\(\int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx\) [22]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3025, 2700, 29} \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \log (\tan (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 3025

Int[1/(sin[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)]]), x_Symbol] :> Dist[Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), Int[1/(Cos[e + f*
x]*Sin[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \int \csc (e+f x) \sec (e+f x) \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tan (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \log (\tan (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {(\log (\cos (e+f x))-\log (\sin (e+f x))) \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}}{a c f} \]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(42)=84\).

Time = 1.79 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.98

method	result	size
default	\(\frac {\cos \left (f x +e \right ) \left (\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )-\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )\right )}{f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\)	\(91\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 193, normalized size of antiderivative = 4.20 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\left [\frac {\sqrt {a c} \log \left (-\frac {4 \, {\left (2 \, a c \cos \left (f x + e\right )^{5} - 2 \, a c \cos \left (f x + e\right )^{3} + a c \cos \left (f x + e\right ) - \sqrt {a c} {\left (2 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{\cos \left (f x + e\right )^{5} - \cos \left (f x + e\right )^{3}}\right )}{2 \, a c f}, \frac {\sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2 \, a c \cos \left (f x + e\right )^{3} - a c \cos \left (f x + e\right )}\right )}{a c f}\right ] \]

[In]

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

[Out]

[1/2*sqrt(a*c)*log(-4*(2*a*c*cos(f*x + e)^5 - 2*a*c*cos(f*x + e)^3 + a*c*cos(f*x + e) - sqrt(a*c)*(2*cos(f*x +
e)^2 - 1)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(cos(f*x + e)^5 - cos(f*x + e)^3))/(a*c*f), sqr
t(-a*c)*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(2*a*c*cos(f*x + e)^3 - a*c*cos(f
*x + e)))/(a*c*f)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 190, normalized size of antiderivative = 4.13 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {\left (-1\right )^{4 \, \cos \left (2 \, f x + 2 \, e\right )} \cosh \left (4 \, \pi \sin \left (2 \, f x + 2 \, e\right )\right ) \log \left (\frac {16 \, {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}{a c {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}^{2}}\right ) - 2 i \, \left (-1\right )^{4 \, \cos \left (2 \, f x + 2 \, e\right )} \arctan \left (\frac {4 \, \sin \left (2 \, f x + 2 \, e\right )}{\sqrt {a} \sqrt {c} {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}}, \frac {4 \, {\left (\cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}{\sqrt {a} \sqrt {c} {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}}\right ) \sinh \left (4 \, \pi \sin \left (2 \, f x + 2 \, e\right )\right )}{2 \, \sqrt {a} \sqrt {c} f} \]

[In]

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

[Out]

-1/2*((-1)^(4*cos(2*f*x + 2*e))*cosh(4*pi*sin(2*f*x + 2*e))*log(16*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 +
2*cos(2*f*x + 2*e) + 1)/(a*c*abs(e^(2*I*f*x + 2*I*e) - 1)^2)) - 2*I*(-1)^(4*cos(2*f*x + 2*e))*arctan2(4*sin(2*
f*x + 2*e)/(sqrt(a)*sqrt(c)*abs(e^(2*I*f*x + 2*I*e) - 1)), 4*(cos(2*f*x + 2*e) + 1)/(sqrt(a)*sqrt(c)*abs(e^(2*
I*f*x + 2*I*e) - 1)))*sinh(4*pi*sin(2*f*x + 2*e)))/(sqrt(a)*sqrt(c)*f)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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