3.7.73 \(\int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx\) [673]

3.7.73.1 Optimal result

Integrand size = 16, antiderivative size = 19 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=2 \text {arctanh}\left (\frac {\sin (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}}\right ) \]

output

2*arctanh(sin(x)/(2*sin(x)+sin(x)^2)^(1/2))
 

3.7.73.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(40\) vs. \(2(19)=38\).

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\frac {2 \text {arcsinh}\left (\frac {\sqrt {\sin (x)}}{\sqrt {2}}\right ) \sqrt {\sin (x)} \sqrt {2+\sin (x)}}{\sqrt {\sin (x) (2+\sin (x))}} \]

input

Integrate[Cos[x]/Sqrt[2*Sin[x] + Sin[x]^2],x]
 

output

(2*ArcSinh[Sqrt[Sin[x]]/Sqrt[2]]*Sqrt[Sin[x]]*Sqrt[2 + Sin[x]])/Sqrt[Sin[x 
]*(2 + Sin[x])]
 

3.7.73.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3739, 1091, 219}

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.

\(\displaystyle \int \frac {\cos (x)}{\sqrt {\sin ^2(x)+2 \sin (x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (x)}{\sqrt {\sin (x)^2+2 \sin (x)}}dx\)

\(\Big \downarrow \) 3739

\(\displaystyle \int \frac {1}{\sqrt {\sin ^2(x)+2 \sin (x)}}d\sin (x)\)

\(\Big \downarrow \) 1091

\(\displaystyle 2 \int \frac {1}{1-\frac {\sin ^2(x)}{\sin ^2(x)+2 \sin (x)}}d\frac {\sin (x)}{\sqrt {\sin ^2(x)+2 \sin (x)}}\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \text {arctanh}\left (\frac {\sin (x)}{\sqrt {\sin ^2(x)+2 \sin (x)}}\right )\)

input

Int[Cos[x]/Sqrt[2*Sin[x] + Sin[x]^2],x]
 

output

2*ArcTanh[Sin[x]/Sqrt[2*Sin[x] + Sin[x]^2]]
 

3.7.73.3.1 Defintions of rubi rules used

rule 219

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

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3739

Int[cos[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*sin[(d_.) + (e_.)*( 
x_)])^(n_.) + (c_.)*((f_.)*sin[(d_.) + (e_.)*(x_)])^(n2_.))^(p_.), x_Symbol 
] :> Module[{g = FreeFactors[Sin[d + e*x], x]}, Simp[g/e   Subst[Int[(1 - g 
^2*x^2)^((m - 1)/2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p, x], x, Sin[d + e 
*x]/g], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && Intege 
rQ[(m - 1)/2]
 

3.7.73.4 Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\ln \left (1+\sin \left (x \right )+\sqrt {2 \sin \left (x \right )+\sin \left (x \right )^{2}}\right )\)	\(17\)
default	\(\ln \left (1+\sin \left (x \right )+\sqrt {2 \sin \left (x \right )+\sin \left (x \right )^{2}}\right )\)	\(17\)

input

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

output

ln(1+sin(x)+(2*sin(x)+sin(x)^2)^(1/2))
 

3.7.73.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).

Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\frac {1}{2} \, \log \left (-2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {-\cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1} {\left (\sin \left (x\right ) + 1\right )} + 4 \, \sin \left (x\right ) + 3\right ) \]

input

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

output

1/2*log(-2*cos(x)^2 + 2*sqrt(-cos(x)^2 + 2*sin(x) + 1)*(sin(x) + 1) + 4*si 
n(x) + 3)
 

3.7.73.6 Sympy [F]

\[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\int \frac {\cos {\left (x \right )}}{\sqrt {\left (\sin {\left (x \right )} + 2\right ) \sin {\left (x \right )}}}\, dx \]

input

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

output

Integral(cos(x)/sqrt((sin(x) + 2)*sin(x)), x)
 

3.7.73.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\log \left (2 \, \sqrt {\sin \left (x\right )^{2} + 2 \, \sin \left (x\right )} + 2 \, \sin \left (x\right ) + 2\right ) \]

input

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

output

log(2*sqrt(sin(x)^2 + 2*sin(x)) + 2*sin(x) + 2)
 

3.7.73.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=-\log \left (-\sqrt {\sin \left (x\right )^{2} + 2 \, \sin \left (x\right )} + \sin \left (x\right ) + 1\right ) \]

input

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

output

-log(-sqrt(sin(x)^2 + 2*sin(x)) + sin(x) + 1)
 

3.7.73.9 Mupad [B] (verification not implemented)

Time = 26.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\ln \left (\sin \left (x\right )+\sqrt {\sin \left (x\right )\,\left (\sin \left (x\right )+2\right )}+1\right ) \]

input

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

output

log(sin(x) + (sin(x)*(sin(x) + 2))^(1/2) + 1)