Integrand size = 27, antiderivative size = 693 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {2 (b c-3 d)^2 \cos (e+f x) (3+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac {8 (b c-3 d)^2 \left (9 c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac {2 (b c-3 d) \left (9 d^2 \left (71 c^2+25 d^2\right )+3 b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (432 c d^3 \left (11 c^2+13 d^2\right )-18 b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-81 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (432 c d^3 \left (11 c^2+13 d^2\right )-18 b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-81 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d^3 \left (c^2-d^2\right )^4 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 (b c-3 d) \left (9 d^2 \left (71 c^2+25 d^2\right )+3 b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d^3 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}} \]
2/7*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e) )^(7/2)+8/35*(-a*d+b*c)^2*(3*a*c*d+b*(c^2-4*d^2))*cos(f*x+e)/d^2/(c^2-d^2) ^2/f/(c+d*sin(f*x+e))^(5/2)-2/105*(-a*d+b*c)*(a^2*d^2*(71*c^2+25*d^2)+a*b* (26*c^3*d-218*c*d^3)+b^2*(8*c^4-17*c^2*d^2+105*d^4))*cos(f*x+e)/d^2/(c^2-d ^2)^3/f/(c+d*sin(f*x+e))^(3/2)+2/105*(16*a^3*c*d^3*(11*c^2+13*d^2)-6*a*b^2 *c*d*(3*c^4-62*c^2*d^2-133*d^4)-9*a^2*b*d^2*(5*c^4+102*c^2*d^2+21*d^4)-b^3 *(8*c^6-23*c^4*d^2+294*c^2*d^4+105*d^6))*cos(f*x+e)/d^2/(c^2-d^2)^4/f/(c+d *sin(f*x+e))^(1/2)-2/105*(16*a^3*c*d^3*(11*c^2+13*d^2)-6*a*b^2*c*d*(3*c^4- 62*c^2*d^2-133*d^4)-9*a^2*b*d^2*(5*c^4+102*c^2*d^2+21*d^4)-b^3*(8*c^6-23*c ^4*d^2+294*c^2*d^4+105*d^6))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e +1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/ 2))*(c+d*sin(f*x+e))^(1/2)/d^3/(c^2-d^2)^4/f/((c+d*sin(f*x+e))/(c+d))^(1/2 )-2/105*(-a*d+b*c)*(a^2*d^2*(71*c^2+25*d^2)+a*b*(26*c^3*d-218*c*d^3)+b^2*( 8*c^4-17*c^2*d^2+105*d^4))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1 /4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2) )*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/(c^2-d^2)^3/f/(c+d*sin(f*x+e))^(1/2)
Time = 8.72 (sec) , antiderivative size = 651, normalized size of antiderivative = 0.94 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {2 \left (\frac {\left (-d^2 \left (27 d \left (105 c^4+254 c^2 d^2+25 d^4\right )-1296 b \left (5 c^3 d^2+3 c d^4\right )-2 b^3 \left (c^5+86 c^3 d^2+105 c d^4\right )+9 b^2 \left (51 c^4 d+298 c^2 d^3+35 d^5\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (18 b^2 \left (3 c^5 d-62 c^3 d^3-133 c d^5\right )-432 \left (11 c^3 d^3+13 c d^5\right )+81 b \left (5 c^4 d^2+102 c^2 d^4+21 d^6\right )+b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) (c+d \sin (e+f x))^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d)^4 (c+d)^4}-\frac {d \cos (e+f x) \left (15 (b c-3 d)^3 \left (c^2-d^2\right )^3-9 (b c-3 d)^2 \left (c^2-d^2\right )^2 \left (3 b c^2+12 c d-7 b d^2\right ) (c+d \sin (e+f x))+\left (c^2-d^2\right ) \left (81 b \left (5 c^3 d^2+27 c d^4\right )+b^3 \left (8 c^5-17 c^3 d^2+105 c d^4\right )+9 b^2 \left (6 c^4 d-67 c^2 d^3-35 d^5\right )-27 \left (71 c^2 d^3+25 d^5\right )\right ) (c+d \sin (e+f x))^2+\left (18 b^2 \left (3 c^5 d-62 c^3 d^3-133 c d^5\right )-432 \left (11 c^3 d^3+13 c d^5\right )+81 b \left (5 c^4 d^2+102 c^2 d^4+21 d^6\right )+b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) (c+d \sin (e+f x))^3\right )}{\left (c^2-d^2\right )^4}\right )}{105 d^3 f (c+d \sin (e+f x))^{7/2}} \]
