Integrand size = 27, antiderivative size = 752 \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {d \left (972 b c d^2-2835 d^3+b^4 d \left (45 c^2-8 d^2\right )-54 b^3 c \left (c^2+5 d^2\right )+9 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (9-b^2\right )^2 f}+\frac {(b c-3 d)^2 \left (18 b c+63 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (9-b^2\right )^2 f (3+b \sin (e+f x))}+\frac {(b c-3 d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (9-b^2\right ) f (3+b \sin (e+f x))^2}+\frac {\left (14985 b c d^3-25515 d^4-b^5 c d \left (51 c^2-104 d^2\right )-405 b^2 d^2 \left (3 c^2-13 d^2\right )-9 b^3 c d \left (21 c^2+361 d^2\right )+27 b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{12 b^4 \left (9-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (36450 b c d^4-76545 d^5-324 b^3 c d^2 \left (4 c^2+29 d^2\right )+81 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+18 b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-9 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{12 b^5 \left (9-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-3 d)^3 \left (540 b c d-132 b^3 c d+2835 d^2+18 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (3-b)^2 b^5 (3+b)^3 f \sqrt {c+d \sin (e+f x)}} \]
1/4*(-a*d+b*c)^2*(7*a^2*d+6*a*b*c-13*b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3 /2)/b^2/(a^2-b^2)^2/f/(a+b*sin(f*x+e))+1/2*(-a*d+b*c)^2*cos(f*x+e)*(c+d*si n(f*x+e))^(5/2)/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2+1/12*d*(36*a^3*b*c*d^2-35 *a^4*d^3+b^4*d*(45*c^2-8*d^2)-18*a*b^3*c*(c^2+5*d^2)+a^2*b^2*d*(9*c^2+61*d ^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b^3/(a^2-b^2)^2/f-1/12*(185*a^4*b*c *d^3-105*a^5*d^4-b^5*c*d*(51*c^2-104*d^2)-15*a^3*b^2*d^2*(3*c^2-13*d^2)-a^ 2*b^3*c*d*(21*c^2+361*d^2)+9*a*b^4*(2*c^4+17*c^2*d^2-8*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)*EllipticE(cos(1/2*e+1/4*P i+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/b^4/(a^2-b^2)^2 /f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/12*(150*a^5*b*c*d^4-105*a^6*d^5-12*a^3 *b^3*c*d^2*(4*c^2+29*d^2)+a^4*b^2*d^3*(26*c^2+223*d^2)-b^6*d*(57*c^4+136*c ^2*d^2+8*d^4)+6*a*b^5*c*(3*c^4+38*c^2*d^2+48*d^4)-a^2*b^4*d*(33*c^4+70*c^2 *d^2+128*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)/b^5/(a^2-b^2)^2/f/(c+d*sin(f*x+e))^(1/2)-1/4*(-a*d+b* c)^3*(20*a^3*b*c*d-44*a*b^3*c*d+35*a^4*d^2+2*a^2*b^2*(4*c^2-43*d^2)+b^4*(4 *c^2+63*d^2))*(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 )*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2))* ((c+d*sin(f*x+e))/(c+d))^(1/2)/(a-b)^2/b^5/(a+b)^3/f/(c+d*sin(f*x+e))^(1/2 )
Result contains complex when optimal does not.
