\(\int \frac {1}{\sqrt {\sin (x)} \sqrt {1+\sin (x)}} \, dx\) [87]

Optimal result

Integrand size = 15, antiderivative size = 17 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {1+\sin (x)}} \, dx=-\sqrt {2} \arcsin \left (\frac {\cos (x)}{1+\sin (x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2860, 222} \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {1+\sin (x)}} \, dx=-\sqrt {2} \arcsin \left (\frac {\cos (x)}{\sin (x)+1}\right ) \]

[In]

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

[Out]

-(Sqrt[2]*ArcSin[Cos[x]/(1 + Sin[x])])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 2860

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[-Sqr
t[2]/(Sqrt[a]*f), Subst[Int[1/Sqrt[1 - x^2], x], x, b*(Cos[e + f*x]/(a + b*Sin[e + f*x]))], x] /; FreeQ[{a, b,
d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d, a/b] && GtQ[a, 0]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(17)=34\).

Time = 0.39 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {1+\sin (x)}} \, dx=\frac {2 \arctan \left (\sqrt {\tan \left (\frac {x}{2}\right )}\right ) \sqrt {\frac {\sin (x)}{1+\sin (x)}} \left (1+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right )}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(37\) vs. \(2(15)=30\).

Time = 2.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.24

method	result	size
default	\(\frac {2 \left (\cos \left (x \right )+1+\sin \left (x \right )\right ) \sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\right )}{\sqrt {\sin \left (x \right )}\, \sqrt {1+\sin \left (x \right )}}\)	\(38\)

[In]

int(1/sin(x)^(1/2)/(1+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))/(1+sin(x))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {1+\sin (x)}} \, dx=2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\sin \left (x\right ) + 1} \sqrt {\sin \left (x\right )}}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ) \]

[In]

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

[Out]

2*sqrt(2)*arctan(sqrt(2)*sqrt(sin(x) + 1)*sqrt(sin(x))/(cos(x) + sin(x) + 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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