\(\int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx\) [88]

Optimal result

Integrand size = 17, antiderivative size = 42 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a+a \sin (x)}}\right )}{\sqrt {a}} \]

[Out]

-arctan(1/2*cos(x)*a^(1/2)*2^(1/2)/sin(x)^(1/2)/(a+a*sin(x))^(1/2))*2^(1/2)/a^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2861, 211} \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a \sin (x)+a}}\right )}{\sqrt {a}} \]

[In]

Int[1/(Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]]),x]

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[a]*Cos[x])/(Sqrt[2]*Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]])])/Sqrt[a])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2861

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[-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*Sin[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]

Rubi steps \begin{align*} \text {integral}& = -\left ((2 a) \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,\frac {a \cos (x)}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}}\right )\right ) \\ & = -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cos (x)}{\sqrt {2} \sqrt {\sin (x)} \sqrt {a+a \sin (x)}}\right )}{\sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\frac {2 \arctan \left (\sqrt {\tan \left (\frac {x}{2}\right )}\right ) \sqrt {\sin (x)} \left (1+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {a (1+\sin (x))} \sqrt {\tan \left (\frac {x}{2}\right )}} \]

[In]

Integrate[1/(Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]]),x]

[Out]

(2*ArcTan[Sqrt[Tan[x/2]]]*Sqrt[Sin[x]]*(1 + Tan[x/2]))/(Sqrt[a*(1 + Sin[x])]*Sqrt[Tan[x/2]])

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {2 \left (\cos \left (x \right )+1+\sin \left (x \right )\right ) \sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\right )}{\sqrt {\sin \left (x \right )}\, \sqrt {a \left (1+\sin \left (x \right )\right )}}\)	\(40\)

[In]

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

[Out]

2*(cos(x)+1+sin(x))*(csc(x)-cot(x))^(1/2)/sin(x)^(1/2)*arctan((csc(x)-cot(x))^(1/2))/(a*(1+sin(x)))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.88 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{a}} \log \left (\frac {17 \, \cos \left (x\right )^{3} - 4 \, \sqrt {2} {\left (3 \, \cos \left (x\right )^{2} + {\left (3 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - \cos \left (x\right ) - 4\right )} \sqrt {a \sin \left (x\right ) + a} \sqrt {-\frac {1}{a}} \sqrt {\sin \left (x\right )} + 3 \, \cos \left (x\right )^{2} + {\left (17 \, \cos \left (x\right )^{2} + 14 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 18 \, \cos \left (x\right ) - 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ), \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a \sin \left (x\right ) + a} {\left (3 \, \sin \left (x\right ) - 1\right )}}{4 \, \sqrt {a} \cos \left (x\right ) \sqrt {\sin \left (x\right )}}\right )}{2 \, \sqrt {a}}\right ] \]

[In]

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

[Out]

[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*sqrt(2)*arctan(1/4*sqrt(2)*sqrt(a*s
in(x) + a)*(3*sin(x) - 1)/(sqrt(a)*cos(x)*sqrt(sin(x))))/sqrt(a)]

Sympy [F]

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

[In]

integrate(1/sin(x)**(1/2)/(a+a*sin(x))**(1/2),x)

[Out]

Integral(1/(sqrt(a*(sin(x) + 1))*sqrt(sin(x))), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right ) + a} \sqrt {\sin \left (x\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(a*sin(x) + a)*sqrt(sin(x))), x)

Giac [F]

\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a+a \sin (x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right ) + a} \sqrt {\sin \left (x\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(a*sin(x) + a)*sqrt(sin(x))), x)

Mupad [F(-1)]

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

[In]

int(1/(sin(x)^(1/2)*(a + a*sin(x))^(1/2)),x)

[Out]

int(1/(sin(x)^(1/2)*(a + a*sin(x))^(1/2)), x)