Integrand size = 29, antiderivative size = 1237 \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} \left (9720 b c d^3-3645 d^4+18 b^2 d^2 \left (1877 c^2+846 d^2\right )+24 b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{1920 b^2 (b c-3 d) d^2 f}-\frac {\sqrt {c+d} (b c+3 d) \left (756 b c d^3-243 d^4+84 b^3 c d \left (c^2-20 d^2\right )-18 b^2 d^2 \left (89 c^2+20 d^2\right )-b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{128 b^3 \sqrt {3+b} d^3 f}-\frac {\left (9720 b c d^3-3645 d^4+18 b^2 d^2 \left (1877 c^2+846 d^2\right )+24 b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {3+b \sin (e+f x)}}-\frac {\left (8253 b c d^2+405 d^3+3 b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}+\frac {(3+b)^{3/2} \left (3645 d^4-810 b d^3 (11 c+3 d)+270 b^2 d^2 \left (64 c^2+23 c d+22 d^2\right )+6 b^3 d \left (165 c^3+917 c^2 d+2392 c d^2+516 d^3\right )-b^4 \left (45 c^4-30 c^3 d-1692 c^2 d^2-1544 c d^3-1024 d^4\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt {c+d} f}-\frac {\left (330 b c d+837 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-21 d) \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f} \]
-1/128*(a*d+b*c)*(28*a^3*b*c*d^3-3*a^4*d^4+28*a*b^3*c*d*(c^2-20*d^2)-2*a^2 *b^2*d^2*(89*c^2+20*d^2)-b^4*(3*c^4+40*c^2*d^2+240*d^4))*EllipticPi((a+b)^ (1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),b*(c+d)/(a +b)/d,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^( 1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c) *(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^3/d^3/f/(a+b)^(1/2)+1/1920 *(c-d)*(360*a^3*b*c*d^3-45*a^4*d^4+2*a^2*b^2*d^2*(1877*c^2+846*d^2)+8*a*b^ 3*d*(45*c^3+791*c*d^2)-b^4*(45*c^4-1692*c^2*d^2-1024*d^4))*EllipticE((a+b) ^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),((a-b)*(c +d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(a+b)^(1/2)*(c+d)^(1/2 )*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1 +sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^2/d^2/(-a*d+b*c)/f-1/240*(110 *a*b*c*d+93*a^2*d^2-b^2*(15*c^2-64*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2) *(a+b*sin(f*x+e))^(1/2)/d/f+3/40*b*(-7*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e) )^(5/2)*(a+b*sin(f*x+e))^(1/2)/d/f-1/5*b^2*cos(f*x+e)*(c+d*sin(f*x+e))^(7/ 2)*(a+b*sin(f*x+e))^(1/2)/d/f+1/1920*(a+b)^(3/2)*(45*a^4*d^4-30*a^3*b*d^3* (11*c+3*d)+30*a^2*b^2*d^2*(64*c^2+23*c*d+22*d^2)+2*a*b^3*d*(165*c^3+917*c^ 2*d+2392*c*d^2+516*d^3)-b^4*(45*c^4-30*c^3*d-1692*c^2*d^2-1544*c*d^3-1024* d^4))*EllipticF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f* x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*x+e)...
Time = 12.50 (sec) , antiderivative size = 2195, normalized size of antiderivative = 1.77 \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Result too large to show} \]
((-4*(-(b*c) + 3*d)*(-15*b^4*c^4 + 103680*b*c^3*d + 13368*b^3*c^3*d + 1323 18*b^2*c^2*d^2 + 3236*b^4*c^2*d^2 + 120312*b*c*d^3 + 31320*b^3*c*d^3 - 121 5*d^4 + 29124*b^2*d^4 + 1024*b^4*d^4)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/ 2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2 *(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(- e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)* Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b) *(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(-180*b^3*c^4 + 57276*b^2*c^3*d + 2292*b^4*c^3*d + 171828*b*c^2*d^2 + 51060*b^3*c^2*d^2 - 4860*c*d^3 + 153180*b^2*c*d^3 + 4624*b^4*c*d^3 + 6188 4*b*d^4 + 13872*b^3*d^4)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin [e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d) )]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + P i/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sq rt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSi n[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*...
