\(\int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx\) [516]

Optimal result

Integrand size = 27, antiderivative size = 244 \[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {d (c+3 d) \cos (e+f x)}{a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {(c+3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{a (c-d)^2 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a (c-d) f \sqrt {c+d \sin (e+f x)}} \] Output:

-d*(c+3*d)*cos(f*x+e)/a/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^(1/2)-cos(f*x+e)/ 
(c-d)/f/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2)+(c+3*d)*EllipticE(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/(c-d)^ 
2/(c+d)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+InverseJacobiAM(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/(c-d)/f/(c+d 
*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 1.89 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (-\frac {2 \left ((c+d)^2 \cos \left (\frac {1}{2} (e+f x)\right )+d (2 (c+d)+(c+3 d) \cos (e+f x)) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}+(c+3 d) (c+d \sin (e+f x))+\left (c^2+4 c d+3 d^2\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-\left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{a (c-d)^2 (c+d) f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3247, 27, 3042, 3233, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3247

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

Input:

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

Output:

-(Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])) 
- (d*((2*a*(c + 3*d)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]] 
) + ((2*a*(c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + 
d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(c^2 - d^ 
2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/ 
(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(c^2 - d^2)))/(2*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 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 3233

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

rule 3247

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( 
n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) 
   Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre 
eQ[{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[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(924\) vs. \(2(236)=472\).

Time = 1.07 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.79

method	result	size
default	\(\frac {\sqrt {\cos \left (f x +e \right )^{2} \sin \left (f x +e \right ) d +\cos \left (f x +e \right )^{2} c}\, \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}+3 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}-3 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-4 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d +4 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-c \,d^{2} \cos \left (f x +e \right )^{2}-3 d^{3} \cos \left (f x +e \right )^{2}+\sin \left (f x +e \right ) c^{2} d -d^{3} \sin \left (f x +e \right )-c^{2} d +d^{3}\right )}{d \left (c^{2}-d^{2}\right ) \sqrt {-\left (c +d \sin \left (f x +e \right )\right ) \left (-1+\sin \left (f x +e \right )\right ) \left (1+\sin \left (f x +e \right )\right )}\, \left (c -d \right ) a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(925\)

Input:

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

Output:

(cos(f*x+e)^2*sin(f*x+e)*d+cos(f*x+e)^2*c)^(1/2)*((d/(c-d)*sin(f*x+e)+1/(c 
-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c 
-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1 
/2))*c^3+3*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+ 
d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e 
)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c^2*d-(d/(c-d)*sin(f*x+e)+1/(c-d)* 
c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d)) 
^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2)) 
*c*d^2-3*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d) 
)^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+ 
1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*d^3-4*(d/(c-d)*sin(f*x+e)+1/(c-d)*c) 
^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^( 
1/2)*EllipticF((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c 
^2*d+4*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^ 
(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticF((d/(c-d)*sin(f*x+e)+1/ 
(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*d^3-c*d^2*cos(f*x+e)^2-3*d^3*cos(f*x+e 
)^2+sin(f*x+e)*c^2*d-d^3*sin(f*x+e)-c^2*d+d^3)/d/(c^2-d^2)/(-(c+d*sin(f*x+ 
e))*(-1+sin(f*x+e))*(1+sin(f*x+e)))^(1/2)/(c-d)/a/cos(f*x+e)/(c+d*sin(f*x+ 
e))^(1/2)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 1092, normalized size of antiderivative = 4.48 \[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*((2*c^3 - c^2*d - 6*c*d^2 - 3*d^3 - (2*c^2*d - 3*c*d^2 - 3*d^3)*cos(f 
*x + e)^2 + (2*c^3 - 3*c^2*d - 3*c*d^2)*cos(f*x + e) + (2*c^3 - c^2*d - 6* 
c*d^2 - 3*d^3 + (2*c^2*d - 3*c*d^2 - 3*d^3)*cos(f*x + e))*sin(f*x + e))*sq 
rt(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^3 - c^2*d - 6*c*d^2 - 3*d^3 - (2*c^2*d - 3*c*d^2 - 3*d^3)*cos(f*x + e 
)^2 + (2*c^3 - 3*c^2*d - 3*c*d^2)*cos(f*x + e) + (2*c^3 - c^2*d - 6*c*d^2 
- 3*d^3 + (2*c^2*d - 3*c*d^2 - 3*d^3)*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^2*d - 4*I*c*d^2 - 3*I*d^3 + (I*c*d^2 + 3*I*d^3)*cos(f*x + e)^2 + (-I*c 
^2*d - 3*I*c*d^2)*cos(f*x + e) + (-I*c^2*d - 4*I*c*d^2 - 3*I*d^3 + (-I*c*d 
^2 - 3*I*d^3)*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, weierstrassPInver 
se(-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^2*d + 4*I*c*d^2 + 3*I 
*d^3 + (-I*c*d^2 - 3*I*d^3)*cos(f*x + e)^2 + (I*c^2*d + 3*I*c*d^2)*cos(f*x 
 + e) + (I*c^2*d + 4*I*c*d^2 + 3*I*d^3 + (I*c*d^2 + 3*I*d^3)*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...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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