\(\int \frac {(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx\) [519]

Optimal result

Integrand size = 27, antiderivative size = 237 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx=-\frac {(c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a+a \sin (e+f x))^2}-\frac {(c+3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) (c+2 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a^2 f \sqrt {c+d \sin (e+f x)}} \] Output:

-1/3*(c+3*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a^2/f/(1+sin(f*x+e))-1/3*(c 
-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^2+1/3*(c+3*d)*Ell 
ipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e)) 
^(1/2)/a^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/3*(c+d)*(c+2*d)*InverseJacob 
iAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d)) 
^(1/2)/a^2/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 2.44 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.19 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (-((c+3 d) (c+d \sin (e+f x)))+\frac {\left (4 d \cos \left (\frac {1}{2} (e+f x)\right )-(c+3 d) \cos \left (\frac {3}{2} (e+f x)\right )+(3 c+5 d) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-2 d^2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+(c+3 d) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{3 a^2 f (1+\sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-((c + 3*d)*(c + d*Sin[e + f*x]) 
) + ((4*d*Cos[(e + f*x)/2] - (c + 3*d)*Cos[(3*(e + f*x))/2] + (3*c + 5*d)* 
Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/ 
2])^3 - 2*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d* 
Sin[e + f*x])/(c + d)] + (c + 3*d)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/ 
4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqr 
t[(c + d*Sin[e + f*x])/(c + d)]))/(3*a^2*f*(1 + Sin[e + f*x])^2*Sqrt[c + d 
*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3244, 27, 3042, 3457, 25, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3244

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3457

\(\displaystyle \frac {-\frac {\int -\frac {2 a^2 (c-d) d^2-a^2 (c-d) d (c+3 d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {2 a^2 (c-d) d^2-a^2 (c-d) d (c+3 d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {2 a^2 (c-d) d^2-a^2 (c-d) d (c+3 d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {a^2 (c+2 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 a^2 (c-d) (c+3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2 (c-d)}-\frac {2 (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {a^2 (c+2 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c-d) (c+3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2 (c-d)}-\frac {2 (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {a^2 (c+2 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c-d) (c+3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2 (c-d)}-\frac {2 (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {2 a^2 (c+2 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c-d) (c+3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2 (c-d)}-\frac {2 (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\)

Input:

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

Output:

-1/3*((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x 
])^2) + ((-2*(c + 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + Sin[ 
e + f*x])) + ((-2*a^2*(c - d)*(c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d 
)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)] 
) + (2*a^2*(c + 2*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + 
d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(f*Sqrt[c + d*Sin[e + f*x]]))/(a^2 
*(c - d)))/(6*a^2)
 

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 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3231

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

rule 3244

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1043\) vs. \(2(222)=444\).

Time = 1.42 (sec) , antiderivative size = 1044, normalized size of antiderivative = 4.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1044\)

Input:

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

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/a^2*(2*d^2*(c/d-1)*((c+d*sin(f*x+e 
))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^( 
1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/( 
c-d))^(1/2),((c-d)/(c+d))^(1/2))+(c^2-2*c*d+d^2)*(-1/3/(c-d)*(-(-c-d*sin(f 
*x+e))*cos(f*x+e)^2)^(1/2)/(1+sin(f*x+e))^2-1/3*(-sin(f*x+e)^2*d-c*sin(f*x 
+e)+d*sin(f*x+e)+c)/(c-d)^2*(c-3*d)/((1+sin(f*x+e))*(-1+sin(f*x+e))*(-c-d* 
sin(f*x+e)))^(1/2)+2*d^2/(3*c^2-6*c*d+3*d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c- 
d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/( 
-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^ 
(1/2),((c-d)/(c+d))^(1/2))-1/3*d*(c-3*d)/(c-d)^2*(c/d-1)*((c+d*sin(f*x+e)) 
/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/ 
2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f 
*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d) 
)^(1/2),((c-d)/(c+d))^(1/2))))+2*d*(c-d)*(-(-sin(f*x+e)^2*d-c*sin(f*x+e)+d 
*sin(f*x+e)+c)/(c-d)/((1+sin(f*x+e))*(-1+sin(f*x+e))*(-c-d*sin(f*x+e)))^(1 
/2)-2*d/(2*c-2*d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e)) 
/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+ 
e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))- 
d/(c-d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1 
/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 860, normalized size of antiderivative = 3.63 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/9*(2*((c^2 + 3*c*d + 3*d^2)*cos(f*x + e)^2 - 2*c^2 - 6*c*d - 6*d^2 - (c^ 
2 + 3*c*d + 3*d^2)*cos(f*x + e) - (2*c^2 + 6*c*d + 6*d^2 + (c^2 + 3*c*d + 
3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassPInverse(-4/3* 
(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e 
) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + 2*((c^2 + 3*c*d + 3*d^2)*cos(f*x + e) 
^2 - 2*c^2 - 6*c*d - 6*d^2 - (c^2 + 3*c*d + 3*d^2)*cos(f*x + e) - (2*c^2 + 
 6*c*d + 6*d^2 + (c^2 + 3*c*d + 3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-1 
/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9* 
I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*( 
(I*c*d + 3*I*d^2)*cos(f*x + e)^2 - 2*I*c*d - 6*I*d^2 + (-I*c*d - 3*I*d^2)* 
cos(f*x + e) + (-2*I*c*d - 6*I*d^2 + (-I*c*d - 3*I*d^2)*cos(f*x + e))*sin( 
f*x + e))*sqrt(1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8 
*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/ 
27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 
 2*I*c)/d)) + 3*((-I*c*d - 3*I*d^2)*cos(f*x + e)^2 + 2*I*c*d + 6*I*d^2 + ( 
I*c*d + 3*I*d^2)*cos(f*x + e) + (2*I*c*d + 6*I*d^2 + (I*c*d + 3*I*d^2)*cos 
(f*x + e))*sin(f*x + e))*sqrt(-1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^ 
2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 
- 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3* 
I*d*sin(f*x + e) + 2*I*c)/d)) + 3*((c*d + 3*d^2)*cos(f*x + e)^2 + c*d -...
 

Sympy [F]

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

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

Output:

(Integral(c*sqrt(c + d*sin(e + f*x))/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1 
), x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)/(sin(e + f*x)**2 
+ 2*sin(e + f*x) + 1), x))/a**2
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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