Time = 7.51 (sec) , antiderivative size = 1329, normalized size of antiderivative = 1.07, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.793, Rules used = {3042, 3272, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 25, 3042, 3532, 3042, 3290, 3477, 3042, 3297, 3475}
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.
| \(\displaystyle \int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^{5/2} \left (10 d a^3+7 b^2 d a-3 b^2 (b c-7 a d) \sin ^2(e+f x)+b^3 c-2 b \left (-15 d a^2+b c a-4 b^2 d\right ) \sin (e+f x)\right )}{2 \sqrt {a+b \sin (e+f x)}}dx}{5 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^{5/2} \left (10 d a^3+7 b^2 d a-3 b^2 (b c-7 a d) \sin ^2(e+f x)+b^3 c-2 b \left (-15 d a^2+b c a-4 b^2 d\right ) \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}}dx}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^{5/2} \left (10 d a^3+7 b^2 d a-3 b^2 (b c-7 a d) \sin (e+f x)^2+b^3 c-2 b \left (-15 d a^2+b c a-4 b^2 d\right ) \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}}dx}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (b^2 \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \sin ^2(e+f x)+2 b \left (40 d^2 a^3+99 b c d a^2-b^2 \left (5 c^2-91 d^2\right ) a+27 b^3 c d\right ) \sin (e+f x)+b \left (80 c d a^3+105 b d^2 a^2+62 b^2 c d a+5 b^3 c^2\right )\right )}{2 \sqrt {a+b \sin (e+f x)}}dx}{4 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (b^2 \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \sin ^2(e+f x)+2 b \left (40 d^2 a^3+99 b c d a^2-b^2 \left (5 c^2-91 d^2\right ) a+27 b^3 c d\right ) \sin (e+f x)+b \left (80 c d a^3+105 b d^2 a^2+62 b^2 c d a+5 b^3 c^2\right )\right )}{\sqrt {a+b \sin (e+f x)}}dx}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (b^2 \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \sin (e+f x)^2+2 b \left (40 d^2 a^3+99 b c d a^2-b^2 \left (5 c^2-91 d^2\right ) a+27 b^3 c d\right ) \sin (e+f x)+b \left (80 c d a^3+105 b d^2 a^2+62 b^2 c d a+5 b^3 c^2\right )\right )}{\sqrt {a+b \sin (e+f x)}}dx}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \sin ^2(e+f x) b^2+\left (3 d \left (160 c^2+93 d^2\right ) a^3+1053 b c d^2 a^2+b^2 d \left (437 c^2+192 d^2\right ) a+b^3 \left (15 c^3+64 d^2 c\right )\right ) b^2+2 \left (387 c d^2 a^3+b d \left (484 c^2+501 d^2\right ) a^2-3 b^2 c \left (5 c^2-296 d^2\right ) a+b^3 d \left (147 c^2+128 d^2\right )\right ) \sin (e+f x) b^2\right )}{2 \sqrt {a+b \sin (e+f x)}}dx}{3 b}-\frac {b \left (93 a^2 d^2+110 a b c d-\left (b^2 \left (15 c^2-64 d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \sin ^2(e+f x) b^2+\left (3 d \left (160 c^2+93 d^2\right ) a^3+1053 b c d^2 a^2+b^2 d \left (437 c^2+192 d^2\right ) a+b^3 \left (15 c^3+64 d^2 c\right )\right ) b^2+2 \left (387 c d^2 a^3+b d \left (484 c^2+501 d^2\right ) a^2-3 b^2 c \left (5 c^2-296 d^2\right ) a+b^3 d \left (147 c^2+128 d^2\right )\right ) \sin (e+f x) b^2\right )}{\sqrt {a+b \sin (e+f x)}}dx}{6 b}-\frac {b \left (93 a^2 d^2+110 a b c d-\left (b^2 \left (15 c^2-64 d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \sin (e+f x)^2 b^2+\left (3 d \left (160 c^2+93 d^2\right ) a^3+1053 b c d^2 a^2+b^2 d \left (437 c^2+192 d^2\right ) a+b^3 \left (15 c^3+64 d^2 c\right )\right ) b^2+2 \left (387 c d^2 a^3+b d \left (484 c^2+501 d^2\right ) a^2-3 b^2 c \left (5 c^2-296 d^2\right ) a+b^3 d \left (147 c^2+128 d^2\right )\right ) \sin (e+f x) b^2\right )}{\sqrt {a+b \sin (e+f x)}}dx}{6 b}-\frac {b \left (93 a^2 d^2+110 a b c d-\left (b^2 \left (15 c^2-64 d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \sin ^2(e+f x) b^2+\left (\left (15 c^4+772 d^2 c^2\right ) b^4+8 a c d \left (256 c^2+257 d^2\right ) b^3+2 a^2 d^2 \left (2737 c^2+386 d^2\right ) b^2+128 a^3 c d \left (15 c^2+16 d^2\right ) b+15 a^4 d^4\right ) b^2-2 (b c+a d) \left (15 c d^2 a^3-b d \left (1606 c^2+573 d^2\right ) a^2+b^2 c \left (15 c^2-3682 d^2\right ) a-b^3 d \left (573 c^2+1156 d^2\right )\right ) \sin (e+f x) b^2}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{2 b}-\frac {b \left (15 a^3 d^3+917 a^2 b c d^2+a b^2 