\(\int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx\) [102]

Optimal result

Integrand size = 37, antiderivative size = 37 \[ \int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx=\frac {a \cos (e+f x) \text {Int}\left (\frac {\sec (e+f x)}{g+h x},x\right )}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a \cos (e+f x) \text {Int}\left (\frac {\tan (e+f x)}{g+h x},x\right )}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \] Output:

a*cos(f*x+e)*Defer(Int)(sec(f*x+e)/(h*x+g),x)/(a-a*sin(f*x+e))^(1/2)/(c+c* 
sin(f*x+e))^(1/2)-a*cos(f*x+e)*Defer(Int)(tan(f*x+e)/(h*x+g),x)/(a-a*sin(f 
*x+e))^(1/2)/(c+c*sin(f*x+e))^(1/2)
 

Mathematica [N/A]

Not integrable

Time = 4.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx=\int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx \] Input:

Integrate[Sqrt[a - a*Sin[e + f*x]]/((g + h*x)*Sqrt[c + c*Sin[e + f*x]]),x]
 

Output:

Integrate[Sqrt[a - a*Sin[e + f*x]]/((g + h*x)*Sqrt[c + c*Sin[e + f*x]]), x 
]
 

Rubi [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5115, 7292, 27, 7293, 2009}

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.

\(\displaystyle \int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c \sin (e+f x)+c}} \, dx\)

\(\Big \downarrow \) 5115

\(\displaystyle \frac {\cos (e+f x) \int \frac {\sec (e+f x) (a-a \sin (e+f x))}{g+h x}dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {\cos (e+f x) \int \frac {a \sec (e+f x) (1-\sin (e+f x))}{g+h x}dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \cos (e+f x) \int \frac {\sec (e+f x) (1-\sin (e+f x))}{g+h x}dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {a \cos (e+f x) \int \left (\frac {\sec (e+f x)}{g+h x}-\frac {\tan (e+f x)}{g+h x}\right )dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a \cos (e+f x) \left (\int \frac {\sec (e+f x)}{g+h x}dx-\int \frac {\tan (e+f x)}{g+h x}dx\right )}{\sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}\)

Input:

Int[Sqrt[a - a*Sin[e + f*x]]/((g + h*x)*Sqrt[c + c*Sin[e + f*x]]),x]
 

Output:

$Aborted
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5115

Int[((g_.) + (h_.)*(x_))^(p_.)*((a_) + (b_.)*Sin[(e_.) + (f_.)*(x_)])^(m_)* 
((c_) + (d_.)*Sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^IntPart[m] 
*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*((c + d*Sin[e + f*x])^FracPa 
rt[m]/Cos[e + f*x]^(2*FracPart[m]))   Int[(g + h*x)^p*Cos[e + f*x]^(2*m)*(c 
 + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && 
 EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ[2*m] && 
IGeQ[n - m, 0]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89

\[\int \frac {\sqrt {a -a \sin \left (f x +e \right )}}{\left (h x +g \right ) \sqrt {c +c \sin \left (f x +e \right )}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx=\int \frac {\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}{\sqrt {c \left (\sin {\left (e + f x \right )} + 1\right )} \left (g + h x\right )}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx=\int { \frac {\sqrt {-a \sin \left (f x + e\right ) + a}}{{\left (h x + g\right )} \sqrt {c \sin \left (f x + e\right ) + c}} \,d x } \] Input:

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

Output:

integrate(sqrt(-a*sin(f*x + e) + a)/((h*x + g)*sqrt(c*sin(f*x + e) + c)), 
x)
 

Giac [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx=\int { \frac {\sqrt {-a \sin \left (f x + e\right ) + a}}{{\left (h x + g\right )} \sqrt {c \sin \left (f x + e\right ) + c}} \,d x } \] Input:

integrate((a-a*sin(f*x+e))^(1/2)/(h*x+g)/(c+c*sin(f*x+e))^(1/2),x, algorit 
hm="giac")
 

Output:

integrate(sqrt(-a*sin(f*x + e) + a)/((h*x + g)*sqrt(c*sin(f*x + e) + c)), 
x)
 

Mupad [N/A]

Not integrable

Time = 16.56 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx=\int \frac {\sqrt {a-a\,\sin \left (e+f\,x\right )}}{\left (g+h\,x\right )\,\sqrt {c+c\,\sin \left (e+f\,x\right )}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {a-a \sin (e+f x)}}{(g+h x) \sqrt {c+c \sin (e+f x)}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right ) g +\sin \left (f x +e \right ) h x +g +h x}d x \right )}{c} \] Input:

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

Output:

(sqrt(c)*sqrt(a)*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1))/(s 
in(e + f*x)*g + sin(e + f*x)*h*x + g + h*x),x))/c