Integrand size = 18, antiderivative size = 42 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a-a \sin (x)}}\right )}{\sqrt {a}} \] Output:
2^(1/2)*arctanh(1/2*a^(1/2)*cos(x)*2^(1/2)/sin(x)^(1/2)/(a-a*sin(x))^(1/2) )/a^(1/2)
Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right ) \sqrt {\sin (x)} \left (-1+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {a-a \sin (x)} \sqrt {\tan \left (\frac {x}{2}\right )}} \] Input:
Integrate[1/(Sqrt[Sin[x]]*Sqrt[a - a*Sin[x]]),x]
Output:
(-2*ArcTanh[Sqrt[Tan[x/2]]]*Sqrt[Sin[x]]*(-1 + Tan[x/2]))/(Sqrt[a - a*Sin[ x]]*Sqrt[Tan[x/2]])
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3261, 221}
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.
\(\displaystyle \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}}dx\) |
\(\Big \downarrow \) 3261 |
\(\displaystyle -2 a \int \frac {1}{2 a^2-\frac {a^3 \cos (x) \cot (x)}{a-a \sin (x)}}d\left (-\frac {a \cos (x)}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a-a \sin (x)}}\right )}{\sqrt {a}}\) |
Input:
Int[1/(Sqrt[Sin[x]]*Sqrt[a - a*Sin[x]]),x]
Output:
(Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[x])/(Sqrt[2]*Sqrt[Sin[x]]*Sqrt[a - a*Sin[x]] )])/Sqrt[a]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Time = 0.78 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\left (\cos \left (x \right )+1-\sin \left (x \right )\right ) \sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \operatorname {arctanh}\left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\right ) \sqrt {2}}{\sqrt {\sin \left (x \right )}\, \sqrt {a \cos \left (\frac {\pi }{4}+\frac {x}{2}\right )^{2}}}\) | \(50\) |
Input:
int(1/sin(x)^(1/2)/(a-a*sin(x))^(1/2),x,method=_RETURNVERBOSE)
Output:
(cos(x)+1-sin(x))*(csc(x)-cot(x))^(1/2)/sin(x)^(1/2)*arctanh((csc(x)-cot(x ))^(1/2))/(a*cos(1/4*Pi+1/2*x)^2)^(1/2)*2^(1/2)
Time = 0.13 (sec) , antiderivative size = 168, normalized size of antiderivative = 4.00 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx=\left [\frac {\sqrt {2} \log \left (\frac {17 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + \frac {4 \, \sqrt {2} {\left (3 \, \cos \left (x\right )^{2} - {\left (3 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - \cos \left (x\right ) - 4\right )} \sqrt {-a \sin \left (x\right ) + a} \sqrt {\sin \left (x\right )}}{\sqrt {a}} - {\left (17 \, \cos \left (x\right )^{2} + 14 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 18 \, \cos \left (x\right ) - 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right )}{4 \, \sqrt {a}}, -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {-a \sin \left (x\right ) + a} \sqrt {-\frac {1}{a}} {\left (3 \, \sin \left (x\right ) + 1\right )}}{4 \, \cos \left (x\right ) \sqrt {\sin \left (x\right )}}\right )\right ] \] Input:
integrate(1/sin(x)^(1/2)/(a-a*sin(x))^(1/2),x, algorithm="fricas")
Output:
[1/4*sqrt(2)*log((17*cos(x)^3 + 3*cos(x)^2 + 4*sqrt(2)*(3*cos(x)^2 - (3*co s(x) + 4)*sin(x) - cos(x) - 4)*sqrt(-a*sin(x) + a)*sqrt(sin(x))/sqrt(a) - (17*cos(x)^2 + 14*cos(x) - 4)*sin(x) - 18*cos(x) - 4)/(cos(x)^3 + 3*cos(x) ^2 - (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4))/sqrt(a), -1/2*sqrt( 2)*sqrt(-1/a)*arctan(1/4*sqrt(2)*sqrt(-a*sin(x) + a)*sqrt(-1/a)*(3*sin(x) + 1)/(cos(x)*sqrt(sin(x))))]
\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx=\int \frac {1}{\sqrt {- a \left (\sin {\left (x \right )} - 1\right )} \sqrt {\sin {\left (x \right )}}}\, dx \] Input:
integrate(1/sin(x)**(1/2)/(a-a*sin(x))**(1/2),x)
Output:
Integral(1/(sqrt(-a*(sin(x) - 1))*sqrt(sin(x))), x)
\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx=\int { \frac {1}{\sqrt {-a \sin \left (x\right ) + a} \sqrt {\sin \left (x\right )}} \,d x } \] Input:
integrate(1/sin(x)^(1/2)/(a-a*sin(x))^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(-a*sin(x) + a)*sqrt(sin(x))), x)
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (31) = 62\).
Time = 0.47 (sec) , antiderivative size = 149, normalized size of antiderivative = 3.55 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx=\frac {\sqrt {2} {\left (\log \left (\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} - \sqrt {\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{4} - 6 \, \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + 1} + 1\right ) - \log \left ({\left | -\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + \sqrt {\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{4} - 6 \, \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + 1} + 3 \right |}\right ) - \log \left ({\left | -\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + \sqrt {\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{4} - 6 \, \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + 1} + 1 \right |}\right )\right )}}{2 \, \sqrt {a} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right )} \] Input:
integrate(1/sin(x)^(1/2)/(a-a*sin(x))^(1/2),x, algorithm="giac")
Output:
1/2*sqrt(2)*(log(tan(-1/8*pi + 1/4*x)^2 - sqrt(tan(-1/8*pi + 1/4*x)^4 - 6* tan(-1/8*pi + 1/4*x)^2 + 1) + 1) - log(abs(-tan(-1/8*pi + 1/4*x)^2 + sqrt( tan(-1/8*pi + 1/4*x)^4 - 6*tan(-1/8*pi + 1/4*x)^2 + 1) + 3)) - log(abs(-ta n(-1/8*pi + 1/4*x)^2 + sqrt(tan(-1/8*pi + 1/4*x)^4 - 6*tan(-1/8*pi + 1/4*x )^2 + 1) + 1)))/(sqrt(a)*sgn(sin(-1/4*pi + 1/2*x)))
Timed out. \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx=\int \frac {1}{\sqrt {\sin \left (x\right )}\,\sqrt {a-a\,\sin \left (x\right )}} \,d x \] Input:
int(1/(sin(x)^(1/2)*(a - a*sin(x))^(1/2)),x)
Output:
int(1/(sin(x)^(1/2)*(a - a*sin(x))^(1/2)), x)
\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx=-\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (x \right )}\, \sqrt {-\sin \left (x \right )+1}}{\sin \left (x \right )^{2}-\sin \left (x \right )}d x \right )}{a} \] Input:
int(1/sin(x)^(1/2)/(a-a*sin(x))^(1/2),x)
Output:
( - sqrt(a)*int((sqrt(sin(x))*sqrt( - sin(x) + 1))/(sin(x)**2 - sin(x)),x) )/a