Integrand size = 36, antiderivative size = 100 \[ \int \frac {\csc (e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {c \cos (e+f x) \log (1+\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {\log (\sin (e+f x)) \sec (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{a f} \]
-c*cos(f*x+e)*ln(1+sin(f*x+e))/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^( 1/2)+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)/a/f
Time = 0.77 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.61 \[ \int \frac {\csc (e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {(\log (\sin (e+f x))-\log (1+\sin (e+f x))) \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}}{a f} \]
((Log[Sin[e + f*x]] - Log[1 + Sin[e + f*x]])*Sec[e + f*x]*Sqrt[a*(1 + Sin[ e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(a*f)
Time = 1.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {3042, 3421, 3042, 3216, 3042, 3146, 16, 3427, 3042, 25, 3956}
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.
\(\displaystyle \int \frac {\csc (e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a \sin (e+f x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {c-c \sin (e+f x)}}{\sin (e+f x) \sqrt {a \sin (e+f x)+a}}dx\) |
\(\Big \downarrow \) 3421 |
\(\displaystyle \frac {\int \csc (e+f x) \sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}dx}{a}-\int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}{\sin (e+f x)}dx}{a}-\int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx\) |
\(\Big \downarrow \) 3216 |
\(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}{\sin (e+f x)}dx}{a}-\frac {a c \cos (e+f x) \int \frac {\cos (e+f x)}{\sin (e+f x) a+a}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}{\sin (e+f x)}dx}{a}-\frac {a c \cos (e+f x) \int \frac {\cos (e+f x)}{\sin (e+f x) a+a}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}{\sin (e+f x)}dx}{a}-\frac {c \cos (e+f x) \int \frac {1}{\sin (e+f x) a+a}d(a \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}{\sin (e+f x)}dx}{a}-\frac {c \cos (e+f x) \log (a \sin (e+f x)+a)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3427 |
\(\displaystyle \frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \int \cot (e+f x)dx}{a}-\frac {c \cos (e+f x) \log (a \sin (e+f x)+a)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \int -\tan \left (e+f x+\frac {\pi }{2}\right )dx}{a}-\frac {c \cos (e+f x) \log (a \sin (e+f x)+a)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \int \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )dx}{a}-\frac {c \cos (e+f x) \log (a \sin (e+f x)+a)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \log (-\sin (e+f x))}{a f}-\frac {c \cos (e+f x) \log (a \sin (e+f x)+a)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
-((c*Cos[e + f*x]*Log[a + a*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqr t[c - c*Sin[e + f*x]])) + (Log[-Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[ e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/(a*f)
3.1.21.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[a*c*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x ]]*Sqrt[c + d*Sin[e + f*x]])) Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 ]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*Sqr t[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-d/c Int[Sqrt [a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/c Int[Sqrt [a + b*Sin[e + f*x]]*(Sqrt[c + d*Sin[e + f*x]]/Sin[e + f*x]), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[ b*c + a*d, 0]
Int[(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]])/sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[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]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\left (2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right )}{f \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}\) | \(100\) |
1/f*(2*ln(-cot(f*x+e)+csc(f*x+e)+1)-ln(csc(f*x+e)-cot(f*x+e)))*(-c*(sin(f* x+e)-1))^(1/2)*(cos(f*x+e)+sin(f*x+e)+1)/(-cos(f*x+e)+sin(f*x+e)-1)/(a*(1+ sin(f*x+e)))^(1/2)
\[ \int \frac {\csc (e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\sqrt {-c \sin \left (f x + e\right ) + c}}{\sqrt {a \sin \left (f x + e\right ) + a} \sin \left (f x + e\right )} \,d x } \]
integral(-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*cos(f*x + e)^2 - a*sin(f*x + e) - a), x)
\[ \int \frac {\csc (e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sin {\left (e + f x \right )}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.59 \[ \int \frac {\csc (e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\frac {2 \, \sqrt {c} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{\sqrt {a}} - \frac {\sqrt {c} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{\sqrt {a}}}{f} \]
(2*sqrt(c)*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/sqrt(a) - sqrt(c)*log( sin(f*x + e)/(cos(f*x + e) + 1))/sqrt(a))/f
Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12 \[ \int \frac {\csc (e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\sqrt {2} \sqrt {a} \sqrt {c} {\left (\frac {\sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} \log \left ({\left | 2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1 \right |}\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{2 \, f} \]
1/2*sqrt(2)*sqrt(a)*sqrt(c)*(sqrt(2)*log(-sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 + 1)/(a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - sqrt(2)*log(abs(2*sin(-1/4 *pi + 1/2*f*x + 1/2*e)^2 - 1))/(a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))*sg n(sin(-1/4*pi + 1/2*f*x + 1/2*e))/f
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]