Integrand size = 36, antiderivative size = 46 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \log (\tan (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
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)
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {(\log (\cos (e+f x))-\log (\sin (e+f x))) \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}}{a c f} \] Input:
Integrate[Csc[e + f*x]/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) ,x]
Output:
-(((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))
Time = 0.43 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3042, 3425, 3042, 3100, 14}
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.
\(\displaystyle \int \frac {\csc (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}dx\) |
\(\Big \downarrow \) 3425 |
\(\displaystyle \frac {\cos (e+f x) \int \csc (e+f x) \sec (e+f x)dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos (e+f x) \int \csc (e+f x) \sec (e+f x)dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3100 |
\(\displaystyle \frac {\cos (e+f x) \int \cot (e+f x)d\tan (e+f x)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {\cos (e+f x) \log (\tan (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
Input:
Int[Csc[e + f*x]/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]
Output:
(Cos[e + f*x]*Log[Tan[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Si n[e + f*x]])
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[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]
Int[1/(sin[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*S qrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Cos[e + f*x]/ (Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) Int[1/(Cos[e + f*x]*S in[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]
Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(42)=84\).
Time = 11.55 (sec) , antiderivative size = 218, normalized size of antiderivative = 4.74
method | result | size |
default | \(\frac {\left (\ln \left (-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-\ln \left (-\frac {2 \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\ln \left (-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )-\ln \left (\frac {-2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\ln \left (\csc \left (\frac {f x}{2}+\frac {e}{2}\right )-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right ) \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )}{f \sqrt {\left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, \sqrt {-\left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c}}\) | \(218\) |
Input:
int(csc(f*x+e)/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x,method=_RET URNVERBOSE)
Output:
1/f*(ln(-cot(1/2*f*x+1/2*e)+csc(1/2*f*x+1/2*e)+1)-ln(-2*(cos(1/2*f*x+1/2*e )+sin(1/2*f*x+1/2*e))/(cos(1/2*f*x+1/2*e)+1))+ln(-cot(1/2*f*x+1/2*e)+csc(1 /2*f*x+1/2*e)-1)-ln(2*(-cos(1/2*f*x+1/2*e)+sin(1/2*f*x+1/2*e))/(cos(1/2*f* x+1/2*e)+1))+ln(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)))*(-1+2*cos(1/2*f*x+ 1/2*e)^2)/((2*cos(1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)+1)*a)^(1/2)/(-(2*cos(1 /2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)-1)*c)^(1/2)
Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 4.22 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\left [\frac {\sqrt {a c} \log \left (-\frac {4 \, {\left (2 \, a c \cos \left (f x + e\right )^{5} - 2 \, a c \cos \left (f x + e\right )^{3} + a c \cos \left (f x + e\right ) - \sqrt {a c} {\left (2 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{\cos \left (f x + e\right )^{5} - \cos \left (f x + e\right )^{3}}\right )}{2 \, a c f}, \frac {\sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} {\left (2 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a c \cos \left (f x + e\right )}\right )}{a c f}\right ] \] Input:
integrate(csc(f*x+e)/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algo rithm="fricas")
Output:
[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*c os(f*x + e) - sqrt(a*c)*(2*cos(f*x + e)^2 - 1)*sqrt(a*sin(f*x + e) + a)*sq rt(-c*sin(f*x + e) + c))/(cos(f*x + e)^5 - cos(f*x + e)^3))/(a*c*f), sqrt( -a*c)*arctan(sqrt(-a*c)*(2*cos(f*x + e)^2 - 1)*sqrt(a*sin(f*x + e) + a)*sq rt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)))/(a*c*f)]
\[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {\csc {\left (e + f x \right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \] Input:
integrate(csc(f*x+e)/(a+a*sin(f*x+e))**(1/2)/(c-c*sin(f*x+e))**(1/2),x)
Output:
Integral(csc(e + f*x)/(sqrt(a*(sin(e + f*x) + 1))*sqrt(-c*(sin(e + f*x) - 1))), x)
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 4.13 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {\left (-1\right )^{4 \, \cos \left (2 \, f x + 2 \, e\right )} \cosh \left (4 \, \pi \sin \left (2 \, f x + 2 \, e\right )\right ) \log \left (\frac {16 \, {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}{a c {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}^{2}}\right ) - 2 i \, \left (-1\right )^{4 \, \cos \left (2 \, f x + 2 \, e\right )} \arctan \left (\frac {4 \, \sin \left (2 \, f x + 2 \, e\right )}{\sqrt {a} \sqrt {c} {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}}, \frac {4 \, {\left (\cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}{\sqrt {a} \sqrt {c} {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}}\right ) \sinh \left (4 \, \pi \sin \left (2 \, f x + 2 \, e\right )\right )}{2 \, \sqrt {a} \sqrt {c} f} \] Input:
integrate(csc(f*x+e)/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algo rithm="maxima")
Output:
-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)
Exception generated. \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csc(f*x+e)/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algo rithm="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
Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int(1/(sin(e + f*x)*(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(1/2)) ,x)
Output:
int(1/(sin(e + f*x)*(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(1/2)) , x)
\[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \csc \left (f x +e \right )}{\sin \left (f x +e \right )^{2}-1}d x \right )}{a c} \] Input:
int(csc(f*x+e)/(a+a*sin(f*x+e))^(1/2)/(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)* csc(e + f*x))/(sin(e + f*x)**2 - 1),x))/(a*c)