Time = 11.11 (sec) , antiderivative size = 1403, normalized size of antiderivative = 1.87 \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {2 d^4 \cos (e+f x)}{3 b^3}+\frac {-b^4 c^4 \cos (e+f x)+12 b^3 c^3 d \cos (e+f x)-54 b^2 c^2 d^2 \cos (e+f x)+108 b c d^3 \cos (e+f x)-81 d^4 \cos (e+f x)}{2 b^3 \left (-9+b^2\right ) (3+b \sin (e+f x))^2}+\frac {18 b^4 c^4 \cos (e+f x)-63 b^3 c^3 d \cos (e+f x)-17 b^5 c^3 d \cos (e+f x)-405 b^2 c^2 d^2 \cos (e+f x)+153 b^4 c^2 d^2 \cos (e+f x)+2187 b c d^3 \cos (e+f x)-459 b^3 c d^3 \cos (e+f x)-2673 d^4 \cos (e+f x)+459 b^2 d^4 \cos (e+f x)}{4 b^3 \left (-9+b^2\right )^2 (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (432 b^3 c^5+24 b^5 c^5-918 b^4 c^4 d+1593 b^3 c^3 d^2+327 b^5 c^3 d^2+2835 b^2 c^2 d^3-1503 b^4 c^2 d^3-1053 b c d^4+477 b^3 c d^4+104 b^5 c d^4-8505 d^5+1971 b^2 d^5-168 b^4 d^5\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\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}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (540 b^3 c^4 d+12 b^5 c^4 d-972 b^2 c^3 d^2-756 b^4 c^3 d^2+18468 b c^2 d^3-2484 b^3 c^2 d^3+480 b^5 c^2 d^3-34020 c d^4+9828 b^2 c d^4-1536 b^4 c d^4-4536 b d^5+1008 b^3 d^5+16 b^5 d^5\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-54 b^4 c^4 d+189 b^3 c^3 d^2+51 b^5 c^3 d^2+1215 b^2 c^2 d^3-459 b^4 c^2 d^3-14985 b c d^4+3249 b^3 c d^4-104 b^5 c d^4+25515 d^5-5265 b^2 d^5+216 b^4 d^5\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{48 (-3+b)^2 b^3 (3+b)^2 f} \]
(Sqrt[c + d*Sin[e + f*x]]*((-2*d^4*Cos[e + f*x])/(3*b^3) + (-(b^4*c^4*Cos[ e + f*x]) + 12*b^3*c^3*d*Cos[e + f*x] - 54*b^2*c^2*d^2*Cos[e + f*x] + 108* b*c*d^3*Cos[e + f*x] - 81*d^4*Cos[e + f*x])/(2*b^3*(-9 + b^2)*(3 + b*Sin[e + f*x])^2) + (18*b^4*c^4*Cos[e + f*x] - 63*b^3*c^3*d*Cos[e + f*x] - 17*b^ 5*c^3*d*Cos[e + f*x] - 405*b^2*c^2*d^2*Cos[e + f*x] + 153*b^4*c^2*d^2*Cos[ e + f*x] + 2187*b*c*d^3*Cos[e + f*x] - 459*b^3*c*d^3*Cos[e + f*x] - 2673*d ^4*Cos[e + f*x] + 459*b^2*d^4*Cos[e + f*x])/(4*b^3*(-9 + b^2)^2*(3 + b*Sin [e + f*x]))))/f + ((-2*(432*b^3*c^5 + 24*b^5*c^5 - 918*b^4*c^4*d + 1593*b^ 3*c^3*d^2 + 327*b^5*c^3*d^2 + 2835*b^2*c^2*d^3 - 1503*b^4*c^2*d^3 - 1053*b *c*d^4 + 477*b^3*c*d^4 + 104*b^5*c*d^4 - 8505*d^5 + 1971*b^2*d^5 - 168*b^4 *d^5)*EllipticPi[(2*b)/(3 + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[( c + d*Sin[e + f*x])/(c + d)])/((3 + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)* (540*b^3*c^4*d + 12*b^5*c^4*d - 972*b^2*c^3*d^2 - 756*b^4*c^3*d^2 + 18468* b*c^2*d^3 - 2484*b^3*c^2*d^3 + 480*b^5*c^2*d^3 - 34020*c*d^4 + 9828*b^2*c* d^4 - 1536*b^4*c*d^4 - 4536*b*d^5 + 1008*b^3*d^5 + 16*b^5*d^5)*Cos[e + f*x ]*((b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + 3*d*EllipticPi[(b*(c + d))/(b*c - 3*d), I*ArcSi nh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[( d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + 3*d + b*(c + d*Sin[e + f*x])))/(b*(b*c - 3*d)*d^2*Sqrt[-(c + d)^(-1)...
