\(\int \frac {1}{(a+a \sin (e+f x))^2 \sqrt {c-c \sin (e+f x)}} \, dx\) [337]

Optimal result

Integrand size = 28, antiderivative size = 124 \[ \int \frac {1}{(a+a \sin (e+f x))^2 \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{2 \sqrt {2} a^2 \sqrt {c} f}-\frac {\sec (e+f x) \sqrt {c-c \sin (e+f x)}}{2 a^2 c f}-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c^2 f} \] Output:

1/4*arctanh(1/2*c^(1/2)*cos(f*x+e)*2^(1/2)/(c-c*sin(f*x+e))^(1/2))*2^(1/2) 
/a^2/c^(1/2)/f-1/2*sec(f*x+e)*(c-c*sin(f*x+e))^(1/2)/a^2/c/f-1/3*sec(f*x+e 
)^3*(c-c*sin(f*x+e))^(3/2)/a^2/c^2/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(a+a \sin (e+f x))^2 \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \left (-5-(3+3 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-3 \sin (e+f x)\right )}{6 a^2 f (1+\sin (e+f x))^2 \sqrt {c-c \sin (e+f x)}} \] Input:

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

Output:

(Cos[e + f*x]*(-5 - (3 + 3*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 
+ Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 3*Sin[e + f 
*x]))/(6*a^2*f*(1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {3042, 3215, 3042, 3154, 3042, 3154, 3042, 3128, 219}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3215

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3154

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3154

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3128

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{2} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} f}-\frac {\sec (e+f x) \sqrt {c-c \sin (e+f x)}}{f}\right )-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 f}}{a^2 c^2}\)

Input:

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

Output:

(-1/3*(Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/f + (c*((Sqrt[c]*ArcTanh 
[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[2]*f) - 
 (Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/f))/2)/(a^2*c^2)
 

Defintions of rubi rules used

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 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 3154

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + 
f*x])^m/(a*f*g*(p + 1))), x] + Simp[a*((m + p + 1)/(g^2*(p + 1)))   Int[(g* 
Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, 
e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ 
[m + 1/2, 2*p]
 

rule 3215

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

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {\left (-1+\sin \left (f x +e \right )\right ) \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}-10 c^{\frac {7}{2}}-6 c^{\frac {7}{2}} \sin \left (f x +e \right )\right )}{12 a^{2} c^{\frac {7}{2}} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(109\)

Input:

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

Output:

-1/12/a^2*(-1+sin(f*x+e))/c^(7/2)/(1+sin(f*x+e))*(3*2^(1/2)*arctanh(1/2*(c 
*(1+sin(f*x+e)))^(1/2)*2^(1/2)/c^(1/2))*c^2*(c*(1+sin(f*x+e)))^(3/2)-10*c^ 
(7/2)-6*c^(7/2)*sin(f*x+e))/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(a+a \sin (e+f x))^2 \sqrt {c-c \sin (e+f x)}} \, dx=\frac {3 \, \sqrt {2} {\left (\cos \left (f x + e\right ) \sin \left (f x + e\right ) + \cos \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, \sqrt {-c \sin \left (f x + e\right ) + c} {\left (3 \, \sin \left (f x + e\right ) + 5\right )}}{24 \, {\left (a^{2} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} c f \cos \left (f x + e\right )\right )}} \] Input:

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

Output:

1/24*(3*sqrt(2)*(cos(f*x + e)*sin(f*x + e) + cos(f*x + e))*sqrt(c)*log(-(c 
*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e 
) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x 
+ e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + 
e) - 2)) - 4*sqrt(-c*sin(f*x + e) + c)*(3*sin(f*x + e) + 5))/(a^2*c*f*cos( 
f*x + e)*sin(f*x + e) + a^2*c*f*cos(f*x + e))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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