\(\int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx\) [765]

Optimal result

Integrand size = 27, antiderivative size = 816 \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx=\frac {d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{12 b^4 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{12 b^5 \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^3 \left (20 a^3 b c d-44 a b^3 c d+35 a^4 d^2+2 a^2 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt {c+d \sin (e+f x)}} \] Output:

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+1/4*(-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*sin(f*x+e))^(5/2)/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2-1/12*(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))*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)/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*(3 
3*c^4+70*c^2*d^2+128*d^4))*InverseJacobiAM(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))*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)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 14.25 (sec) , antiderivative size = 1526, normalized size of antiderivative = 1.87 \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

Integrate[(c + d*Sin[e + f*x])^(9/2)/(a + b*Sin[e + f*x])^3,x]
 

Output:

(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]) + 4*a*b^3*c^3*d*Cos[e + f*x] - 6*a^2*b^2*c^2*d^2*Cos[e + f*x] + 
4*a^3*b*c*d^3*Cos[e + f*x] - a^4*d^4*Cos[e + f*x])/(2*b^3*(-a^2 + b^2)*(a 
+ b*Sin[e + f*x])^2) + (6*a*b^4*c^4*Cos[e + f*x] - 7*a^2*b^3*c^3*d*Cos[e + 
 f*x] - 17*b^5*c^3*d*Cos[e + f*x] - 15*a^3*b^2*c^2*d^2*Cos[e + f*x] + 51*a 
*b^4*c^2*d^2*Cos[e + f*x] + 27*a^4*b*c*d^3*Cos[e + f*x] - 51*a^2*b^3*c*d^3 
*Cos[e + f*x] - 11*a^5*d^4*Cos[e + f*x] + 17*a^3*b^2*d^4*Cos[e + f*x])/(4* 
b^3*(-a^2 + b^2)^2*(a + b*Sin[e + f*x]))))/f - ((-2*(-48*a^2*b^3*c^5 - 24* 
b^5*c^5 + 306*a*b^4*c^4*d - 177*a^2*b^3*c^3*d^2 - 327*b^5*c^3*d^2 - 105*a^ 
3*b^2*c^2*d^3 + 501*a*b^4*c^2*d^3 + 13*a^4*b*c*d^4 - 53*a^2*b^3*c*d^4 - 10 
4*b^5*c*d^4 + 35*a^5*d^5 - 73*a^3*b^2*d^5 + 56*a*b^4*d^5)*EllipticPi[(2*b) 
/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c 
 + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-60*a^2*b^3*c^4*d - 1 
2*b^5*c^4*d + 36*a^3*b^2*c^3*d^2 + 252*a*b^4*c^3*d^2 - 228*a^4*b*c^2*d^3 + 
 276*a^2*b^3*c^2*d^3 - 480*b^5*c^2*d^3 + 140*a^5*c*d^4 - 364*a^3*b^2*c*d^4 
 + 512*a*b^4*c*d^4 + 56*a^4*b*d^5 - 112*a^2*b^3*d^5 - 16*b^5*d^5)*Cos[e + 
f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e 
 + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*Ar 
cSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqr 
t[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(...
 

Rubi [A] (verified)

Time = 7.44 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.01, 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

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\frac {\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\frac {2 d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {-\frac {2 \left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\frac {6 d (b c-a d)^3 \left (35 d^2 a^4+20 b c d a^3+2 b^2 \left (4 c^2-43 d^2\right ) a^2-44 b^3 c d a+b^4 \left (4 c^2+63 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b (a+b) f \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b f \sqrt {c+d \sin (e+f x)}}}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\)

Input:

Int[(c + d*Sin[e + f*x])^(9/2)/(a + b*Sin[e + f*x])^3,x]
 

Output:

((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))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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]
 

rule 3134

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]
 

rule 3140

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]
 

rule 3142

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]
 

rule 3271

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])
 

rule 3284

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]
 

rule 3286

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]
 

rule 3481

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]
 

rule 3526

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]
 

rule 3528

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])))
 

rule 3538

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2758\) vs. \(2(794)=1588\).

Time = 128.96 (sec) , antiderivative size = 2759, normalized size of antiderivative = 3.38

method	result	size
default	\(\text {Expression too large to display}\)	\(2759\)

Input:

int((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
 

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(d^3/b^5*(-2*b*d*(3*a*d-5*b*c)*(c/ 
d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+ 
sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*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)))+d^2*b^2*(-2/3/d*(-(-c- 
d*sin(f*x+e))*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)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d* 
sin(f*x+e))*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-s 
in(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e) 
)*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))))+12*a^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)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f 
*x+e)^2)^(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)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*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)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate((c+d*sin(f*x+e))**(9/2)/(a+b*sin(f*x+e))**3,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((d*sin(f*x + e) + c)^(9/2)/(b*sin(f*x + e) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \] Input:

int((c + d*sin(e + f*x))^(9/2)/(a + b*sin(e + f*x))^3,x)
 

Output:

int((c + d*sin(e + f*x))^(9/2)/(a + b*sin(e + f*x))^3, x)
 

Reduce [F]

\[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{3} b^{3}+3 \sin \left (f x +e \right )^{2} a \,b^{2}+3 \sin \left (f x +e \right ) a^{2} b +a^{3}}d x \right ) c^{4}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{3} b^{3}+3 \sin \left (f x +e \right )^{2} a \,b^{2}+3 \sin \left (f x +e \right ) a^{2} b +a^{3}}d x \right ) d^{4}+4 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{3} b^{3}+3 \sin \left (f x +e \right )^{2} a \,b^{2}+3 \sin \left (f x +e \right ) a^{2} b +a^{3}}d x \right ) c \,d^{3}+6 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3} b^{3}+3 \sin \left (f x +e \right )^{2} a \,b^{2}+3 \sin \left (f x +e \right ) a^{2} b +a^{3}}d x \right ) c^{2} d^{2}+4 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3} b^{3}+3 \sin \left (f x +e \right )^{2} a \,b^{2}+3 \sin \left (f x +e \right ) a^{2} b +a^{3}}d x \right ) c^{3} d \] Input:

int((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x)
 

Output:

int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**3*b**3 + 3*sin(e + f*x)**2*a*b 
**2 + 3*sin(e + f*x)*a**2*b + a**3),x)*c**4 + int((sqrt(sin(e + f*x)*d + c 
)*sin(e + f*x)**4)/(sin(e + f*x)**3*b**3 + 3*sin(e + f*x)**2*a*b**2 + 3*si 
n(e + f*x)*a**2*b + a**3),x)*d**4 + 4*int((sqrt(sin(e + f*x)*d + c)*sin(e 
+ f*x)**3)/(sin(e + f*x)**3*b**3 + 3*sin(e + f*x)**2*a*b**2 + 3*sin(e + f* 
x)*a**2*b + a**3),x)*c*d**3 + 6*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x) 
**2)/(sin(e + f*x)**3*b**3 + 3*sin(e + f*x)**2*a*b**2 + 3*sin(e + f*x)*a** 
2*b + a**3),x)*c**2*d**2 + 4*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/( 
sin(e + f*x)**3*b**3 + 3*sin(e + f*x)**2*a*b**2 + 3*sin(e + f*x)*a**2*b + 
a**3),x)*c**3*d