Integrand size = 31, antiderivative size = 401 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}-\frac {16 \left (60 a^2-49 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^4 d}+\frac {2 \left (80 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^3 d}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {8 \left (320 a^4-318 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{315 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {16 a \left (160 a^4-199 a^2 b^2+39 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{315 b^6 d \sqrt {a+b \sin (c+d x)}} \]
-2*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^3/a/b^2/d/(a+b*sin(d*x+c))^(1/2)+8/315* a*(160*a^2-139*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d-16/315*(60*a^2 -49*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^4/d+2/63*(80*a^2-6 3*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/2)/a/b^3/d-2/9*cos(d*x+ c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)/b^2/d-8/315*(320*a^4-318*a^2*b^2+21 *b^4)*(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)*Ellipt icE(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)+16/315*a*(160*a^4-199*a^2*b^2+39* b^4)*(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)*Ellipti cF(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 = 3.94 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {-32 \left (320 a^5+320 a^4 b-318 a^3 b^2-318 a^2 b^3+21 a b^4+21 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+64 a \left (160 a^4-199 a^2 b^2+39 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b \cos (c+d x) \left (-5120 a^4+4768 a^2 b^2-203 b^4-8 \left (40 a^2 b^2-21 b^4\right ) \cos (2 (c+d x))+35 b^4 \cos (4 (c+d x))-1280 a^3 b \sin (c+d x)+1012 a b^3 \sin (c+d x)+100 a b^3 \sin (3 (c+d x))\right )}{1260 b^6 d \sqrt {a+b \sin (c+d x)}} \]
(-32*(320*a^5 + 320*a^4*b - 318*a^3*b^2 - 318*a^2*b^3 + 21*a*b^4 + 21*b^5) *EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x]) /(a + b)] + 64*a*(160*a^4 - 199*a^2*b^2 + 39*b^4)*EllipticF[(-2*c + Pi - 2 *d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*Cos[c + d*x ]*(-5120*a^4 + 4768*a^2*b^2 - 203*b^4 - 8*(40*a^2*b^2 - 21*b^4)*Cos[2*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)] - 1280*a^3*b*Sin[c + d*x] + 1012*a*b^3*Si n[c + d*x] + 100*a*b^3*Sin[3*(c + d*x)]))/(1260*b^6*d*Sqrt[a + b*Sin[c + d *x]])
Time = 2.36 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.07, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.677, Rules used = {3042, 3371, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^4}{(a+b \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3371 |
| \(\displaystyle \frac {4 \int \frac {\sin ^2(c+d x) \left (-\left (\left (80 a^2-63 b^2\right ) \sin ^2(c+d x)\right )-2 a b \sin (c+d x)+15 \left (4 a^2-3 b^2\right )\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin ^2(c+d x) \left (-\left (\left (80 a^2-63 b^2\right ) \sin ^2(c+d x)\right )-2 a b \sin (c+d x)+15 \left (4 a^2-3 b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2 \left (-\left (\left (80 a^2-63 b^2\right ) \sin (c+d x)^2\right )-2 a b \sin (c+d x)+15 \left (4 a^2-3 b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int -\frac {2 \sin (c+d x) \left (-5 b \sin (c+d x) a^2-2 \left (60 a^2-49 b^2\right ) \sin ^2(c+d x) a+\left (80 a^2-63 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}+\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \int \frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^2-2 \left (60 a^2-49 b^2\right ) \sin ^2(c+d x) a+\left (80 a^2-63 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \int \frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^2-2 \left (60 a^2-49 b^2\right ) \sin (c+d x)^2 a+\left (80 a^2-63 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 \int -\frac {-3 \left (160 a^2-139 b^2\right ) \sin ^2(c+d x) a^2+4 \left (60 a^2-49 b^2\right ) a^2-b \left (40 a^2-21 b^2\right ) \sin (c+d x) a}{2 \sqrt {a+b \sin (c+d x)}}dx}{5 b}+\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\int \frac {-3 \left (160 a^2-139 b^2\right ) \sin ^2(c+d x) a^2+4 \left (60 a^2-49 b^2\right ) a^2-b \left (40 a^2-21 b^2\right ) \sin (c+d x) a}{\sqrt {a+b \sin (c+d x)}}dx}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\int \frac {-3 \left (160 a^2-139 b^2\right ) \sin (c+d x)^2 a^2+4 \left (60 a^2-49 b^2\right ) a^2-b \left (40 a^2-21 b^2\right ) \sin (c+d x) a}{\sqrt {a+b \sin (c+d x)}}dx}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {2 \int \frac {3 \left (b \left (80 a^2-57 b^2\right ) a^2+\left (320 a^4-318 b^2 a^2+21 b^4\right ) \sin (c+d x) a\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{3 b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\int \frac {b \left (80 a^2-57 b^2\right ) a^2+\left (320 a^4-318 b^2 a^2+21 b^4\right ) \sin (c+d x) a}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\int \frac {b \left (80 a^2-57 b^2\right ) a^2+\left (320 a^4-318 b^2 a^2+21 b^4\right ) \sin (c+d x) a}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {2 a^2 \left (160 a^4-199 a^2 b^2+39 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {2 a^2 \left (160 a^4-199 a^2 b^2+39 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a^2 \left (160 a^4-199 a^2 b^2+39 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a^2 \left (160 a^4-199 a^2 b^2+39 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a^2 \left (160 a^4-199 a^2 b^2+39 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a^2 \left (160 a^4-199 a^2 b^2+39 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a^2 \left (160 a^4-199 a^2 b^2+39 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {4 a \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {2 a^2 \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}+\frac {\frac {2 a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a^2 \left (160 a^4-199 a^2 b^2+39 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}}{b}}{5 b}\right )}{7 b}}{9 a b^2}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}\) |
