\(\int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx\) [19]

Optimal result

Integrand size = 36, antiderivative size = 46 \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\log (\sin (e+f x)) \sec (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{f} \]

[Out]

ln(sin(f*x+e))*sec(f*x+e)*(a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {3027, 3556} \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \log (\sin (e+f x))}{f} \]

[In]

Int[Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

(Log[Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/f

Rule 3027

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

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \left (\sec (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}\right ) \int \cot (e+f x) \, dx \\ & = \frac {\log (\sin (e+f x)) \sec (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.35 \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}}{f} \]

[In]

Integrate[Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

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

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {\sec \left (f x +e \right ) \left (\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )-\ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}{f}\)	\(68\)

[In]

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

[Out]

1/f*sec(f*x+e)*(ln(csc(f*x+e)-cot(f*x+e))-ln(2/(1+cos(f*x+e))))*(a*(1+sin(f*x+e)))^(1/2)*(-c*(sin(f*x+e)-1))^(
1/2)

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 202, normalized size of antiderivative = 4.39 \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\left [\frac {\sqrt {a c} \log \left (\frac {4 \, {\left (256 \, a c \cos \left (f x + e\right )^{5} - 512 \, a c \cos \left (f x + e\right )^{3} + 337 \, a c \cos \left (f x + e\right ) + {\left (256 \, \cos \left (f x + e\right )^{4} - 512 \, \cos \left (f x + e\right )^{2} + 175\right )} \sqrt {a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )}\right )}{2 \, f}, -\frac {\sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} {\left (16 \, \cos \left (f x + e\right )^{2} - 7\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{16 \, a c \cos \left (f x + e\right )^{3} - 25 \, a c \cos \left (f x + e\right )}\right )}{f}\right ] \]

[In]

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

[Out]

[1/2*sqrt(a*c)*log(4*(256*a*c*cos(f*x + e)^5 - 512*a*c*cos(f*x + e)^3 + 337*a*c*cos(f*x + e) + (256*cos(f*x +
e)^4 - 512*cos(f*x + e)^2 + 175)*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(cos(f*x + e)^3
- cos(f*x + e)))/f, -sqrt(-a*c)*arctan(sqrt(-a*c)*(16*cos(f*x + e)^2 - 7)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*si
n(f*x + e) + c)/(16*a*c*cos(f*x + e)^3 - 25*a*c*cos(f*x + e)))/f]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.24 \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {\sqrt {a} \sqrt {c} \log \left ({\left | 2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1 \right |}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \]

[In]

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

[Out]

-sqrt(a)*sqrt(c)*log(abs(2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(
-1/4*pi + 1/2*f*x + 1/2*e))/f

Mupad [F(-1)]

Timed out. \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )} \,d x \]

[In]

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

[Out]

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