\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx\) [326]

Optimal result

Integrand size = 37, antiderivative size = 395 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=-\frac {\left (B \left (5 c^2-58 c d-43 d^2\right )+A \left (3 c^2-22 c d+115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^4 f}+\frac {d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} (c-d)^4 (c+d)^{3/2} f}-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))}-\frac {(3 A c+5 B c-15 A d+7 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \] Output:

-1/32*(B*(5*c^2-58*c*d-43*d^2)+A*(3*c^2-22*c*d+115*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)^4/f+d^ 
(3/2)*(A*d*(7*c+5*d)-B*(5*c^2+5*c*d+2*d^2))*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)^4/(c+d)^(3/2)/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))-1/16*(3* 
A*c-15*A*d+5*B*c+7*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))-1/16*d*(A*(3*c^2-16*c*d-35*d^2)+B*(5*c^2+32*c*d+11*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))
 

Mathematica [C] (warning: unable to verify)

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

Time = 19.92 (sec) , antiderivative size = 1680, normalized size of antiderivative = 4.25 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((1 + I)*(3*A*c^2 + 5*B*c^2 - 22*A*c*d - 58*B*c*d + 115*A*d^2 - 43*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^4 
- 64*(-1)^(1/4)*c^3*d + 96*(-1)^(1/4)*c^2*d^2 - 64*(-1)^(1/4)*c*d^3 + 16*( 
-1)^(1/4)*d^4)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(A*d*(7*c + 5*d) 
 - B*(5*c^2 + 5*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + RootS 
um[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 + T 
an[(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 + Ta 
n[(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)/(4*(c - d)^ 
4*(c + d)^(3/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(-(A*d*(7*c + 5 
*d)) + B*(5*c^2 + 5*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + R 
ootSum[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]*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)/(4*(...
 

Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.11, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {3042, 3457, 27, 3042, 3457, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3457

\(\displaystyle -\frac {\int -\frac {a (3 A c+5 B c-10 A d+2 B d)+5 a (A-B) d \sin (e+f x)}{2 (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2}dx}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (3 A c+5 B c-10 A d+2 B d)+5 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2}dx}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (3 A c+5 B c-10 A d+2 B d)+5 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2}dx}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {-\frac {\int -\frac {\left (B \left (5 c^2-43 d c-22 d^2\right )+A \left (3 c^2-13 d c+70 d^2\right )\right ) a^2+3 d (3 A (c-5 d)+B (5 c+7 d)) \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{2 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (B \left (5 c^2-43 d c-22 d^2\right )+A \left (3 c^2-13 d c+70 d^2\right )\right ) a^2+3 d (3 A (c-5 d)+B (5 c+7 d)) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (B \left (5 c^2-43 d c-22 d^2\right )+A \left (3 c^2-13 d c+70 d^2\right )\right ) a^2+3 d (3 A (c-5 d)+B (5 c+7 d)) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3463

\(\displaystyle \frac {\frac {-\frac {\int -\frac {\left (B \left (5 c^3-48 d c^2-69 d^2 c-32 d^3\right )+A \left (3 c^3-16 d c^2+77 d^2 c+80 d^3\right )\right ) a^3+d \left (A \left (3 c^2-16 d c-35 d^2\right )+B \left (5 c^2+32 d c+11 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (B \left (5 c^3-48 d c^2-69 d^2 c-32 d^3\right )+A \left (3 c^3-16 d c^2+77 d^2 c+80 d^3\right )\right ) a^3+d \left (A \left (3 c^2-16 d c-35 d^2\right )+B \left (5 c^2+32 d c+11 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (B \left (5 c^3-48 d c^2-69 d^2 c-32 d^3\right )+A \left (3 c^3-16 d c^2+77 d^2 c+80 d^3\right )\right ) a^3+d \left (A \left (3 c^2-16 d c-35 d^2\right )+B \left (5 c^2+32 d c+11 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3464

\(\displaystyle \frac {\frac {\frac {\frac {a^3 (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {16 a^2 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {a^3 (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {16 a^2 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3128

\(\displaystyle \frac {\frac {\frac {-\frac {16 a^2 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {2 a^3 (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {-\frac {16 a^2 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {\sqrt {2} a^{5/2} (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {\frac {\frac {32 a^3 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {1}{a (c+d)-\frac {a^2 d \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}-\frac {\sqrt {2} a^{5/2} (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {32 a^{5/2} d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d) \sqrt {c+d}}-\frac {\sqrt {2} a^{5/2} (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

Input:

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^ 
2),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])) + (-1/2*(a*(3*A*c + 5*B*c - 15*A*d + 7*B*d)*Cos[e + f*x])/(( 
c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])) + ((-((Sqrt[2]*a 
^(5/2)*(c + d)*(B*(5*c^2 - 58*c*d - 43*d^2) + A*(3*c^2 - 22*c*d + 115*d^2) 
)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/((c 
- d)*f)) + (32*a^(5/2)*d^(3/2)*(A*d*(7*c + 5*d) - B*(5*c^2 + 5*c*d + 2*d^2 
))*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^2*d*(A*(3*c^2 - 1 
6*c*d - 35*d^2) + B*(5*c^2 + 32*c*d + 11*d^2))*Cos[e + f*x])/((c^2 - d^2)* 
f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])))/(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. \(4091\) vs. \(2(358)=716\).

Time = 0.56 (sec) , antiderivative size = 4092, normalized size of antiderivative = 10.36

method	result	size
default	\(\text {Expression too large to display}\)	\(4092\)

Input:

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

Output:

1/32*(6*A*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*((c+d)*a*d)^(1/2)*c^4+101*B*2^ 
(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^( 
1/2)*sin(f*x+e)*a^2*c^3*d+255*B*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1 
/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e)*a^2*c^2*d^2+43*B*2^(1/2) 
*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)* 
sin(f*x+e)^2*a^2*c^3*d-3*A*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2 
^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e)^3*a^2*c^3*d+19*A*2^(1/2)*arct 
anh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*sin(f 
*x+e)^3*a^2*c^2*d^2-93*A*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^( 
1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e)^3*a^2*c*d^3-5*B*2^(1/2)*arctanh 
(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+ 
e)^3*a^2*c^3*d+53*B*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/ 
a^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e)^3*a^2*c^2*d^2+101*B*2^(1/2)*arctanh( 
1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e 
)^3*a^2*c*d^3+13*A*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a 
^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e)^2*a^2*c^3*d-55*A*2^(1/2)*arctanh(1/2* 
(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e)^2* 
a^2*c^2*d^2-301*A*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^ 
(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e)^2*a^2*c*d^3+187*B*2^(1/2)*arctanh(1/2* 
(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*sin(f*x+e)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2433 vs. \(2 (358) = 716\).

Time = 7.95 (sec) , antiderivative size = 5151, normalized size of antiderivative = 13.04 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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