\(\int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx\) [802]

Optimal result

Integrand size = 18, antiderivative size = 44 \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=-\log (\sin (x))+2 \log \left (-\sqrt {\cos (x)}+\sqrt {\cos (x)+\sin (x)}\right )+\frac {2 \sqrt {\cos (x)+\sin (x)}}{\sqrt {\cos (x)}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.83 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.55 \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\frac {2 \left (\cos (x)+\sin (x)-\text {arctanh}\left (\frac {\sqrt {\cos (x)}}{\sqrt {\cos (x)+\sqrt {\sin ^2(x)}}}\right ) \sqrt {\cos (x)} \sqrt {\cos (x)+\sqrt {\sin ^2(x)}}\right )}{\sqrt {\cos (x)} \sqrt {\cos (x)+\sin (x)}} \] Input:

Integrate[(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2),x]
 

Output:

(2*(Cos[x] + Sin[x] - ArcTanh[Sqrt[Cos[x]]/Sqrt[Cos[x] + Sqrt[Sin[x]^2]]]* 
Sqrt[Cos[x]]*Sqrt[Cos[x] + Sqrt[Sin[x]^2]]))/(Sqrt[Cos[x]]*Sqrt[Cos[x] + S 
in[x]])
 

Rubi [F]

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.

\(\displaystyle \int \frac {\csc (x) \sqrt {\sin (x)+\cos (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {\sin (x)+\cos (x)}}{\sin (x) \cos (x)^{3/2}}dx\)

\(\Big \downarrow \) 4902

\(\displaystyle 2 \int \frac {\cot \left (\frac {x}{2}\right ) \sqrt {\frac {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}{\tan ^2\left (\frac {x}{2}\right )+1}}}{2 \left (\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}\right )^{3/2}}d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {\sqrt {\frac {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}{\tan ^2\left (\frac {x}{2}\right )+1}} \cot \left (\frac {x}{2}\right )}{\left (\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}\right )^{3/2}}d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \int \frac {\cot \left (\frac {x}{2}\right ) \sqrt {\frac {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}{\tan ^2\left (\frac {x}{2}\right )+1}} \left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{3/2}}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right )^{3/2}}d\tan \left (\frac {x}{2}\right )}{\sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}} \sqrt {\tan ^2\left (\frac {x}{2}\right )+1}}\)

\(\Big \downarrow \) 7270

\(\displaystyle \frac {\sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \sqrt {\frac {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}{\tan ^2\left (\frac {x}{2}\right )+1}} \int \frac {\cot \left (\frac {x}{2}\right ) \sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1} \left (\tan ^2\left (\frac {x}{2}\right )+1\right )}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right )^{3/2}}d\tan \left (\frac {x}{2}\right )}{\sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1} \sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \sqrt {\frac {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}{\tan ^2\left (\frac {x}{2}\right )+1}} \int \left (\frac {\sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1} \cot \left (\frac {x}{2}\right )}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right )^{3/2}}+\frac {\tan \left (\frac {x}{2}\right ) \sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right )^{3/2}}\right )d\tan \left (\frac {x}{2}\right )}{\sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1} \sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \sqrt {\frac {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}{\tan ^2\left (\frac {x}{2}\right )+1}} \left (\int \frac {\cot \left (\frac {x}{2}\right ) \sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right )^{3/2}}d\tan \left (\frac {x}{2}\right )-\frac {2^{3/4} \sqrt {\sqrt {2}-1} \sqrt {1-\tan \left (\frac {x}{2}\right )} \sqrt {\left (1+\sqrt {2}\right ) \tan \left (\frac {x}{2}\right )+1} \left (-\tan \left (\frac {x}{2}\right )+\sqrt {2}+1\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt [4]{2} \sqrt {\left (1+\sqrt {2}\right ) \tan \left (\frac {x}{2}\right )+1}}{\sqrt {1-\tan \left (\frac {x}{2}\right )}}\right ),\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt {\frac {-\tan \left (\frac {x}{2}\right )+\sqrt {2}+1}{\tan \left (\frac {x}{2}\right )+1}} \sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}}+\frac {\sqrt [4]{2} \sqrt {5 \sqrt {2}-7} \sqrt {1-\tan \left (\frac {x}{2}\right )} \sqrt {\left (1+\sqrt {2}\right ) \tan \left (\frac {x}{2}\right )+1} \left (-\tan \left (\frac {x}{2}\right )+\sqrt {2}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{\sqrt {2}},\arctan \left (\frac {\sqrt [4]{2} \sqrt {\left (1+\sqrt {2}\right ) \tan \left (\frac {x}{2}\right )+1}}{\sqrt {1-\tan \left (\frac {x}{2}\right )}}\right ),\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt {\frac {-\tan \left (\frac {x}{2}\right )+\sqrt {2}+1}{\tan \left (\frac {x}{2}\right )+1}} \sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}}+\frac {\sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1}}{\sqrt {1-\tan ^2\left (\frac {x}{2}\right )}}\right )}{\sqrt {-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1} \sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}}}\)

