^7 - 64*a ^6*b - 174*a^5*b^2 + 174*a^4*b^3 + 81*a^3*b^4 - 81*a^2*b^5 - 195*a*b^6 + 1 95*b^7)*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)]*(4096*a^6 - 12416*a^4*b^2 + 8100*a^2*b ^4 + 6786*b^6 + (-1280*a^4*b^2 + 3168*a^2*b^4 + 21723*b^6)*Cos[2*(c + d*x) ] + 42*(6*a^2*b^4 - 13*b^6)*Cos[4*(c + d*x)] - 3003*b^6*Cos[6*(c + d*x)] - 3072*a^5*b*Sin[c + d*x] + 8432*a^3*b^3*Sin[c + d*x] - 41424*a*b^5*Sin[c + d*x] + 560*a^3*b^3*Sin[3*(c + d*x)] + 13776*a*b^5*Sin[3*(c + d*x)] + 7392 *a*b^5*Sin[5*(c + d*x)])))/(1441440*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b )])
Time = 3.32 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.06, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.871, Rules used = {3042, 3374, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 27, 3042, 3232, 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.
| \(\displaystyle \int \sin ^2(c+d x) \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^4 (a+b \sin (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3374 |
| \(\displaystyle -\frac {4 \int \frac {1}{4} \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (-\left (\left (80 a^2-221 b^2\right ) \sin ^2(c+d x)\right )+6 a b \sin (c+d x)+15 \left (4 a^2-13 b^2\right )\right )dx}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (-\left (\left (80 a^2-221 b^2\right ) \sin ^2(c+d x)\right )+6 a b \sin (c+d x)+15 \left (4 a^2-13 b^2\right )\right )dx}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sin (c+d x)^2 (a+b \sin (c+d x))^{3/2} \left (-\left (\left (80 a^2-221 b^2\right ) \sin (c+d x)^2\right )+6 a b \sin (c+d x)+15 \left (4 a^2-13 b^2\right )\right )dx}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2 \int -2 \sin (c+d x) (a+b \sin (c+d x))^{3/2} \left (-15 a \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)+3 b \left (5 a^2+13 b^2\right ) \sin (c+d x)+a \left (80 a^2-221 b^2\right )\right )dx}{11 b}+\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \int \sin (c+d x) (a+b \sin (c+d x))^{3/2} \left (-15 a \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)+3 b \left (5 a^2+13 b^2\right ) \sin (c+d x)+a \left (80 a^2-221 b^2\right )\right )dx}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \int \sin (c+d x) (a+b \sin (c+d x))^{3/2} \left (-15 a \left (8 a^2-21 b^2\right ) \sin (c+d x)^2+3 b \left (5 a^2+13 b^2\right ) \sin (c+d x)+a \left (80 a^2-221 b^2\right )\right )dx}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {2 \int -\frac {3}{2} (a+b \sin (c+d x))^{3/2} \left (10 \left (8 a^2-21 b^2\right ) a^2+8 b \left (5 a^2-9 b^2\right ) \sin (c+d x) a-\left (160 a^4-375 b^2 a^2+117 b^4\right ) \sin ^2(c+d x)\right )dx}{9 b}+\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\int (a+b \sin (c+d x))^{3/2} \left (10 \left (8 a^2-21 b^2\right ) a^2+8 b \left (5 a^2-9 b^2\right ) \sin (c+d x) a-\left (160 a^4-375 b^2 a^2+117 b^4\right ) \sin ^2(c+d x)\right )dx}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\int (a+b \sin (c+d x))^{3/2} \left (10 \left (8 a^2-21 b^2\right ) a^2+8 b \left (5 a^2-9 b^2\right ) \sin (c+d x) a-\left (160 a^4-375 b^2 a^2+117 b^4\right ) \sin (c+d x)^2\right )dx}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \int -\frac {5}{2} (a+b \sin (c+d x))^{3/2} \left (3 b \left (16 a^4-27 b^2 a^2+39 b^4\right )-2 a \left (32 a^4-47 b^2 a^2-27 b^4\right ) \sin (c+d x)\right )dx}{7 b}+\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \int (a+b \sin (c+d x))^{3/2} \left (3 b \left (16 a^4-27 b^2 a^2+39 b^4\right )-2 a \left (32 a^4-47 b^2 a^2-27 b^4\right ) \sin (c+d x)\right )dx}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \int (a+b \sin (c+d x))^{3/2} \left (3 b \left (16 a^4-27 b^2 a^2+39 b^4\right )-2 a \left (32 a^4-47 b^2 a^2-27 b^4\right ) \sin (c+d x)\right )dx}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \sin (c+d x)} \left (a b \left (16 a^4-41 b^2 a^2+249 b^4\right )-\left (64 a^6-174 b^2 a^4+81 b^4 a^2-195 b^6\right ) \sin (c+d x)\right )dx+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \int \sqrt {a+b \sin (c+d x)} \left (a b \left (16 a^4-41 b^2 a^2+249 b^4\right )-\left (64 a^6-174 b^2 a^4+81 b^4 a^2-195 b^6\right ) \sin (c+d x)\right )dx+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \int \sqrt {a+b \sin (c+d x)} \left (a b \left (16 a^4-41 b^2 a^2+249 b^4\right )-\left (64 a^6-174 b^2 a^4+81 b^4 a^2-195 b^6\right ) \sin (c+d x)\right )dx+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {2}{3} \int -\frac {b \left (16 a^6-51 b^2 a^4-666 b^4 a^2-195 b^6\right )+2 a \left (32 a^6-111 b^2 a^4+102 b^4 a^2-471 b^6\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx+\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}-\frac {1}{3} \int \frac {b \left (16 a^6-51 b^2 a^4-666 b^4 a^2-195 b^6\right )+2 a \left (32 a^6-111 b^2 a^4+102 b^4 a^2-471 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}-\frac {1}{3} \int \frac {b \left (16 a^6-51 b^2 a^4-666 b^4 