Integrand size = 17, antiderivative size = 42 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a+a \sin (x)}}\right )}{\sqrt {a}} \] Output:
-2^(1/2)*arctan(1/2*a^(1/2)*cos(x)*2^(1/2)/sin(x)^(1/2)/(a+a*sin(x))^(1/2) )/a^(1/2)
Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\frac {2 \arctan \left (\sqrt {\tan \left (\frac {x}{2}\right )}\right ) \sqrt {\sin (x)} \left (1+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {a (1+\sin (x))} \sqrt {\tan \left (\frac {x}{2}\right )}} \] Input:
Integrate[1/(Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]]),x]
Output:
(2*ArcTan[Sqrt[Tan[x/2]]]*Sqrt[Sin[x]]*(1 + Tan[x/2]))/(Sqrt[a*(1 + 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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3042, 3261, 218}
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.
\(\displaystyle \int \frac {1}{\sqrt {\sin (x)} \sqrt {a \sin (x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {\sin (x)} \sqrt {a \sin (x)+a}}dx\) |
\(\Big \downarrow \) 3261 |
\(\displaystyle -2 a \int \frac {1}{\frac {\cos (x) \cot (x) a^3}{\sin (x) a+a}+2 a^2}d\frac {a \cos (x)}{\sqrt {\sin (x)} \sqrt {\sin (x) a+a}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a \sin (x)+a}}\right )}{\sqrt {a}}\) |
Input:
Int[1/(Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]]),x]
Output:
-((Sqrt[2]*ArcTan[(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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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 = 0.77 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {\left (\cos \left (x \right )+1+\sin \left (x \right )\right ) \sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\right ) \sqrt {2}}{\sqrt {\sin \left (x \right )}\, \sqrt {a \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )^{2}}}\) | \(48\) |
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)*arctan((csc(x)-cot(x) )^(1/2))/(a*sin(1/4*Pi+1/2*x)^2)^(1/2)*2^(1/2)
Time = 0.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.88 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{a}} \log \left (\frac {17 \, \cos \left (x\right )^{3} - 4 \, \sqrt {2} {\left (3 \, \cos \left (x\right )^{2} + {\left (3 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - \cos \left (x\right ) - 4\right )} \sqrt {a \sin \left (x\right ) + a} \sqrt {-\frac {1}{a}} \sqrt {\sin \left (x\right )} + 3 \, \cos \left (x\right )^{2} + {\left (17 \, \cos \left (x\right )^{2} + 14 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 18 \, \cos \left (x\right ) - 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ), \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a \sin \left (x\right ) + a} {\left (3 \, \sin \left (x\right ) - 1\right )}}{4 \, \sqrt {a} \cos \left (x\right ) \sqrt {\sin \left (x\right )}}\right )}{2 \, \sqrt {a}}\right ] \] Input:
integrate(1/sin(x)^(1/2)/(a+a*sin(x))^(1/2),x, algorithm="fricas")
Output:
[1/4*sqrt(2)*sqrt(-1/a)*log((17*cos(x)^3 - 4*sqrt(2)*(3*cos(x)^2 + (3*cos( x) + 4)*sin(x) - cos(x) - 4)*sqrt(a*sin(x) + a)*sqrt(-1/a)*sqrt(sin(x)) + 3*cos(x)^2 + (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)), 1/2*s qrt(2)*arctan(1/4*sqrt(2)*sqrt(a*sin(x) + a)*(3*sin(x) - 1)/(sqrt(a)*cos(x )*sqrt(sin(x))))/sqrt(a)]
\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\int \frac {1}{\sqrt {a \left (\sin {\left (x \right )} + 1\right )} \sqrt {\sin {\left (x \right )}}}\, dx \] 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)
\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right ) + a} \sqrt {\sin \left (x\right )}} \,d x } \] Input:
integrate(1/sin(x)^(1/2)/(a+a*sin(x))^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(a*sin(x) + a)*sqrt(sin(x))), 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