Time = 7.11 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.10, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 3271, 27, 25, 3042, 3526, 27, 3042, 3528, 27, 3042, 3538, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (5 d (b c-a d)^2+d \left (-\left (\left (3 c^2-4 d^2\right ) b^2\right )+6 a c d b-7 a^2 d^2\right ) \sin ^2(e+f x)+4 b c \left (2 b c d-a \left (c^2+d^2\right )\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)\right )}{2 (a+b \sin (e+f x))^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {(c+d \sin (e+f x))^{3/2} \left (4 a b c^3-13 b^2 d c^2+14 a b d^2 c-5 a^2 d^3-d \left (-\left (\left (3 c^2-4 d^2\right ) b^2\right )+6 a c d b-7 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)\right )}{(a+b \sin (e+f x))^2}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (4 a b c^3-13 b^2 d c^2+14 a b d^2 c-5 a^2 d^3-d \left (-\left (\left (3 c^2-4 d^2\right ) b^2\right )+6 a c d b-7 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)\right )}{(a+b \sin (e+f x))^2}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (4 a b c^3-13 b^2 d c^2+14 a b d^2 c-5 a^2 d^3-d \left (-\left (\left (3 c^2-4 d^2\right ) b^2\right )+6 a c d b-7 a^2 d^2\right ) \sin (e+f x)^2+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)\right )}{(a+b \sin (e+f x))^2}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {(b c-a d)^2 \left (7 a^2 d+6 a b c-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\int -\frac {\sqrt {c+d \sin (e+f x)} \left (c^2 \left (4 c^2+63 d^2\right ) b^4-2 a c d \left (27 c^2+47 d^2\right ) b^3+a^2 \left (8 c^4+27 d^2 c^2+39 d^4\right ) b^2+28 a^3 c d^3 b-21 a^4 d^4-d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \sin ^2(e+f x)+2 d \left (7 c d^2 a^4-2 b d \left (3 c^2-d^2\right ) a^3+b^2 c \left (7 c^2-5 d^2\right ) a^2-4 b^3 d \left (3 c^2+2 d^2\right ) a-b^4 c \left (c^2-16 d^2\right )\right ) \sin (e+f x)\right )}{2 (a+b \sin (e+f x))}dx}{b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (c^2 \left (4 c^2+63 d^2\right ) b^4-2 a c d \left (27 c^2+47 d^2\right ) b^3+a^2 \left (8 c^4+27 d^2 c^2+39 d^4\right ) b^2+28 a^3 c d^3 b-21 a^4 d^4-d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \sin ^2(e+f x)+2 d \left (7 c d^2 a^4-2 b d \left (3 c^2-d^2\right ) a^3+b^2 c \left (7 c^2-5 d^2\right ) a^2-4 b^3 d \left (3 c^2+2 d^2\right ) a-b^4 c \left (c^2-16 d^2\right )\right ) \sin (e+f x)\right )}{a+b \sin (e+f x)}dx}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (c^2 \left (4 c^2+63 d^2\right ) b^4-2 a c d \left (27 c^2+47 d^2\right ) b^3+a^2 \left (8 c^4+27 d^2 c^2+39 d^4\right ) b^2+28 a^3 c d^3 b-21 a^4 d^4-d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \sin (e+f x)^2+2 d \left (7 c d^2 a^4-2 b d \left (3 c^2-d^2\right ) a^3+b^2 c \left (7 c^2-5 d^2\right ) a^2-4 b^3 d \left (3 c^2+2 d^2\right ) a-b^4 c \left (c^2-16 d^2\right )\right ) \sin (e+f x)\right )}{a+b \sin (e+f x)}dx}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {-3 c^3 \left (4 c^2+63 d^2\right ) b^5+a d \left (162 c^4+327 d^2 c^2-8 d^4\right ) b^4-3 a^2 c \left (8 c^4+33 d^2 c^2+69 d^4\right ) b^3-a^3 d^3 \left (75 c^2-61 d^2\right ) b^2+99 a^4 c d^4 b-35 a^5 d^5-d \left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sin ^2(e+f x)+2 d \left (35 c d^3 a^5-b \left (57 c^2 d^2-14 d^4\right ) a^4+b^2 c d \left (9 c^2-91 d^2\right ) a^3-b^3 \left (15 c^4-69 d^2 c^2+28 d^4\right ) a^2+b^4 c d \left (63 c^2+128 d^2\right ) a-b^5 \left (3 c^4+120 d^2 c^2+4 d^4\right )\right ) \sin (e+f x)}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{3 b}+\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {\int \frac {-3 c^3 \left (4 c^2+63 d^2\right ) b^5+a d \left (162 c^4+327 d^2 c^2-8 d^4\right ) b^4-3 a^2 c \left (8 c^4+33 d^2 c^2+69 d^4\right ) b^3-a^3 d^3 \left (75 c^2-61 d^2\right ) b^2+99 a^4 c d^4 b-35 a^5 d^5-d \left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sin ^2(e+f x)+2 d \left (35 c d^3 a^5-b \left (57 c^2 d^2-14 d^4\right ) a^4+b^2 c d \left (9 c^2-91 d^2\right ) a^3-b^3 \left (15 c^4-69 d^2 c^2+28 d^4\right ) a^2+b^4 c d \left (63 c^2+128 d^2\right ) a-b^5 \left (3 c^4+120 d^2 c^2+4 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {\int \frac {-3 c^3 \left (4 c^2+63 d^2\right ) b^5+a d \left (162 c^4+327 d^2 c^2-8 d^4\right ) b^4-3 a^2 c \left (8 c^4+33 d^2 c^2+69 d^4\right ) b^3-a^3 d^3 \left (75 c^2-61 d^2\right ) b^2+99 a^4 c d^4 b-35 a^5 d^5-d \left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sin (e+f x)^2+2 d \left (35 c d^3 a^5-b \left (57 c^2 d^2-14 d^4\right ) a^4+b^2 c d \left (9 c^2-91 d^2\right ) a^3-b^3 \left (15 c^4-69 d^2 c^2+28 d^4\right ) a^2+b^4 c d \left (63 c^2+128 d^2\right ) a-b^5 \left (3 c^4+120 d^2 c^2+4 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\left (-105 a^5 d^4+185 a^4 b c d^3-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )-b^5 c d \left (51 c^2-104 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{b}-\frac {\int \frac {d \left (105 c d^4 a^6-5 b d^3 \left (37 c^2-7 d^2\right ) a^5+3 b^2 c d^2 \left (15 c^2-98 d^2\right ) a^4+b^3 d \left (21 c^4+436 d^2 c^2-61 d^4\right ) a^3+3 b^4 c \left (2 c^4-18 d^2 c^2+93 d^4\right ) a^2-b^5 d \left (111 c^4+431 d^2 c^2-8 d^4\right ) a+3 b^6 c^3 \left (4 c^2+63 d^2\right )\right )-d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\left (-105 a^5 d^4+185 a^4 b c d^3-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )-b^5 c d \left (51 c^2-104 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{b}-\frac {\int \frac {d \left (105 c d^4 a^6-5 b d^3 \left (37 c^2-7 d^2\right ) a^5+3 b^2 c d^2 \left (15 c^2-98 d^2\right ) a^4+b^3 d \left (21 c^4+436 d^2 c^2-61 d^4\right ) a^3+3 b^4 c \left (2 c^4-18 d^2 c^2+93 d^4\right ) a^2-b^5 d \left (111 c^4+431 d^2 c^2-8 d^4\right ) a+3 b^6 c^3 \left (4 c^2+63 d^2\right )\right )-d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\frac {\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\frac {2 d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\int \frac {d \left (105 c d^4 a^6-5 b d^3 \left (37 c^2-7 d^2\right ) a^5+3 b^2 c d^2 \left (15 c^2-98 d^2\right ) a^4+b^3 d \left (21 c^4+436 d^2 c^2-61 d^4\right ) a^3+3 b^4 c \left (2 c^4-18 d^2 c^2+93 d^4\right ) a^2-b^5 d \left (111 c^4+431 d^2 c^2-8 d^4\right ) a+3 b^6 c^3 \left (4 c^2+63 d^2\right )\right )-d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {\left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\frac {\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\frac {2 d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {d \left (105 c d^4 a^6-5 b d^3 \left (37 c^2-7 d^2\right ) a^5+3 b^2 c d^2 \left (15 c^2-98 d^2\right ) a^4+b^3 d \left (21 c^4+436 d^2 c^2-61 d^4\right ) a^3+3 b^4 c \left (2 c^4-18 d^2 c^2+93 d^4\right ) a^2-b^5 d \left (111 c^4+431 d^2 c^2-8 d^4\right ) a+3 b^6 c^3 \left (4 c^2+63 d^2\right )\right )-d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\frac {\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\frac {2 d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {2 \left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {d \left (105 c d^4 a^6-5 b d^3 \left (37 c^2-7 d^2\right ) a^5+3 b^2 c d^2 \left (15 c^2-98 d^2\right ) a^4+b^3 d \left (21 c^4+436 d^2 c^2-61 d^4\right ) a^3+3 b^4 c \left (2 c^4-18 d^2 c^2+93 d^4\right ) a^2-b^5 d \left (111 c^4+431 d^2 c^2-8 d^4\right ) a+3 b^6 c^3 \left (4 c^2+63 d^2\right )\right )-d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\frac {3 d (b c-a d)^3 \left (35 a^4 d^2+20 a^3 b c d+2 a^2 b^2 \left (4 c^2-43 d^2\right )-44 a b^3 c d+b^4 \left (4 c^2+63 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {d \left (-105 a^6 d^5+150 a^5 b c d^4+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 \left (-105 a^5 d^4+185 a^4 b c d^3-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\frac {3 d (b c-a d)^3 \left (35 a^4 d^2+20 a^3 b c d+2 a^2 b^2 \left (4 c^2-43 d^2\right )-44 a b^3 c d+b^4 \left (4 c^2+63 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {d \left (-105 a^6 d^5+150 a^5 b c d^4+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 \left (-105 a^5 d^4+185 a^4 b c d^3-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\frac {3 d (b c-a d)^3 \left (35 a^4 d^2+20 a^3 b c d+2 a^2 b^2 \left (4 c^2-43 d^2\right )-44 a b^3 c d+b^4 \left (4 c^2+63 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {d \left (-105 a^6 d^5+150 a^5 b c d^4+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (-105 a^5 d^4+185 a^4 b c d^3-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\frac {3 d (b c-a d)^3 \left (35 a^4 d^2+20 a^3 b c d+2 a^2 b^2 \left (4 c^2-43 d^2\right )-44 a b^3 c d+b^4 \left (4 c^2+63 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {d \left (-105 a^6 d^5+150 a^5 b c d^4+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (-105 a^5 d^4+185 a^4 b c d^3-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 d \left (-35 a^4 d^3+36 a^3 b c d^2+a^2 b^2 d \left (9 c^2+61 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {\frac {3 d (b c-a d)^3 \left (35 a^4 d^2+20 a^3 b c d+2 a^2 b^2 \left (4 c^2-43 d^2\right )-44 a b^3 c d+b^4 \left (4 c^2+63 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {2 d \left (-105 a^6 d^5+150 a^5 b c d^4+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (-105 a^5 d^4+185 a^4 b c d^3-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}}{2 b \left (a^2-b^2\right )}+\frac {\left (7 a^2 d+6 a b c-13 b^2 d\right ) (b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\frac {\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\frac {2 d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {2 \left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {3 d (b c-a d)^3 \left (35 d^2 a^4+20 b c d a^3+2 b^2 \left (4 c^2-43 d^2\right ) a^2-44 b^3 c d a+b^4 \left (4 c^2+63 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b f \sqrt {c+d \sin (e+f x)}}}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\frac {\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\frac {2 d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {2 \left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {3 d (b c-a d)^3 \left (35 d^2 a^4+20 b c d a^3+2 b^2 \left (4 c^2-43 d^2\right ) a^2-44 b^3 c d a+b^4 \left (4 c^2+63 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b f \sqrt {c+d \sin (e+f x)}}}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\frac {\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\frac {2 