(2*(((-(d^2*(27*d*(105*c^4 + 254*c^2*d^2 + 25*d^4) - 1296*b*(5*c^3*d^2 + 3 *c*d^4) - 2*b^3*(c^5 + 86*c^3*d^2 + 105*c*d^4) + 9*b^2*(51*c^4*d + 298*c^2 *d^3 + 35*d^5))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) + (18*b^2 *(3*c^5*d - 62*c^3*d^3 - 133*c*d^5) - 432*(11*c^3*d^3 + 13*c*d^5) + 81*b*( 5*c^4*d^2 + 102*c^2*d^4 + 21*d^6) + b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*E llipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*(c + d*Sin[e + f*x])^3*Sq rt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)^4*(c + d)^4) - (d*Cos[e + f*x]* (15*(b*c - 3*d)^3*(c^2 - d^2)^3 - 9*(b*c - 3*d)^2*(c^2 - d^2)^2*(3*b*c^2 + 12*c*d - 7*b*d^2)*(c + d*Sin[e + f*x]) + (c^2 - d^2)*(81*b*(5*c^3*d^2 + 2 7*c*d^4) + b^3*(8*c^5 - 17*c^3*d^2 + 105*c*d^4) + 9*b^2*(6*c^4*d - 67*c^2* d^3 - 35*d^5) - 27*(71*c^2*d^3 + 25*d^5))*(c + d*Sin[e + f*x])^2 + (18*b^2 *(3*c^5*d - 62*c^3*d^3 - 133*c*d^5) - 432*(11*c^3*d^3 + 13*c*d^5) + 81*b*( 5*c^4*d^2 + 102*c^2*d^4 + 21*d^6) + b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*(c + d*Sin[e + f*x])^3))/(c^2 - d^2)^4))/(105*d^3*f*(c + d*Sin [e + f*x])^(7/2))
Time = 3.68 (sec) , antiderivative size = 768, normalized size of antiderivative = 1.11, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3271, 27, 3042, 3500, 27, 3042, 3233, 27, 3042, 3233, 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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {2 \int \frac {-7 c d a^3+16 b d^2 a^2-11 b^2 c d a+2 b^3 c^2-b \left (\left (4 c^2-7 d^2\right ) b^2+6 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x)+\left (5 a (b c-a d)^2-7 b \left (a b c^2+\left (a^2+b^2\right ) d c-3 a b d^2\right )\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{7/2}}dx}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\int \frac {-7 c d a^3+16 b d^2 a^2-11 b^2 c d a+2 b^3 c^2-b \left (\left (4 c^2-7 d^2\right ) b^2+6 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x)+\left (5 a (b c-a d)^2-7 b \left (a b c^2+\left (a^2+b^2\right ) d c-3 a b d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}}dx}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\int \frac {-7 c d a^3+16 b d^2 a^2-11 b^2 c d a+2 b^3 c^2-b \left (\left (4 c^2-7 d^2\right ) b^2+6 a c d b-3 a^2 d^2\right ) \sin (e+f x)^2+\left (5 a (b c-a d)^2-7 b \left (a b c^2+\left (a^2+b^2\right ) d c-3 a b d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}}dx}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {-\frac {2 \int -\frac {5 d \left (-d \left (7 c^2+5 d^2\right ) a^3+36 b c d^2 a^2-3 b^2 d \left (5 c^2+7 d^2\right ) a-2 b^3 \left (c^3-7 c d^2\right )\right )+\left (-\left (\left (8 c^4-7 d^2 c^2+35 d^4\right ) b^3\right )-18 a c d \left (c^2-7 d^2\right ) b^2-9 a^2 d^2 \left (5 c^2+7 d^2\right ) b+36 a^3 c d^3\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}-\frac {8 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\int \frac {5 d \left (-d \left (7 c^2+5 d^2\right ) a^3+36 b c d^2 a^2-3 b^2 d \left (5 c^2+7 d^2\right ) a-2 b^3 \left (c^3-7 c d^2\right )\right )+\left (-\left (\left (8 c^4-7 d^2 c^2+35 d^4\right ) b^3\right )-18 a c d \left (c^2-7 d^2\right ) b^2-9 a^2 d^2 \left (5 c^2+7 d^2\right ) b+36 a^3 c d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\int \frac {5 d \left (-d \left (7 c^2+5 d^2\right ) a^3+36 b c d^2 a^2-3 b^2 d \left (5 c^2+7 d^2\right ) a-2 b^3 \left (c^3-7 c d^2\right )\right )+\left (-\left (\left (8 c^4-7 d^2 c^2+35 d^4\right ) b^3\right )-18 a c d \left (c^2-7 d^2\right ) b^2-9 a^2 d^2 \left (5 c^2+7 d^2\right ) b+36 a^3 c d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {2 \int -\frac {3 d \left (-c d \left (35 c^2+61 d^2\right ) a^3+9 b d^2 \left (25 c^2+7 d^2\right ) a^2-3 b^2 c d \left (19 c^2+77 d^2\right ) a-b^3 \left (2 c^4-63 d^2 c^2-35 d^4\right )\right )-(b c-a d) \left (\left (8 c^4-17 d^2 c^2+105 d^4\right ) b^2+a \left (26 c^3 d-218 c d^3\right ) b+a^2 d^2 \left (71 c^2+25 d^2\right )\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\int \frac {3 d \left (-c d \left (35 c^2+61 d^2\right ) a^3+9 b d^2 \left (25 c^2+7 d^2\right ) a^2-3 b^2 c d \left (19 c^2+77 d^2\right ) a-b^3 \left (2 c^4-63 d^2 c^2-35 d^4\right )\right )-(b c-a d) \left (\left (8 c^4-17 d^2 c^2+105 d^4\right ) b^2+a \left (26 c^3 d-218 c d^3\right ) b+a^2 d^2 \left (71 c^2+25 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\int \frac {3 d \left (-c d \left (35 c^2+61 d^2\right ) a^3+9 b d^2 \left (25 c^2+7 d^2\right ) a^2-3 b^2 c d \left (19 c^2+77 d^2\right ) a-b^3 \left (2 c^4-63 d^2 c^2-35 d^4\right )\right )-(b c-a d) \left (\left (8 c^4-17 d^2 c^2+105 d^4\right ) b^2+a \left (26 c^3 d-218 c d^3\right ) b+a^2 d^2 \left (71 c^2+25 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {-\frac {2 \int -\frac {d \left (-d \left (105 c^4+254 d^2 c^2+25 d^4\right ) a^3+144 b c d^2 \left (5 c^2+3 d^2\right ) a^2-3 b^2 d \left (51 c^4+298 d^2 c^2+35 d^4\right ) a+2 b^3 \left (c^5+86 d^2 c^3+105 d^4 c\right )\right )-\left (-\left (\left (8 c^6-23 d^2 c^4+294 d^4 c^2+105 d^6\right ) b^3\right )-6 a c d \left (3 c^4-62 d^2 c^2-133 d^4\right ) b^2-9 a^2 d^2 \left (5 c^4+102 d^2 c^2+21 d^4\right ) b+16 a^3 c d^3 \left (11 c^2+13 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {\int \frac {d \left (-d \left (105 c^4+254 d^2 c^2+25 d^4\right ) a^3+144 b c d^2 \left (5 c^2+3 d^2\right ) a^2-3 b^2 d \left (51 c^4+298 d^2 c^2+35 d^4\right ) a+2 b^3 \left (c^5+86 d^2 c^3+105 d^4 c\right )\right )-\left (-\left (\left (8 c^6-23 d^2 c^4+294 d^4 c^2+105 d^6\right ) b^3\right )-6 a c d \left (3 c^4-62 d^2 c^2-133 d^4\right ) b^2-9 a^2 d^2 \left (5 c^4+102 d^2 c^2+21 d^4\right ) b+16 a^3 c d^3 \left (11 c^2+13 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {\int \frac {d \left (-d \left (105 c^4+254 d^2 c^2+25 d^4\right ) a^3+144 b c d^2 \left (5 c^2+3 d^2\right ) a^2-3 b^2 d \left (51 c^4+298 d^2 c^2+35 d^4\right ) a+2 b^3 \left (c^5+86 d^2 c^3+105 d^4 c\right )\right )-\left (-\left (\left (8 c^6-23 d^2 c^4+294 d^4 c^2+105 d^6\right ) b^3\right )-6 a c d \left (3 c^4-62 d^2 c^2-133 d^4\right ) b^2-9 a^2 d^2 \left (5 c^4+102 d^2 c^2+21 d^4\right ) b+16 a^3 c d^3 \left (11 c^2+13 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}}-\frac {\frac {\frac {2 (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}+\frac {\frac {-\frac {2 \left (c^2-d^2\right ) (b c-a d) \left (71 a^2 c^2 d^2+25 a^2 d^4+26 a b c^3 d-218 a b c d^3+8 b^2 c^4-17 b^2 c^2 d^2+105 b^2 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-\left (b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}}{5 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}}{7 d \left (c^2-d^2\right )}\) |
(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(7*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(7/2)) - ((-8*(b*c - a*d)^2*(3*a*c*d + b*(c^2 - 4*d^2))* Cos[e + f*x])/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) + ((2*(b*c - a*d)*(8*b^2*c^4 + 26*a*b*c^3*d + 71*a^2*c^2*d^2 - 17*b^2*c^2*d^2 - 218*a*b *c*d^3 + 25*a^2*d^4 + 105*b^2*d^4)*Cos[e + f*x])/(3*(c^2 - d^2)*f*(c + d*S in[e + f*x])^(3/2)) + ((-2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d*( 3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2 *d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*Ellipti cE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[ (c + d*Sin[e + f*x])/(c + d)]) - (2*(b*c - a*d)*(c^2 - d^2)*(8*b^2*c^4 + 2 6*a*b*c^3*d + 71*a^2*c^2*d^2 - 17*b^2*c^2*d^2 - 218*a*b*c*d^3 + 25*a^2*d^4 + 105*b^2*d^4)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*S in[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(c^2 - d^2))/(3*(c^ 2 - d^2)))/(5*d*(c^2 - d^2)))/(7*d*(c^2 - d^2))
3.8.44.3.1 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[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*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] && LtQ[m, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* (n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin [e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], 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] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2110\) vs. \(2(754)=1508\).