d \left (345 c^2+772 d^2\right )-\left (b^3 \left (45 c^3-516 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (93 a^2 d^2+110 a b c d-\left (b^2 \left (15 c^2-64 d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \sin ^2(e+f x) b^2+\left (\left (15 c^4+772 d^2 c^2\right ) b^4+8 a c d \left (256 c^2+257 d^2\right ) b^3+2 a^2 d^2 \left (2737 c^2+386 d^2\right ) b^2+128 a^3 c d \left (15 c^2+16 d^2\right ) b+15 a^4 d^4\right ) b^2-2 (b c+a d) \left (15 c d^2 a^3-b d \left (1606 c^2+573 d^2\right ) a^2+b^2 c \left (15 c^2-3682 d^2\right ) a-b^3 d \left (573 c^2+1156 d^2\right )\right ) \sin (e+f x) b^2}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 b}-\frac {b \left (15 a^3 d^3+917 a^2 b c d^2+a b^2 d \left (345 c^2+772 d^2\right )-\left (b^3 \left (45 c^3-516 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (93 a^2 d^2+110 a b c d-\left (b^2 \left (15 c^2-64 d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \sin (e+f x)^2 b^2+\left (\left (15 c^4+772 d^2 c^2\right ) b^4+8 a c d \left (256 c^2+257 d^2\right ) b^3+2 a^2 d^2 \left (2737 c^2+386 d^2\right ) b^2+128 a^3 c d \left (15 c^2+16 d^2\right ) b+15 a^4 d^4\right ) b^2-2 (b c+a d) \left (15 c d^2 a^3-b d \left (1606 c^2+573 d^2\right ) a^2+b^2 c \left (15 c^2-3682 d^2\right ) a-b^3 d \left (573 c^2+1156 d^2\right )\right ) \sin (e+f x) b^2}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 b}-\frac {b \left (15 a^3 d^3+917 a^2 b c d^2+a b^2 d \left (345 c^2+772 d^2\right )-\left (b^3 \left (45 c^3-516 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (93 a^2 d^2+110 a b c d-\left (b^2 \left (15 c^2-64 d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}+\frac {\frac {\frac {\frac {\int -\frac {15 (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \sin ^2(e+f x) b^2+\left (-\left (\left (45 c^5-1692 d^2 c^3-1024 d^4 c\right ) b^5\right )+a d \left (375 c^4+3092 d^2 c^2-1024 d^4\right ) b^4-54 a^2 c d^2 \left (13 c^2+162 d^2\right ) b^3-2 a^3 d^3 \left (7171 c^2+1618 d^2\right ) b^2-a^4 c d^2 \left (3840 c^2+4501 d^2\right ) b+15 a^5 d^5\right ) b^2-2 \left (15 c d^4 a^5+b d^3 \left (2822 c^2+1161 d^2\right ) a^4+2 b^2 c d^2 \left (674 c^2+4433 d^2\right ) a^3-6 b^3 d \left (65 c^4-1276 d^2 c^2-514 d^4\right ) a^2+b^4 c \left (45 c^4+1502 d^2 c^2+3344 d^4\right ) a+b^5 c^2 d \left (15 c^2+772 d^2\right )\right ) \sin (e+f x) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}+\frac {\frac {\frac {-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b^2}{d f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {15 (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \sin ^2(e+f x) b^2+\left (-\left (\left (45 c^5-1692 d^2 c^3-1024 d^4 c\right ) b^5\right )+a d \left (375 c^4+3092 d^2 c^2-1024 d^4\right ) b^4-54 a^2 c d^2 \left (13 c^2+162 d^2\right ) b^3-2 a^3 d^3 \left (7171 c^2+1618 d^2\right ) b^2-a^4 c d^2 \left (3840 c^2+4501 d^2\right ) b+15 a^5 d^5\right ) b^2-2 \left (15 c d^4 a^5+b d^3 \left (2822 c^2+1161 d^2\right ) a^4+2 b^2 c d^2 \left (674 c^2+4433 d^2\right ) a^3-6 b^3 d \left (65 c^4-1276 d^2 c^2-514 d^4\right ) a^2+b^4 c \left (45 c^4+1502 d^2 c^2+3344 d^4\right ) a+b^5 c^2 d \left (15 c^2+772 d^2\right )\right ) \sin (e+f x) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}}{4 b}-\frac {b \left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}+\frac {\frac {\frac {-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b^2}{d f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {15 (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \sin (e+f x)^2 b^2+\left (-\left (\left (45 c^5-1692 d^2 c^3-1024 d^4 c\right ) b^5\right )+a d \left (375 c^4+3092 d^2 c^2-1024 d^4\right ) b^4-54 a^2 c d^2 \left (13 c^2+162 d^2\right ) b^3-2 a^3 d^3 \left (7171 c^2+1618 d^2\right ) b^2-a^4 c d^2 \left (3840 c^2+4501 d^2\right ) b+15 a^5 d^5\right ) b^2-2 \left (15 c d^4 a^5+b d^3 \left (2822 c^2+1161 d^2\right ) a^4+2 b^2 c d^2 \left (674 c^2+4433 d^2\right ) a^3-6 b^3 d \left (65 c^4-1276 d^2 c^2-514 d^4\right ) a^2+b^4 c \left (45 c^4+1502 d^2 c^2+3344 d^4\right ) a+b^5 c^2 d \left (15 c^2+772 