\(\int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx\) [570]

Optimal result

Integrand size = 27, antiderivative size = 313 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=-\frac {\left (3 c^2-22 c d+115 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^4 f}+\frac {d^{5/2} (7 c+5 d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} (c-d)^4 (c+d)^{3/2} f}-\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))}-\frac {3 (c-5 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))}-\frac {(c-7 d) d (3 c+5 d) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \] Output:

-1/32*(3*c^2-22*c*d+115*d^2)*arctanh(1/2*a^(1/2)*cos(f*x+e)*2^(1/2)/(a+a*s 
in(f*x+e))^(1/2))*2^(1/2)/a^(5/2)/(c-d)^4/f+d^(5/2)*(7*c+5*d)*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*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+ 
e))-3/16*(c-5*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*(c-7*d)*d*(3*c+5*d)*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 = 7.79 (sec) , antiderivative size = 932, normalized size of antiderivative = 2.98 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(c - d)^2*Sin[(e + f*x)/2] - 4*( 
c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(3*c - 19*d)*(c - d)*Si 
n[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (c - d)*(-3*c + 1 
9*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (1 + I)*(-1)^(3/4)*(3*c^2 - 
 22*c*d + 115*d^2)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] 
*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (4*d^(5/2)*(7*c + 5*d)*(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])^4)/(c + d)^(3/2) - (4*d^(5/2)*(7*c + 5*d)*(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])^4)/(c + d)^(3/2) + (16*(c - d)*d^3*(Cos[(e + f*x)/2...
 

Rubi [A] (verified)

Time = 1.95 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.15, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 3245, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3245

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3457

\(\displaystyle \frac {-\frac {\int -\frac {\left (3 c^2-13 d c+70 d^2\right ) a^2+9 (c-5 d) d \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{2 a^2 (c-d)}-\frac {3 a (c-5 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (3 c^2-13 d c+70 d^2\right ) a^2+9 (c-5 d) d \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{4 a^2 (c-d)}-\frac {3 a (c-5 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (3 c^2-13 d c+70 d^2\right ) a^2+9 (c-5 d) d \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{4 a^2 (c-d)}-\frac {3 a (c-5 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\)

\(\Big \downarrow \) 3463

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3464

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3128

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 3252

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/4*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x 
])) + ((-3*a*(c - 5*d)*Cos[e + f*x])/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/ 
2)*(c + d*Sin[e + f*x])) + ((-((Sqrt[2]*a^(5/2)*(c + d)*(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^(5/2)*(7*c + 5*d)*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*(3*c^2 - 16*c*d - 35*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 3245

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*((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*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ 
[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] &&  !GtQ[n, 0] && (Intege 
rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 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. \(1971\) vs. \(2(276)=552\).

Time = 1.13 (sec) , antiderivative size = 1972, normalized size of antiderivative = 6.30

method	result	size
default	\(\text {Expression too large to display}\)	\(1972\)

Input:

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

Output:

-1/32/a^(9/2)*(-608*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2) 
)*a^(5/2)*sin(f*x+e)^2*c*d^4-38*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/ 
2)*a^(1/2)*sin(f*x+e)*d^4+3*2^(1/2)*(a*(c+d)*d)^(1/2)*arctanh(1/2*(-a*(-1+ 
sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^4-13*2^(1/2)*(a*(c+d)*d)^(1/2)*a 
rctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^ 
3*d+55*2^(1/2)*(a*(c+d)*d)^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^ 
(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^2*d^2+301*2^(1/2)*(a*(c+d)*d)^(1/2)*arct 
anh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c*d^3 
-35*2^(1/2)*(a*(c+d)*d)^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/ 
2)/a^(1/2))*sin(f*x+e)*a^2*c^3*d+52*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d) 
^(1/2)*a^(3/2)*c*d^3-160*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^ 
(1/2))*a^(5/2)*c*d^4+20*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/ 
2)*c^4+32*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*d^4-6*(-a*( 
-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^4-160*arctanh((-a*(-1+si 
n(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*d^5+167*2^(1/2) 
*(a*(c+d)*d)^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2)) 
*sin(f*x+e)*a^2*c^2*d^2+323*2^(1/2)*(a*(c+d)*d)^(1/2)*arctanh(1/2*(-a*(-1+ 
sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c*d^3+3*2^(1/2)*(a*(c+d 
)*d)^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x 
+e)^3*a^2*c^3*d-19*2^(1/2)*(a*(c+d)*d)^(1/2)*arctanh(1/2*(-a*(-1+sin(f*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1717 vs. \(2 (276) = 552\).

Time = 0.97 (sec) , antiderivative size = 3719, normalized size of antiderivative = 11.88 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

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

int(1/(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**3