(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(a*b^2*d*Sqrt[a + b*Sin[c + d *x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(9*b^2*d) + ((2*(80*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d* x]])/(7*b*d) - (4*((4*a*(60*a^2 - 49*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(5*b*d) - ((2*a^2*(160*a^2 - 139*b^2)*Cos[c + d*x]*Sqr t[a + b*Sin[c + d*x]])/(b*d) + ((2*a*(320*a^4 - 318*a^2*b^2 + 21*b^4)*Elli pticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b*d*Sq rt[(a + b*Sin[c + d*x])/(a + b)]) - (4*a^2*(160*a^4 - 199*a^2*b^2 + 39*b^4 )*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]]))/b)/(5*b)))/(7*b))/(9*a*b^2)
3.12.77.3.1 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_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1))), x] + (-Simp[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*((d*Sin[e + f* x])^(n + 1)/(b^2*d*f*(m + n + 4))), x] - Simp[1/(a*b^2*(m + 1)*(m + n + 4)) Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 2 )*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; Fre eQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && Lt Q[m, -1] && !LtQ[n, -1] && 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. \(1189\) vs. \(2(437)=874\).
Time = 2.50 (sec) , antiderivative size = 1190, normalized size of antiderivative = 2.97
2/315*(35*b^6*sin(d*x+c)^6-50*a*b^5*sin(d*x+c)^5+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^5*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^4*b^2-1592*((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^3+1044*((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^2*b^4+312*((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 *b^5-84*((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^6-1280*((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^6+2552*((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^4*b^2-1356*((a+b*si n(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.22 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.80 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (\sqrt {2} {\left (640 \, a^{5} b - 876 \, a^{3} b^{3} + 213 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (640 \, a^{6} - 876 \, a^{4} b^{2} + 213 \, a^{2} b^{4}\right )}\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 \, {\left (\sqrt {2} {\left (640 \, a^{5} b - 876 \, a^{3} b^{3} + 213 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (640 \, a^{6} - 876 \, a^{4} b^{2} + 213 \, a^{2} b^{4}\right )}\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 \, {\left (\sqrt {2} {\left (320 i \, a^{4} b^{2} - 318 i \, a^{2} b^{4} + 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (320 i \, a^{5} b - 318 i \, a^{3} b^{3} + 21 i \, a b^{5}\right )}\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 \, {\left (\sqrt {2} {\left (-320 i \, a^{4} b^{2} + 318 i \, a^{2} b^{4} - 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-320 i \, a^{5} b + 318 i \, a^{3} b^{3} - 21 i \, a b^{5}\right )}\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 (35 \, b^{6} \cos \left (d x + c\right )^{5} - {\left (80 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (320 \, a^{4} b^{2} - 318 \, a^{2} b^{4} + 21 \, b^{6}\right )} \cos \left (d x + c\right ) + 2 \, {\left (25 \, a b^{5} \cos \left (d x + c\right )^{3} - {\left (80 \, a^{3} b^{3} - 57 \, a b^{5}\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 )}}{945 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \]
-2/945*(2*(sqrt(2)*(640*a^5*b - 876*a^3*b^3 + 213*a*b^5)*sin(d*x + c) + sq rt(2)*(640*a^6 - 876*a^4*b^2 + 213*a^2*b^4))*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)*(640*a^5*b - 876*a^3 *b^3 + 213*a*b^5)*sin(d*x + c) + sqrt(2)*(640*a^6 - 876*a^4*b^2 + 213*a^2* b^4))*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) + 6*(sqrt(2)*(320*I*a^4*b^2 - 318*I*a^2*b^4 + 21*I*b^6)*sin(d*x + c) + sqrt(2)*(320*I*a^5*b - 318*I*a^3*b^3 + 21*I*a*b^5))*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^ 4*b^2 + 318*I*a^2*b^4 - 21*I*b^6)*sin(d*x + c) + sqrt(2)*(-320*I*a^5*b + 3 18*I*a^3*b^3 - 21*I*a*b^5))*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*(35*b^6*cos(d*x + c)^5 - (80*a^2*b^4 - 7* b^6)*cos(d*x + c)^3 - 2*(320*a^4*b^2 - 318*a^2*b^4 + 21*b^6)*cos(d*x + c) + 2*(25*a*b^5*cos(d*x + c)^3 - (80*a^3*b^3 - 57*a*b^5)*cos(d*x + c))*sin(d *x + c))*sqrt(b*sin(d*x + c) + a))/(b^8*d*sin(d*x + c) + a*b^7*d)
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]