Time = 72.35 (sec) , antiderivative size = 2111, normalized size of antiderivative = 3.05
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2111\) |
| parts | \(\text {Expression too large to display}\) | \(5606\) |
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(b^3/d^3*(2*d*cos(f*x+e)^2/(c^2-d^ 2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*sin (f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e) -1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f *x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f *x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1 )*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c +d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e ))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+3*b^2*(a*d-b*c)/d^3*(2/3/(c^2-d^2)/ d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x +e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/ (3*c^4-6*c^2*d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f *x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)* cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d)) ^(1/2))+8/3*c*d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-s in(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e) -c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2) ,((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d ))^(1/2))))+3*b*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^3*(2/5/(c^2-d^2)/d^2*(-(-d*s in(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^3+16/15*c/(c^2-d^2)^2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.52 (sec) , antiderivative size = 4693, normalized size of antiderivative = 6.77 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Too large to display} \]
1/315*((sqrt(2)*(16*b^3*c^7*d^4 + 36*a*b^2*c^6*d^5 + 2*(45*a^2*b - 26*b^3) *c^5*d^6 - (37*a^3 + 285*a*b^2)*c^4*d^7 - 36*(9*a^2*b - 2*b^3)*c^3*d^8 + 2 *(173*a^3 + 543*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70*b^3)*c*d^10 + 15*(5*a^3 + 21*a*b^2)*d^11)*cos(f*x + e)^4 - 2*sqrt(2)*(48*b^3*c^9*d^2 + 108*a*b^2* c^8*d^3 + 10*(27*a^2*b - 14*b^3)*c^7*d^4 - 3*(37*a^3 + 273*a*b^2)*c^6*d^5 - 2*(441*a^2*b - 82*b^3)*c^5*d^6 + (1001*a^3 + 2973*a*b^2)*c^4*d^7 - 54*(5 7*a^2*b + 22*b^3)*c^3*d^8 + (571*a^3 + 2031*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70*b^3)*c*d^10 + 15*(5*a^3 + 21*a*b^2)*d^11)*cos(f*x + e)^2 - 4*(sqrt(2) *(16*b^3*c^8*d^3 + 36*a*b^2*c^7*d^4 + 2*(45*a^2*b - 26*b^3)*c^6*d^5 - (37* a^3 + 285*a*b^2)*c^5*d^6 - 36*(9*a^2*b - 2*b^3)*c^4*d^7 + 2*(173*a^3 + 543 *a*b^2)*c^3*d^8 - 6*(153*a^2*b + 70*b^3)*c^2*d^9 + 15*(5*a^3 + 21*a*b^2)*c *d^10)*cos(f*x + e)^2 - sqrt(2)*(16*b^3*c^10*d + 36*a*b^2*c^9*d^2 + 18*(5* a^2*b - 2*b^3)*c^8*d^3 - (37*a^3 + 249*a*b^2)*c^7*d^4 - 2*(117*a^2*b - 10* b^3)*c^6*d^5 + 3*(103*a^3 + 267*a*b^2)*c^5*d^6 - 6*(207*a^2*b + 58*b^3)*c^ 4*d^7 + (421*a^3 + 1401*a*b^2)*c^3*d^8 - 6*(153*a^2*b + 70*b^3)*c^2*d^9 + 15*(5*a^3 + 21*a*b^2)*c*d^10))*sin(f*x + e) + sqrt(2)*(16*b^3*c^11 + 36*a* b^2*c^10*d + 2*(45*a^2*b + 22*b^3)*c^9*d^2 - (37*a^3 + 69*a*b^2)*c^8*d^3 + 8*(27*a^2*b - 28*b^3)*c^7*d^4 + 4*(31*a^3 - 147*a*b^2)*c^6*d^5 - 4*(693*a ^2*b + 10*b^3)*c^5*d^6 + 2*(1057*a^3 + 3273*a*b^2)*c^4*d^7 - 72*(81*a^2*b + 34*b^3)*c^3*d^8 + 4*(199*a^3 + 744*a*b^2)*c^2*d^9 - 6*(153*a^2*b + 70...
Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]