a^2-195 b^6\right )+2 a \left (32 a^6-111 b^2 a^4+102 b^4 a^2-471 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {4 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}-\frac {4 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}-\frac {4 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{11 b d}-\frac {4 \left (\frac {10 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{3 b d}-\frac {\frac {2 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {5 \left (\frac {4 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}+\frac {3}{5} \left (\frac {2 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}-\frac {4 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )\right )\right )}{7 b}}{3 b}\right )}{11 b}}{195 b^2}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}\) |
(4*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(39*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(15*b*d) - ((2* (80*a^2 - 221*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2)) /(11*b*d) - (4*((10*a*(8*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x]*(a + b*Si n[c + d*x])^(5/2))/(3*b*d) - ((2*(160*a^4 - 375*a^2*b^2 + 117*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(7*b*d) - (5*((4*a*(32*a^4 - 47*a^2*b^2 - 27*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*d) + (3*((2*(64*a^6 - 174*a^4*b^2 + 81*a^2*b^4 - 195*b^6)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x] ])/(3*d) + ((-4*a*(32*a^6 - 111*a^4*b^2 + 102*a^2*b^4 - 471*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*(64*a^8 - 238*a^6*b^2 + 255*a^4*b^4 - 27 6*a^2*b^6 + 195*b^8)*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))/(7*b)) /(3*b)))/(11*b))/(195*b^2)
3.12.51.3.1 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. \(1800\) vs. \(2(554)=1108\).
Time = 2.94 (sec) , antiderivative size = 1801, normalized size of antiderivative = 3.41
-2/45045*(780*((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^9-256*((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^9-428*a^5*b^4+128*a^7*b^2+256*(( 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^8*b-192*((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^2-952*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si n(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^3+684*((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^ 4+1020*((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^5-3480*((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^6-1104*((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)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.22 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.30 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} {\left (128 \, a^{8} - 492 \, a^{6} b^{2} + 561 \, a^{4} b^{4} + 114 \, a^{2} b^{6} + 585 \, b^{8}\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 (128 \, a^{8} - 492 \, a^{6} b^{2} + 561 \, a^{4} b^{4} + 114 \, a^{2} b^{6} + 585 \, b^{8}\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 ) - 12 \, \sqrt {2} {\left (-32 i \, a^{7} b + 111 i \, a^{5} b^{3} - 102 i \, a^{3} b^{5} + 471 i \, a 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 ) - 12 \, \sqrt {2} {\left (32 i \, a^{7} b - 111 i \, a^{5} b^{3} + 102 i \, a^{3} b^{5} - 471 i \, a 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 (3003 \, b^{8} \cos \left (d x + c\right )^{7} - 21 \, {\left (3 \, a^{2} b^{6} + 208 \, b^{8}\right )} \cos \left (d x + c\right )^{5} + 5 \, {\left (16 \, a^{4} b^{4} - 27 \, a^{2} b^{6} + 39 \, b^{8}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (64 \, a^{6} b^{2} - 174 \, a^{4} b^{4} + 81 \, a^{2} b^{6} - 195 \, b^{8}\right )} \cos \left (d x + c\right ) - 2 \, {\left (1848 \, a b^{7} \cos \left (d x + c\right )^{5} + 35 \, {\left (a^{3} b^{5} - 15 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (16 \, a^{5} b^{3} - 41 \, a^{3} b^{5} + 249 \, a 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)*(128*a^8 - 492*a^6*b^2 + 561*a^4*b^4 + 114*a^2*b^6 + 5 85*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) + 2*sqrt(2)*(128*a^8 - 492*a^6*b^2 + 561*a^4*b^4 + 114*a^2*b^6 + 585*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 ) - 12*sqrt(2)*(-32*I*a^7*b + 111*I*a^5*b^3 - 102*I*a^3*b^5 + 471*I*a*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)) - 12*sqrt(2)*(32*I*a^7*b - 111*I*a^5*b^3 + 102*I*a^3*b^5 - 471*I*a*b^7)*sqr t(-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*(3003*b^8*cos(d*x + c)^7 - 21*(3*a^2*b^6 + 208*b^8)*cos(d*x + c)^5 + 5*( 16*a^4*b^4 - 27*a^2*b^6 + 39*b^8)*cos(d*x + c)^3 - 2*(64*a^6*b^2 - 174*a^4 *b^4 + 81*a^2*b^6 - 195*b^8)*cos(d*x + c) - 2*(1848*a*b^7*cos(d*x + c)^5 + 35*(a^3*b^5 - 15*a*b^7)*cos(d*x + c)^3 - 3*(16*a^5*b^3 - 41*a^3*b^5 + 249 *a*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^7*d)
Timed out. \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2} \,d x } \]
\[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]