\(\int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx\) [806]

Optimal result

Integrand size = 29, antiderivative size = 688 \[ \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 b^2 \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}+\frac {8 b^2 \left (a b c-2 a^2 d+b^2 d\right ) \cos (e+f x)}{3 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (3 a^4 d^3-b^4 d \left (5 c^2-8 d^2\right )+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {a+b} \left (a^2-b^2\right ) (c-d) \sqrt {c+d} (b c-a d)^4 f}-\frac {2 \left (3 a^2 b (2 c-3 d) d-3 a^3 d^2-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {a+b} \left (a^2-b^2\right ) (c-d) \sqrt {c+d} (b c-a d)^3 f} \] Output:

2/3*b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^(3/2)/(c+d*sin( 
f*x+e))^(1/2)+8/3*b^2*(-2*a^2*d+a*b*c+b^2*d)*cos(f*x+e)/(a^2-b^2)^2/(-a*d+ 
b*c)^2/f/(a+b*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)+2/3*(3*a^4*d^3-b^4* 
d*(5*c^2-8*d^2)+3*a^2*b^2*d*(3*c^2-5*d^2)-4*a*b^3*c*(c^2-d^2))*EllipticE(( 
c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b 
)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c 
+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^ 
(1/2)*(c+d*sin(f*x+e))/(a+b)^(1/2)/(a^2-b^2)/(c-d)/(c+d)^(1/2)/(-a*d+b*c)^ 
4/f-2/3*(3*a^2*b*(2*c-3*d)*d-3*a^3*d^2-3*a*b^2*(c^2-2*d^2)+b^3*(c^2-6*c*d+ 
8*d^2))*EllipticF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin( 
f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*((-a*d+b*c)*(1-s 
in(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b) 
/(c+d*sin(f*x+e)))^(1/2)*(c+d*sin(f*x+e))/(a+b)^(1/2)/(a^2-b^2)/(c-d)/(c+d 
)^(1/2)/(-a*d+b*c)^3/f
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2352\) vs. \(2(688)=1376\).

Time = 7.25 (sec) , antiderivative size = 2352, normalized size of antiderivative = 3.42 \[ \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]
 

Output:

(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((2*b^3*Cos[e + f*x])/( 
3*(a^2 - b^2)*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x])^2) - (2*(4*a*b^4*c*Cos 
[e + f*x] - 9*a^2*b^3*d*Cos[e + f*x] + 5*b^5*d*Cos[e + f*x]))/(3*(a^2 - b^ 
2)^2*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((b*c - 
 a*d)^3*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(-3*a^ 
2*b^3*c^4 - b^5*c^4 + 9*a^3*b^2*c^3*d - 5*a*b^4*c^3*d - 9*a^4*b*c^2*d^2 + 
20*a^2*b^3*c^2*d^2 - 7*b^5*c^2*d^2 + 3*a^5*c*d^3 - 15*a^3*b^2*c*d^3 + 8*a* 
b^4*c*d^3 + 9*a^4*b*d^4 - 17*a^2*b^3*d^4 + 8*b^5*d^4)*Sqrt[((c + d)*Cot[(- 
e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + 
Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c 
) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt 
[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)] 
*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 
 a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]] 
) - 4*(-(b*c) + a*d)*(-4*a*b^4*c^4 + 5*a^2*b^3*c^3*d - 5*b^5*c^3*d + 9*a^3 
*b^2*c^2*d^2 - a*b^4*c^2*d^2 + 3*a^4*b*c*d^3 - 11*a^2*b^3*c*d^3 + 8*b^5*c* 
d^3 + 3*a^5*d^4 - 15*a^3*b^2*d^4 + 8*a*b^4*d^4)*((Sqrt[((c + d)*Cot[(-e + 
Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 
 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + 
a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[...
 

Rubi [A] (verified)

Time = 2.63 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 3281, 27, 3042, 3534, 27, 3042, 3477, 3042, 3297, 3475}

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.

