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

Optimal. Leaf size=108 \[ \frac{a \cos (e+f x) \text{Unintegrable}\left (\frac{\sec (e+f x)}{g+h x},x\right )}{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{a \cos (e+f x) \text{Unintegrable}\left (\frac{\tan (e+f x)}{g+h x},x\right )}{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}} \]

[Out]

(a*Cos[e + f*x]*Unintegrable[Sec[e + f*x]/(g + h*x), x])/(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) -
(a*Cos[e + f*x]*Unintegrable[Tan[e + f*x]/(g + h*x), x])/(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.646446, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{a-a \sin (e+f x)}}{(g+h x) \sqrt{c+c \sin (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sqrt{a-a \sin (e+f x)}}{(g+h x) \sqrt{c+c \sin (e+f x)}} \, dx &=\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+c \sin (e+f x)}}\\ &=\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+c \sin (e+f x)}}\\ &=\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+c \sin (e+f x)}}\\ &=\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+c \sin (e+f x)}}\\ &=\frac{(a \cos (e+f x)) \int \frac{\sec (e+f x)}{g+h x} \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(a \cos (e+f x)) \int \frac{\tan (e+f x)}{g+h x} \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 3.5796, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a-a \sin (e+f x)}}{(g+h x) \sqrt{c+c \sin (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{hx+g}\sqrt{a-a\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{c+c\sin \left ( fx+e \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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