Input:

Int[(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 3042

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

rule 4902

Int[u_, x_Symbol] :> With[{w = Block[{$ShowSteps = False, $StepCounter = Nu 
ll}, Int[SubstFor[1/(1 + FreeFactors[Tan[FunctionOfTrig[u, x]/2], x]^2*x^2) 
, Tan[FunctionOfTrig[u, x]/2]/FreeFactors[Tan[FunctionOfTrig[u, x]/2], x], 
u, x], x]]}, Module[{v = FunctionOfTrig[u, x], d}, Simp[d = FreeFactors[Tan 
[v/2], x]; 2*(d/Coefficient[v, x, 1])   Subst[Int[SubstFor[1/(1 + d^2*x^2), 
 Tan[v/2]/d, u, x], x], x, Tan[v/2]/d], x]] /; CalculusFreeQ[w, x]] /; Inve 
rseFunctionFreeQ[u, x] &&  !FalseQ[FunctionOfTrig[u, x]]
 

rule 7270

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p 
]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])))   Int[u*v 
^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !Free 
Q[v, x] &&  !FreeQ[w, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(124\) vs. \(2(36)=72\).

Time = 2.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.84

method	result	size
default	\(\frac {\left (2 \cos \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \left (\cos \left (x \right )+\sin \left (x \right )\right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right ) \ln \left (-\frac {\cos \left (x \right ) \cot \left (x \right )-2 \cot \left (x \right )+2 \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \left (\cos \left (x \right )+\sin \left (x \right )\right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right )-\sin \left (x \right )+\csc \left (x \right )-1}{\cos \left (x \right )-1}\right )+2 \sqrt {\frac {\cos \left (x \right ) \left (\cos \left (x \right )+\sin \left (x \right )\right )}{\left (1+\cos \left (x \right )\right )^{2}}}\right ) \sqrt {\cos \left (x \right )+\sin \left (x \right )}}{\sqrt {\cos \left (x \right )}\, \left (1+\cos \left (x \right )\right ) \sqrt {\frac {\cos \left (x \right ) \left (\cos \left (x \right )+\sin \left (x \right )\right )}{\left (1+\cos \left (x \right )\right )^{2}}}}\)	\(125\)

Input:

int(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

(2*cos(x)*(cos(x)*(cos(x)+sin(x))/(1+cos(x))^2)^(1/2)+cos(x)*ln(-(cos(x)*c 
ot(x)-2*cot(x)+2*sin(x)*(cos(x)*(cos(x)+sin(x))/(1+cos(x))^2)^(1/2)+cos(x) 
-sin(x)+csc(x)-1)/(cos(x)-1))+2*(cos(x)*(cos(x)+sin(x))/(1+cos(x))^2)^(1/2 
))*(cos(x)+sin(x))^(1/2)/cos(x)^(1/2)/(1+cos(x))/(cos(x)*(cos(x)+sin(x))/( 
1+cos(x))^2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (36) = 72\).

Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.18 \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=-\frac {\cos \left (x\right ) \log \left ({\left (2 \, \cos \left (x\right ) + \sin \left (x\right )\right )} \sqrt {\cos \left (x\right ) + \sin \left (x\right )} \sqrt {\cos \left (x\right )} + \frac {7}{4} \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4}\right ) - \cos \left (x\right ) \log \left (-{\left (2 \, \cos \left (x\right ) + \sin \left (x\right )\right )} \sqrt {\cos \left (x\right ) + \sin \left (x\right )} \sqrt {\cos \left (x\right )} + \frac {7}{4} \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4}\right ) - 8 \, \sqrt {\cos \left (x\right ) + \sin \left (x\right )} \sqrt {\cos \left (x\right )}}{4 \, \cos \left (x\right )} \] Input:

integrate(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x, algorithm="fricas")
 

Output:

-1/4*(cos(x)*log((2*cos(x) + sin(x))*sqrt(cos(x) + sin(x))*sqrt(cos(x)) + 
7/4*cos(x)^2 + 2*cos(x)*sin(x) + 1/4) - cos(x)*log(-(2*cos(x) + sin(x))*sq 
rt(cos(x) + sin(x))*sqrt(cos(x)) + 7/4*cos(x)^2 + 2*cos(x)*sin(x) + 1/4) - 
 8*sqrt(cos(x) + sin(x))*sqrt(cos(x)))/cos(x)
 

Sympy [F]

\[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\int \frac {\sqrt {\sin {\left (x \right )} + \cos {\left (x \right )}} \csc {\left (x \right )}}{\cos ^{\frac {3}{2}}{\left (x \right )}}\, dx \] Input:

integrate(csc(x)*(cos(x)+sin(x))**(1/2)/cos(x)**(3/2),x)
 

Output:

Integral(sqrt(sin(x) + cos(x))*csc(x)/cos(x)**(3/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (36) = 72\).

Time = 0.26 (sec) , antiderivative size = 518, normalized size of antiderivative = 11.77 \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx =\text {Too large to display} \] Input:

integrate(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x, algorithm="maxima")
 

Output:

4*((2*cos(2*x) + sin(2*x))*cos(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2* 
x) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1))^3 + (2*cos(2*x) + sin(2*x)) 
*cos(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2 
*x) + sin(4*x) + 1))*sin(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1 
, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1))^2 - (cos(2*x) - 2*sin(2*x) + 1)*s 
in(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x 
) + sin(4*x) + 1))^3 - (cos(2*x) - sin(2*x) - 1)*cos(1/2*arctan2(-cos(4*x) 
 + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1)) - ((c 
os(2*x) - 2*sin(2*x) + 1)*cos(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x 
) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1))^2 + cos(2*x) + sin(2*x) - 1) 
*sin(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2 
*x) + sin(4*x) + 1)))/((4*(cos(2*x) - sin(2*x))*cos(4*x) + 2*cos(4*x)^2 + 
4*cos(2*x)^2 + 4*(cos(2*x) + sin(2*x) + 1)*sin(4*x) + 2*sin(4*x)^2 + 4*sin 
(2*x)^2 + 4*cos(2*x) + 4*sin(2*x) + 2)^(1/4)*(cos(1/2*arctan2(-cos(4*x) + 
sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1))^2 + sin( 
1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x) + 
 sin(4*x) + 1))^2))
 

Giac [F]

\[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\int { \frac {\sqrt {\cos \left (x\right ) + \sin \left (x\right )} \csc \left (x\right )}{\cos \left (x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x, algorithm="giac")
 

Output:

integrate(sqrt(cos(x) + sin(x))*csc(x)/cos(x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\int \frac {\sqrt {\cos \left (x\right )+\sin \left (x\right )}}{{\cos \left (x\right )}^{3/2}\,\sin \left (x\right )} \,d x \] Input:

int((cos(x) + sin(x))^(1/2)/(cos(x)^(3/2)*sin(x)),x)
 

Output:

int((cos(x) + sin(x))^(1/2)/(cos(x)^(3/2)*sin(x)), x)
 

Reduce [F]

\[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\int \frac {\sqrt {\cos \left (x \right )+\sin \left (x \right )}\, \sqrt {\cos \left (x \right )}\, \csc \left (x \right )}{\cos \left (x \right )^{2}}d x \] Input:

int(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x)
 

Output:

int((sqrt(cos(x) + sin(x))*sqrt(cos(x))*csc(x))/cos(x)**2,x)