d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {2 \left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {6 d (b c-a d)^3 \left (35 d^2 a^4+20 b c d a^3+2 b^2 \left (4 c^2-43 d^2\right ) a^2-44 b^3 c d a+b^4 \left (4 c^2+63 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b (a+b) f \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b f \sqrt {c+d \sin (e+f x)}}}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(2*b*(a^2 - b^2)*f *(a + b*Sin[e + f*x])^2) + (((b*c - a*d)^2*(6*a*b*c + 7*a^2*d - 13*b^2*d)* Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(b*(a^2 - b^2)*f*(a + b*Sin[e + f *x])) + ((2*d*(36*a^3*b*c*d^2 - 35*a^4*d^3 + b^4*d*(45*c^2 - 8*d^2) - 18*a *b^3*c*(c^2 + 5*d^2) + a^2*b^2*d*(9*c^2 + 61*d^2))*Cos[e + f*x]*Sqrt[c + d *Sin[e + f*x]])/(3*b*f) - ((-2*(185*a^4*b*c*d^3 - 105*a^5*d^4 - b^5*c*d*(5 1*c^2 - 104*d^2) - 15*a^3*b^2*d^2*(3*c^2 - 13*d^2) - a^2*b^3*c*d*(21*c^2 + 361*d^2) + 9*a*b^4*(2*c^4 + 17*c^2*d^2 - 8*d^4))*EllipticE[(e - Pi/2 + f* x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b*f*Sqrt[(c + d*Sin[e + f* x])/(c + d)]) - ((-2*d*(150*a^5*b*c*d^4 - 105*a^6*d^5 - 12*a^3*b^3*c*d^2*( 4*c^2 + 29*d^2) + a^4*b^2*d^3*(26*c^2 + 223*d^2) - b^6*d*(57*c^4 + 136*c^2 *d^2 + 8*d^4) + 6*a*b^5*c*(3*c^4 + 38*c^2*d^2 + 48*d^4) - a^2*b^4*d*(33*c^ 4 + 70*c^2*d^2 + 128*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sq rt[(c + d*Sin[e + f*x])/(c + d)])/(b*f*Sqrt[c + d*Sin[e + f*x]]) + (6*d*(b *c - a*d)^3*(20*a^3*b*c*d - 44*a*b^3*c*d + 35*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 43*d^2) + b^4*(4*c^2 + 63*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x )/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a + b)*f*Sqrt[ c + d*Sin[e + f*x]]))/(b*d))/(3*b))/(2*b*(a^2 - b^2)))/(4*b*(a^2 - b^2))
3.8.58.3.1 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[((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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, 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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((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 - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*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] && GtQ[m, 0] && LtQ[n, -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])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{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]
Leaf count of result is larger than twice the leaf count of optimal. \(2774\) vs. \(2(885)=1770\).
Time = 90.07 (sec) , antiderivative size = 2775, normalized size of antiderivative = 3.69
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(-20*d^2/b^6*(a^3*d^3-3*a^2*b*c*d^ 2+3*a*b^2*c^2*d-b^3*c^3)*(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+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2), (-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))+d^3/b^5*(12*d^2*a^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*s in(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+d^2*b^2*(-2/3/d*(-(-d*sin(f*x +e)-c)*cos(f*x+e)^2)^(1/2)+2/3*(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))-4/3*c/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)*c os(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))))+20*b^2*c^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) )-30*a*b*c*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+...
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
\[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]