\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx\) [327]

Optimal result
Mathematica [C] (warning: unable to verify)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F(-1)]
Giac [F(-2)]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 37, antiderivative size = 519 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=-\frac {\left (B \left (5 c^2-82 c d-115 d^2\right )+3 A \left (c^2-10 c d+73 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^5 f}+\frac {d^{3/2} \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{5/2} (c-d)^5 (c+d)^{5/2} f}-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \] Output:

-1/32*(B*(5*c^2-82*c*d-115*d^2)+3*A*(c^2-10*c*d+73*d^2))*arctanh(1/2*a^(1/ 
2)*cos(f*x+e)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))*2^(1/2)/a^(5/2)/(c-d)^5/f+1/ 
4*d^(3/2)*(3*A*d*(21*c^2+30*c*d+13*d^2)-B*(35*c^3+70*c^2*d+67*c*d^2+20*d^3 
))*arctanh(a^(1/2)*d^(1/2)*cos(f*x+e)/(c+d)^(1/2)/(a+a*sin(f*x+e))^(1/2))/ 
a^(5/2)/(c-d)^5/(c+d)^(5/2)/f-1/4*(A-B)*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e) 
)^(5/2)/(c+d*sin(f*x+e))^2-1/16*(3*A*c-19*A*d+5*B*c+11*B*d)*cos(f*x+e)/a/( 
c-d)^2/f/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^2-1/16*d*(A*(3*c^2-20*c*d 
-31*d^2)+B*(5*c^2+28*c*d+15*d^2))*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(a+a*sin( 
f*x+e))^(1/2)/(c+d*sin(f*x+e))^2-1/16*d*(3*A*(c^3-7*c^2*d-37*c*d^2-21*d^3) 
+B*(5*c^3+73*c^2*d+79*c*d^2+35*d^3))*cos(f*x+e)/a^2/(c-d)^4/(c+d)^2/f/(a+a 
*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 19.39 (sec) , antiderivative size = 2465, normalized size of antiderivative = 4.75 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + 
f*x])^3),x]
 

Output:

((1 + I)*(3*A*c^2 + 5*B*c^2 - 30*A*c*d - 82*B*c*d + 219*A*d^2 - 115*B*d^2) 
*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(e + f*x)/4]*(Cos[(e + f*x)/4] - Sin[( 
e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((16*(-1)^(1/4)*c^5 
 - 80*(-1)^(1/4)*c^4*d + 160*(-1)^(1/4)*c^3*d^2 - 160*(-1)^(1/4)*c^2*d^3 + 
 80*(-1)^(1/4)*c*d^4 - 16*(-1)^(1/4)*d^5)*f*(a*(1 + Sin[e + f*x]))^(5/2)) 
- (d^(3/2)*(-3*A*d*(21*c^2 + 30*c*d + 13*d^2) + B*(35*c^3 + 70*c^2*d + 67* 
c*d^2 + 20*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + RootSum[c + 4*d*#1 
 + 2*c*#1^2 - 4*d*#1^3 + c*#1^4 & , (-(d*Log[-#1 + Tan[(e + f*x)/4]]) + Sq 
rt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]] - c*Log[-#1 + Tan[(e + f*x)/ 
4]]*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1 + 3*d*Log[-# 
1 + Tan[(e + f*x)/4]]*#1^2 - Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4 
]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)/4]]*#1^3)/(-d - c*#1 + 3*d*#1^2 - c*#1 
^3) & ])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(16*(c - d)^5*(c + d)^(5 
/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(-3*A*d*(21*c^2 + 30*c*d + 
13*d^2) + B*(35*c^3 + 70*c^2*d + 67*c*d^2 + 20*d^3))*(e + f*x - 2*Log[Sec[ 
(e + f*x)/4]^2] + RootSum[c + 4*d*#1 + 2*c*#1^2 - 4*d*#1^3 + c*#1^4 & , (- 
(d*Log[-#1 + Tan[(e + f*x)/4]]) - Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f 
*x)/4]] - c*Log[-#1 + Tan[(e + f*x)/4]]*#1 - 2*Sqrt[d]*Sqrt[c + d]*Log[-#1 
 + Tan[(e + f*x)/4]]*#1 + 3*d*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 + Sqrt[d]*S 
qrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)...
 

