\(\int \frac {1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx\) [564]

Optimal result

Integrand size = 27, antiderivative size = 318 \[ \int \frac {1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=-\frac {(c-13 d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} (c-d)^4 f}-\frac {d^{3/2} \left (35 c^2+42 c d+19 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{3/2} (c-d)^4 (c+d)^{5/2} f}-\frac {\cos (e+f x)}{2 (c-d) f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d (c+2 d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d (2 c+d) (c+7 d) \cos (e+f x)}{4 a (c-d)^3 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \] Output:

-1/4*(c-13*d)*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^(3/2)/(c-d)^4/f-1/4*d^(3/2)*(35*c^2+42*c*d+19*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^(3/2)/(c-d) 
^4/(c+d)^(5/2)/f-1/2*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f* 
x+e))^2-1/2*d*(c+2*d)*cos(f*x+e)/a/(c-d)^2/(c+d)/f/(a+a*sin(f*x+e))^(1/2)/ 
(c+d*sin(f*x+e))^2-1/4*d*(2*c+d)*(c+7*d)*cos(f*x+e)/a/(c-d)^3/(c+d)^2/f/(a 
+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))
 

Mathematica [C] (warning: unable to verify)

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

Time = 7.99 (sec) , antiderivative size = 932, normalized size of antiderivative = 2.93 \[ \int \frac {1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(16*(c - d)*Sin[(e + f*x)/2] - 8*(c 
 - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (8 + 8*I)*(-1)^(3/4)*(c - 13 
*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x) 
/2] + Sin[(e + f*x)/2])^2 - (d^(3/2)*(35*c^2 + 42*c*d + 19*d^2)*(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])^2)/(c + d)^(5/2) + (d^(3/2)*(35*c^2 + 42*c*d + 19*d^2) 
*(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])^2)/(c + d)^(5/2) - (8*(c - d)^2*d^2*(Cos[(e 
 + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(( 
c + d)*(c + d*Sin[e + f*x])^2) - (4*(c - d)*d^2*(11*c + 5*d)*(Cos[(e + ...
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.14, 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, 3463, 27, 3042, 3463, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3245

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3463

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3463

\(\displaystyle \frac {\frac {-\frac {\int -\frac {\left (2 c^3-20 d c^2-35 d^2 c-19 d^3\right ) a^3+d (2 c+d) (c+7 d) \sin (e+f x) a^3}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (2 c^3-20 d c^2-35 d^2 c-19 d^3\right ) a^3+d (2 c+d) (c+7 d) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (2 c^3-20 d c^2-35 d^2 c-19 d^3\right ) a^3+d (2 c+d) (c+7 d) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3464

\(\displaystyle \frac {\frac {\frac {\frac {2 a^3 (c-13 d) (c+d)^2 \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}+\frac {a^2 d^2 \left (35 c^2+42 c d+19 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{2 a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {2 a^3 (c-13 d) (c+d)^2 \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}+\frac {a^2 d^2 \left (35 c^2+42 c d+19 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{2 a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3128

\(\displaystyle \frac {\frac {\frac {\frac {a^2 d^2 \left (35 c^2+42 c d+19 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {4 a^3 (c-13 d) (c+d)^2 \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}}{2 a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {a^2 d^2 \left (35 c^2+42 c d+19 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {2 \sqrt {2} a^{5/2} (c-13 d) (c+d)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{2 a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {\frac {-\frac {2 a^3 d^2 \left (35 c^2+42 c d+19 d^2\right ) \int \frac {1}{a (c+d)-\frac {a^2 d \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}-\frac {2 \sqrt {2} a^{5/2} (c-13 d) (c+d)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{2 a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {-\frac {2 a^{5/2} d^{3/2} \left (35 c^2+42 c d+19 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d) \sqrt {c+d}}-\frac {2 \sqrt {2} a^{5/2} (c-13 d) (c+d)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{2 a \left (c^2-d^2\right )}-\frac {a^2 d (2 c+d) (c+7 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a d (c+2 d) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\)

Input:

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

Output:

-1/2*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x 
])^2) + ((-2*a*d*(c + 2*d)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + 
 f*x]]*(c + d*Sin[e + f*x])^2) + (((-2*Sqrt[2]*a^(5/2)*(c - 13*d)*(c + d)^ 
2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/((c 
- d)*f) - (2*a^(5/2)*d^(3/2)*(35*c^2 + 42*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*S 
qrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/((c - d)*Sqr 
t[c + d]*f))/(2*a*(c^2 - d^2)) - (a^2*d*(2*c + d)*(c + 7*d)*Cos[e + f*x])/ 
((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])))/(a*(c^2 - d 
^2)))/(4*a^2*(c - d))
 

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 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 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. \(2218\) vs. \(2(279)=558\).

Time = 1.18 (sec) , antiderivative size = 2219, normalized size of antiderivative = 6.98

method	result	size
default	\(\text {Expression too large to display}\)	\(2219\)

Input:

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

Output:

-1/4*(-a*(-1+sin(f*x+e)))^(1/2)*((-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1 
/2)*a^(3/2)*c^2*d^3-13*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2 
)*c*d^4+80*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^5-2*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2) 
*sin(f*x+e)^2*d^5+35*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2 
))*a^(5/2)*sin(f*x+e)*c^4*d^2+112*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a* 
(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c^3*d^3+35*arctanh((-a*(-1+sin(f*x+e))) 
^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*c^2*d^4+42*arctanh((-a*(- 
1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*c*d^5+70*ar 
ctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2 
*c^3*d^3+119*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^2*d^4+2*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^( 
3/2)*c^4*d+38*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5 
/2)*sin(f*x+e)*c*d^5+5*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2 
)*sin(f*x+e)*d^5-11*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c 
^2*d^3+6*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c*d^4+(a*(c+ 
d)*d)^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2 
)*a^2*c^5+103*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5 
/2)*sin(f*x+e)*c^2*d^4-3*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3 
/2)*sin(f*x+e)*d^5+11*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1923 vs. \(2 (279) = 558\).

Time = 1.06 (sec) , antiderivative size = 4133, normalized size of antiderivative = 13.00 \[ \int \frac {1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

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

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

Output:

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