\(\int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx\) [313]

Optimal result

Integrand size = 37, antiderivative size = 309 \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx=-\frac {\sqrt {2} (A-B) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} (c-d)^3 f}+\frac {\left (A d \left (15 c^2+10 c d+7 d^2\right )-B \left (3 c^3+6 c^2 d+19 c d^2+4 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 \sqrt {a} (c-d)^3 \sqrt {d} (c+d)^{5/2} f}-\frac {(B c-A d) \cos (e+f x)}{2 \left (c^2-d^2\right ) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}+\frac {\left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{4 \left (c^2-d^2\right )^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \] Output:

-2^(1/2)*(A-B)*arctanh(1/2*a^(1/2)*cos(f*x+e)*2^(1/2)/(a+a*sin(f*x+e))^(1/ 
2))/a^(1/2)/(c-d)^3/f+1/4*(A*d*(15*c^2+10*c*d+7*d^2)-B*(3*c^3+6*c^2*d+19*c 
*d^2+4*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^(1/2)/(c-d)^3/d^(1/2)/(c+d)^(5/2)/f-1/2*(-A*d+B*c)*cos(f*x+e)/ 
(c^2-d^2)/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2+1/4*(A*d*(7*c+d)-B*( 
3*c^2+c*d+4*d^2))*cos(f*x+e)/(c^2-d^2)^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 = 14.46 (sec) , antiderivative size = 1209, normalized size of antiderivative = 3.91 \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((2 + 2*I)*(A - B)*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]))/((( 
-1)^(1/4)*c^3 - 3*(-1)^(1/4)*c^2*d + 3*(-1)^(1/4)*c*d^2 - (-1)^(1/4)*d^3)* 
f*Sqrt[a*(1 + Sin[e + f*x])]) - ((-(A*d*(15*c^2 + 10*c*d + 7*d^2)) + B*(3* 
c^3 + 6*c^2*d + 19*c*d^2 + 4*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]*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]))/(16*(c - 
 d)^3*Sqrt[d]*(c + d)^(5/2)*f*Sqrt[a*(1 + Sin[e + f*x])]) + ((-(A*d*(15*c^ 
2 + 10*c*d + 7*d^2)) + B*(3*c^3 + 6*c^2*d + 19*c*d^2 + 4*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]*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]))/(16*(c - d)^3*Sqrt[d]*(c + d)^(5/2)*f*Sqrt[a*(1 + S...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3463

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3463

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^2 \left (8 A c^2-5 B c^2+9 A d c-15 B d c+7 A d^2-4 B d^2\right )-a^2 \left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int -\frac {\left (B \left (5 c^2+15 d c+4 d^2\right )-A \left (8 c^2+9 d c+7 d^2\right )\right ) a^2+\left (A d (7 c+d)-B \left (3 c^2+d c+4 d^2\right )\right ) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a \left (A d (7 c+d)-B \left (3 c^2+c d+4 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a \left (c^2-d^2\right )}-\frac {(B c-A d) \cos (e+f x)}{2 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}\)

Input:

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

Output:

$Aborted
 

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 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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2276\) vs. \(2(276)=552\).

Time = 0.51 (sec) , antiderivative size = 2277, normalized size of antiderivative = 7.37

method	result	size
default	\(\text {Expression too large to display}\)	\(2277\)

Input:

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^3,x,method=_R 
ETURNVERBOSE)
 

Output:

-1/4*(8*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^4*c*d^3-4*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^4*c^2*d^ 
2-8*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^4*c*d^3+8*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)*a^4*c^3*d+16*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)*a^4*c^2*d^2+8*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)*a^4*c*d^3-8*B*2^(1/2)*a 
rctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*((c+d)*a*d)^(1/2)*si 
n(f*x+e)*a^4*c^3*d-16*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^4*c^2*d^2-8*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^4*c*d^3+4*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^4*c^2*d^2+3*B*a^(9/2)*arctanh((-a*(s 
in(f*x+e)-1))^(1/2)*d/((c+d)*a*d)^(1/2))*c^5+4*B*a^(9/2)*arctanh((-a*(sin( 
f*x+e)-1))^(1/2)*d/((c+d)*a*d)^(1/2))*c^2*d^3-A*a^(7/2)*(-a*(sin(f*x+e)-1) 
)^(1/2)*((c+d)*a*d)^(1/2)*d^4+5*B*a^(7/2)*(-a*(sin(f*x+e)-1))^(1/2)*((c+d) 
*a*d)^(1/2)*c^4-4*B*a^(7/2)*(-a*(sin(f*x+e)-1))^(1/2)*((c+d)*a*d)^(1/2)*d^ 
4-A*a^(5/2)*(-a*(sin(f*x+e)-1))^(3/2)*((c+d)*a*d)^(1/2)*d^4+4*B*a^(5/2)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1963 vs. \(2 (276) = 552\).

Time = 4.24 (sec) , antiderivative size = 4180, normalized size of antiderivative = 13.53 \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^3,x, al 
gorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))**3,x)
 

Output:

Timed out
 

Maxima [F(-1)]

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

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^3,x, al 
gorithm="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

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

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

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^ 
3),x)
 

Output:

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^ 
3), x)
 

Reduce [F]

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

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^3,x)
                                                                                    
                                                                                    
 

Output:

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