Rubi [A] (verified)

Time = 3.36 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {3042, 3457, 27, 3042, 3457, 27, 3042, 3463, 27, 3042, 3463, 25, 3042, 3464, 3042, 3128, 219, 3252, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^3}dx\)

\(\Big \downarrow \) 3457

\(\displaystyle -\frac {\int -\frac {a (3 A c+5 B c-12 A d+4 B d)+7 a (A-B) d \sin (e+f x)}{2 (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (3 A c+5 B c-12 A d+4 B d)+7 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (3 A c+5 B c-12 A d+4 B d)+7 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {-\frac {\int -\frac {\left (B \left (5 c^2-57 d c-60 d^2\right )+A \left (3 c^2-15 d c+124 d^2\right )\right ) a^2+5 d (3 A c+5 B c-19 A d+11 B d) \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{2 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (B \left (5 c^2-57 d c-60 d^2\right )+A \left (3 c^2-15 d c+124 d^2\right )\right ) a^2+5 d (3 A c+5 B c-19 A d+11 B d) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (B \left (5 c^2-57 d c-60 d^2\right )+A \left (3 c^2-15 d c+124 d^2\right )\right ) a^2+5 d (3 A c+5 B c-19 A d+11 B d) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3463

\(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \left (\left (B \left (5 c^3-62 d c^2-113 d^2 c-70 d^3\right )+3 A \left (c^3-6 d c^2+43 d^2 c+42 d^3\right )\right ) a^3+3 d \left (A \left (3 c^2-20 d c-31 d^2\right )+B \left (5 c^2+28 d c+15 d^2\right )\right ) \sin (e+f x) a^3\right )}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{2 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (B \left (5 c^3-62 d c^2-113 d^2 c-70 d^3\right )+3 A \left (c^3-6 d c^2+43 d^2 c+42 d^3\right )\right ) a^3+3 d \left (A \left (3 c^2-20 d c-31 d^2\right )+B \left (5 c^2+28 d c+15 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (B \left (5 c^3-62 d c^2-113 d^2 c-70 d^3\right )+3 A \left (c^3-6 d c^2+43 d^2 c+42 d^3\right )\right ) a^3+3 d \left (A \left (3 c^2-20 d c-31 d^2\right )+B \left (5 c^2+28 d c+15 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3463

\(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {\left (B \left (5 c^4-67 d c^3-201 d^2 c^2-233 d^3 c-80 d^4\right )+3 A \left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right )\right ) a^4+d \left (3 A \left (c^3-7 d c^2-37 d^2 c-21 d^3\right )+B \left (5 c^3+73 d c^2+79 d^2 c+35 d^3\right )\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (B \left (5 c^4-67 d c^3-201 d^2 c^2-233 d^3 c-80 d^4\right )+3 A \left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right )\right ) a^4+d \left (3 A \left (c^3-7 d c^2-37 d^2 c-21 d^3\right )+B \left (5 c^3+73 d c^2+79 d^2 c+35 d^3\right )\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (B \left (5 c^4-67 d c^3-201 d^2 c^2-233 d^3 c-80 d^4\right )+3 A \left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right )\right ) a^4+d \left (3 A \left (c^3-7 d c^2-37 d^2 c-21 d^3\right )+B \left (5 c^3+73 d c^2+79 d^2 c+35 d^3\right )\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3464

\(\displaystyle \frac {\frac {\frac {\frac {\frac {a^4 (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {4 a^3 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\frac {a^4 (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {4 a^3 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3128

\(\displaystyle \frac {\frac {\frac {\frac {-\frac {4 a^3 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {2 a^4 (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {-\frac {4 a^3 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {\frac {\frac {\frac {8 a^4 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {1}{a (c+d)-\frac {a^2 d \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {\frac {8 a^{7/2} d^{3/2} \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d) \sqrt {c+d}}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\)

Input:

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^ 
3),x]
 

Output:

-1/4*((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*S 
in[e + f*x])^2) + (-1/2*(a*(3*A*c + 5*B*c - 19*A*d + 11*B*d)*Cos[e + f*x]) 
/((c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) + ((-2*a^2* 
d*(A*(3*c^2 - 20*c*d - 31*d^2) + B*(5*c^2 + 28*c*d + 15*d^2))*Cos[e + f*x] 
)/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + ((-((S 
qrt[2]*a^(7/2)*(c + d)^2*(B*(5*c^2 - 82*c*d - 115*d^2) + 3*A*(c^2 - 10*c*d 
 + 73*d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x 
]])])/((c - d)*f)) + (8*a^(7/2)*d^(3/2)*(3*A*d*(21*c^2 + 30*c*d + 13*d^2) 
- B*(35*c^3 + 70*c^2*d + 67*c*d^2 + 20*d^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[ 
e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/((c - d)*Sqrt[c + d]*f) 
)/(a*(c^2 - d^2)) - (2*a^3*d*(3*A*(c^3 - 7*c^2*d - 37*c*d^2 - 21*d^3) + B* 
(5*c^3 + 73*c^2*d + 79*c*d^2 + 35*d^3))*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[ 
a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])))/(a*(c^2 - d^2)))/(4*a^2*(c - d) 
))/(8*a^2*(c - d))
 