d^2\right )\right ) \sin (e+f x) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}}{4 b}-\frac {b \left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {15 (a d+b c) \left (-3 a^4 d^4+28 a^3 b c d^3-2 a^2 b^2 d^2 \left (89 c^2+20 d^2\right )+28 a b^3 c d \left (c^2-20 d^2\right )-\left (b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx+\frac {\int \frac {2 \left (a^2-b^2\right ) d (b c-a d) \left (15 c^3 b^3+772 c d^2 b^3+516 a d^3 b^2+917 a c^2 d b^2+345 a^2 c d^2 b-45 a^3 d^3\right ) \sin (e+f x) b^3+\left (a^2-b^2\right ) (b c-a d) \left (45 c^4 b^4-1024 d^4 b^4-1692 c^2 d^2 b^4-4784 a c d^3 b^3-330 a c^3 d b^3-660 a^2 d^4 b^2-1920 a^2 c^2 d^2 b^2+330 a^3 c d^3 b-45 a^4 d^4\right ) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {b^2 \left (-45 a^4 d^4+360 a^3 b c d^3+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )+8 a b^3 c d \left (45 c^2+791 d^2\right )-\left (b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (15 a^3 d^3+917 a^2 b c d^2+a b^2 d \left (345 c^2+772 d^2\right )-\left (b^3 \left (45 c^3-516 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (93 a^2 d^2+110 a b c d-\left (b^2 \left (15 c^2-64 d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {15 (a d+b c) \left (-3 a^4 d^4+28 a^3 b c d^3-2 a^2 b^2 d^2 \left (89 c^2+20 d^2\right )+28 a b^3 c d \left (c^2-20 d^2\right )-\left (b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx+\frac {\int \frac {2 \left (a^2-b^2\right ) d (b c-a d) \left (15 c^3 b^3+772 c d^2 b^3+516 a d^3 b^2+917 a c^2 d b^2+345 a^2 c d^2 b-45 a^3 d^3\right ) \sin (e+f x) b^3+\left (a^2-b^2\right ) (b c-a d) \left (45 c^4 b^4-1024 d^4 b^4-1692 c^2 d^2 b^4-4784 a c d^3 b^3-330 a c^3 d b^3-660 a^2 d^4 b^2-1920 a^2 c^2 d^2 b^2+330 a^3 c d^3 b-45 a^4 d^4\right ) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {b^2 \left (-45 a^4 d^4+360 a^3 b c d^3+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )+8 a b^3 c d \left (45 c^2+791 d^2\right )-\left (b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (15 a^3 d^3+917 a^2 b c d^2+a b^2 d \left (345 c^2+772 d^2\right )-\left (b^3 \left (45 c^3-516 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (93 a^2 d^2+110 a b c d-\left (b^2 \left (15 c^2-64 d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}+\frac {\frac {\frac {-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b^2}{d f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {30 \sqrt {c+d} (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {\int \frac {2 \left (a^2-b^2\right ) d (b c-a d) \left (15 c^3 b^3+772 c d^2 b^3+516 a d^3 b^2+917 a c^2 d b^2+345 a^2 c d^2 b-45 a^3 d^3\right ) \sin (e+f x) b^3+\left (a^2-b^2\right ) (b c-a d) \left (45 c^4 b^4-1024 d^4 b^4-1692 c^2 d^2 b^4-4784 a c d^3 b^3-330 a c^3 d b^3-660 a^2 d^4 b^2-1920 a^2 c^2 d^2 b^2+330 a^3 c d^3 b-45 a^4 d^4\right ) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}}{4 b}-\frac {b \left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}+\frac {\frac {\frac {-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b^2}{d f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {30 \sqrt {c+d} (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {b^3 (a+b) (b c-a d) \left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right ) b^4\right )+2 a d \left (165 c^3+917 d c^2+2392 d^2 c+516 d^3\right ) b^3+30 a^2 d^2 \left (64 c^2+23 d c+22 d^2\right ) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}}{b^2}}{2 d}}{4 b}-\frac {b \left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}+\frac {\frac {\frac {-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b^2}{d f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {30 \sqrt {c+d} (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {b^3 (a+b) (b c-a d) \left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right ) b^4\right )+2 a d \left (165 c^3+917 d c^2+2392 d^2 c+516 d^3\right ) b^3+30 a^2 d^2 \left (64 c^2+23 d c+22 d^2\right ) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}}{b^2}}{2 d}}{4 b}-\frac {b \left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}+\frac {\frac {\frac {-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b^2}{d f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {30 \sqrt {c+d} (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {b^3 (a+b) (b c-a d) \left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {2 b^2 \sqrt {a+b} \left (a^2-b^2\right ) \left (-\left (\left (45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right ) b^4\right )+2 a d \left (165 c^3+917 d c^2+2392 d^2 c+516 d^3\right ) b^3+30 a^2 d^2 \left (64 c^2+23 d c+22 d^2\right ) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) \sqrt {c+d} f}}{b^2}}{2 d}}{4 b}-\frac {b \left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 f}+\frac {\frac {\frac {-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)} b^2}{d f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {30 \sqrt {c+d} (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {-\frac {2 \sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) b^3}{(b c-a d) f}-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \left (-\left (\left (45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right ) b^4\right )+2 a d \left (165 c^3+917 d c^2+2392 d^2 c+516 d^3\right ) b^3+30 a^2 d^2 \left (64 c^2+23 d c+22 d^2\right ) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) b^2}{(a-b) \sqrt {c+d} f}}{b^2}}{2 d}}{4 b}-\frac {b \left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {b \left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 b}}{10 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
-1/5*(b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2) )/(d*f) + ((3*b*(b*c - 7*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d *Sin[e + f*x])^(5/2))/(4*f) + (-1/3*(b*(110*a*b*c*d + 93*a^2*d^2 - b^2*(15 *c^2 - 64*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x]) ^(3/2))/f + (-1/2*(b*(917*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(345*c^2 + 77 2*d^2) - b^3*(45*c^3 - 516*c*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*S qrt[c + d*Sin[e + f*x]])/f + (-((b^2*(360*a^3*b*c*d^3 - 45*a^4*d^4 + 8*a*b ^3*c*d*(45*c^2 + 791*d^2) + 2*a^2*b^2*d^2*(1877*c^2 + 846*d^2) - b^4*(45*c ^4 - 1692*c^2*d^2 - 1024*d^4))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f *Sqrt[a + b*Sin[e + f*x]])) - ((30*Sqrt[c + d]*(b*c + a*d)*(28*a^3*b*c*d^3 - 3*a^4*d^4 + 28*a*b^3*c*d*(c^2 - 20*d^2) - 2*a^2*b^2*d^2*(89*c^2 + 20*d^ 2) - b^4*(3*c^4 + 40*c^2*d^2 + 240*d^4))*EllipticPi[(b*(c + d))/((a + b)*d ), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*S in[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(( (b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b* c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*d*f) + ((-2*b^3*Sqrt[a + b]*(c - d)*Sqrt[c + d]*(360 *a^3*b*c*d^3 - 45*a^4*d^4 + 8*a*b^3*c*d*(45*c^2 + 791*d^2) + 2*a^2*b^2*d^2 *(1877*c^2 + 846*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*EllipticE[ ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*S...
3.8.77.3.1 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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/ (c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/( c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*(( c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ .) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d )*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] )/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ ((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) *(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 ]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] /Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
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])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x ]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] & & NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a *d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 40.48 (sec) , antiderivative size = 680680, normalized size of antiderivative = 550.27
Timed out. \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]