\(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 3281

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-5 d^2\right ) a^2-6 b^3 c d a-b^4 \left (c^2+8 d^2\right )+2 b \left (3 d^2 a^3+3 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-5 d^2\right ) a^2-6 b^3 c d a-b^4 \left (c^2+8 d^2\right )+2 b \left (3 d^2 a^3+3 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-5 d^2\right ) a^2-6 b^3 c d a-b^4 \left (c^2+8 d^2\right )+2 b \left (3 d^2 a^3+3 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}+\frac {(a-b) \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}+\frac {(a-b) \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3297

\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}+\frac {2 (a-b) \sqrt {a+b} \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3475

\(\displaystyle \frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}+\frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}-\frac {2 (a-b) \sqrt {a+b} \left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)^2}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}\)

Input:

Int[1/((a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]
 

Output:

(2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/ 
2)*Sqrt[c + d*Sin[e + f*x]]) + ((8*b^2*(a*b*c - 2*a^2*d + b^2*d)*Cos[e + f 
*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e 
+ f*x]]) - ((-2*(a - b)*Sqrt[a + b]*(3*a^4*d^3 - b^4*d*(5*c^2 - 8*d^2) + 3 
*a^2*b^2*d*(3*c^2 - 5*d^2) - 4*a*b^3*c*(c^2 - d^2))*EllipticE[ArcSin[(Sqrt 
[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], 
 ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - 
Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Si 
n[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - 
d)*Sqrt[c + d]*(b*c - a*d)^2*f) + (2*(a - b)*Sqrt[a + b]*(3*a^2*b*(2*c - 3 
*d)*d - 3*a^3*d^2 - 3*a*b^2*(c^2 - 2*d^2) + b^3*(c^2 - 6*c*d + 8*d^2))*Ell 
ipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + 
 d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt 
[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(( 
(b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Si 
n[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f))/((a^2 - b^2)*(b*c - a*d) 
))/(3*(a^2 - b^2)*(b*c - a*d))
 

Defintions of rubi rules used

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 3281

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 
))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n 
 + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2* 
n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))
 

rule 3297

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ 
.) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d 
)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] 
)/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ 
((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S 
qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - 
b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N 
eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
 

rule 3475

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) 
*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim 
p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 
]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e 
 + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + 
 f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] 
/Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr 
eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] 
&& NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
 

rule 3477

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S 
imp[(A - B)/(a - b)   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* 
x]]), x], x] - Simp[(A*b - a*B)/(a - b)   Int[(1 + Sin[e + f*x])/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e 
, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && NeQ[A, B]
 

rule 3534

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(214173\) vs. \(2(642)=1284\).

Time = 20.04 (sec) , antiderivative size = 214174, normalized size of antiderivative = 311.30

method	result	size
default	\(\text {Expression too large to display}\)	\(214174\)

Input:

int(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOS 
E)
 

Output:

result too large to display
 

Fricas [F]

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

integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((2*b^3*c*d + 3 
*a*b^2*d^2)*cos(f*x + e)^4 + (a^3 + 3*a*b^2)*c^2 + 2*(3*a^2*b + b^3)*c*d + 
 (a^3 + 3*a*b^2)*d^2 - (3*a*b^2*c^2 + 2*(3*a^2*b + 2*b^3)*c*d + (a^3 + 6*a 
*b^2)*d^2)*cos(f*x + e)^2 + (b^3*d^2*cos(f*x + e)^4 + (3*a^2*b + b^3)*c^2 
+ 2*(a^3 + 3*a*b^2)*c*d + (3*a^2*b + b^3)*d^2 - (b^3*c^2 + 6*a*b^2*c*d + ( 
3*a^2*b + 2*b^3)*d^2)*cos(f*x + e)^2)*sin(f*x + e)), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate(1/((b*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^(3/2)), x)
 

Giac [F]

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

integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate(1/((b*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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