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 219
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042
Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3128
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d 
Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], 
 x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
 

rule 3252
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f)   Subst[Int[1/(b*c + a*d - d*x^2), 
x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, 
 e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3457
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( 
n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) 
  Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b 
*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 
2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ 
[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] 
 &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

rule 3463
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n 
+ 1)/(f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*(n + 1)*(c^2 - d^2))   Int[(a 
 + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 
1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], 
 x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Eq 
Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m 
 + 1/2, 0])
 

rule 3464
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + 
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A 
*b - a*B)/(b*c - a*d)   Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c 
- A*d)/(b*c - a*d)   Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], 
 x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - 
 b^2, 0] && NeQ[c^2 - d^2, 0]
 
Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7321\) vs. \(2(476)=952\).

Time = 0.64 (sec) , antiderivative size = 7322, normalized size of antiderivative = 14.11

method result size
default \(\text {Expression too large to display}\) \(7322\)

Input:

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x,method=_R 
ETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4135 vs. \(2 (476) = 952\).

Time = 24.95 (sec) , antiderivative size = 8555, normalized size of antiderivative = 16.48 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, al 
gorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**3,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, al 
gorithm="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, al 
gorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\int \frac {A+B\,\sin \left (e+f\,x\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \] Input:

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^ 
3),x)
 

Output:

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^ 
3), x)
 

Reduce [F]

\[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{6} d^{3}+3 \sin \left (f x +e \right )^{5} c \,d^{2}+3 \sin \left (f x +e \right )^{5} d^{3}+3 \sin \left (f x +e \right )^{4} c^{2} d +9 \sin \left (f x +e \right )^{4} c \,d^{2}+3 \sin \left (f x +e \right )^{4} d^{3}+\sin \left (f x +e \right )^{3} c^{3}+9 \sin \left (f x +e \right )^{3} c^{2} d +9 \sin \left (f x +e \right )^{3} c \,d^{2}+\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c^{3}+9 \sin \left (f x +e \right )^{2} c^{2} d +3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{3}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) a +\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{6} d^{3}+3 \sin \left (f x +e \right )^{5} c \,d^{2}+3 \sin \left (f x +e \right )^{5} d^{3}+3 \sin \left (f x +e \right )^{4} c^{2} d +9 \sin \left (f x +e \right )^{4} c \,d^{2}+3 \sin \left (f x +e \right )^{4} d^{3}+\sin \left (f x +e \right )^{3} c^{3}+9 \sin \left (f x +e \right )^{3} c^{2} d +9 \sin \left (f x +e \right )^{3} c \,d^{2}+\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c^{3}+9 \sin \left (f x +e \right )^{2} c^{2} d +3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{3}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) b \right )}{a^{3}} \] Input:

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x)
 

Output:

(sqrt(a)*(int(sqrt(sin(e + f*x) + 1)/(sin(e + f*x)**6*d**3 + 3*sin(e + f*x 
)**5*c*d**2 + 3*sin(e + f*x)**5*d**3 + 3*sin(e + f*x)**4*c**2*d + 9*sin(e 
+ f*x)**4*c*d**2 + 3*sin(e + f*x)**4*d**3 + sin(e + f*x)**3*c**3 + 9*sin(e 
 + f*x)**3*c**2*d + 9*sin(e + f*x)**3*c*d**2 + sin(e + f*x)**3*d**3 + 3*si 
n(e + f*x)**2*c**3 + 9*sin(e + f*x)**2*c**2*d + 3*sin(e + f*x)**2*c*d**2 + 
 3*sin(e + f*x)*c**3 + 3*sin(e + f*x)*c**2*d + c**3),x)*a + int((sqrt(sin( 
e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**6*d**3 + 3*sin(e + f*x)**5*c*d* 
*2 + 3*sin(e + f*x)**5*d**3 + 3*sin(e + f*x)**4*c**2*d + 9*sin(e + f*x)**4 
*c*d**2 + 3*sin(e + f*x)**4*d**3 + sin(e + f*x)**3*c**3 + 9*sin(e + f*x)** 
3*c**2*d + 9*sin(e + f*x)**3*c*d**2 + sin(e + f*x)**3*d**3 + 3*sin(e + f*x 
)**2*c**3 + 9*sin(e + f*x)**2*c**2*d + 3*sin(e + f*x)**2*c*d**2 + 3*sin(e 
+ f*x)*c**3 + 3*sin(e + f*x)*c**2*d + c**3),x)*b))/a**3