Integrand size = 18, antiderivative size = 44 \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=-\log (\sin (x))+2 \log \left (-\sqrt {\cos (x)}+\sqrt {\cos (x)+\sin (x)}\right )+\frac {2 \sqrt {\cos (x)+\sin (x)}}{\sqrt {\cos (x)}} \]
Time = 0.94 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.55 \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\frac {2 \left (\cos (x)+\sin (x)-\text {arctanh}\left (\frac {\sqrt {\cos (x)}}{\sqrt {\cos (x)+\sqrt {\sin ^2(x)}}}\right ) \sqrt {\cos (x)} \sqrt {\cos (x)+\sqrt {\sin ^2(x)}}\right )}{\sqrt {\cos (x)} \sqrt {\cos (x)+\sin (x)}} \]
(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]])
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}}}\) |
3.9.59.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(36)=72\).
Time = 10.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.82
| method | result | size |
| default | \(\frac {\left (2 \cos \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \left (\sin \left (x \right )+\cos \left (x \right )\right )}{\left (\cos \left (x \right )+1\right )^{2}}}+\cos \left (x \right ) \ln \left (2 \cot \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \left (\sin \left (x \right )+\cos \left (x \right )\right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \cot \left (x \right )-1+2 \csc \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \left (\sin \left (x \right )+\cos \left (x \right )\right )}{\left (\cos \left (x \right )+1\right )^{2}}}\right )+2 \sqrt {\frac {\cos \left (x \right ) \left (\sin \left (x \right )+\cos \left (x \right )\right )}{\left (\cos \left (x \right )+1\right )^{2}}}\right ) \sqrt {\sin \left (x \right )+\cos \left (x \right )}}{\left (\cos \left (x \right )+1\right ) \sqrt {\frac {\cos \left (x \right ) \left (\sin \left (x \right )+\cos \left (x \right )\right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {\cos \left (x \right )}}\) | \(124\) |
(2*cos(x)*(cos(x)*(sin(x)+cos(x))/(cos(x)+1)^2)^(1/2)+cos(x)*ln(2*cot(x)*( cos(x)*(sin(x)+cos(x))/(cos(x)+1)^2)^(1/2)-2*cot(x)-1+2*csc(x)*(cos(x)*(si n(x)+cos(x))/(cos(x)+1)^2)^(1/2))+2*(cos(x)*(sin(x)+cos(x))/(cos(x)+1)^2)^ (1/2))*(sin(x)+cos(x))^(1/2)/(cos(x)+1)/(cos(x)*(sin(x)+cos(x))/(cos(x)+1) ^2)^(1/2)/cos(x)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (36) = 72\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.18 \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=-\frac {\cos \left (x\right ) \log \left ({\left (2 \, \cos \left (x\right ) + \sin \left (x\right )\right )} \sqrt {\cos \left (x\right ) + \sin \left (x\right )} \sqrt {\cos \left (x\right )} + \frac {7}{4} \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4}\right ) - \cos \left (x\right ) \log \left (-{\left (2 \, \cos \left (x\right ) + \sin \left (x\right )\right )} \sqrt {\cos \left (x\right ) + \sin \left (x\right )} \sqrt {\cos \left (x\right )} + \frac {7}{4} \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{4}\right ) - 8 \, \sqrt {\cos \left (x\right ) + \sin \left (x\right )} \sqrt {\cos \left (x\right )}}{4 \, \cos \left (x\right )} \]
-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)
\[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\int \frac {\sqrt {\sin {\left (x \right )} + \cos {\left (x \right )}} \csc {\left (x \right )}}{\cos ^{\frac {3}{2}}{\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (36) = 72\).
Time = 0.49 (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=\frac {4 \, {\left ({\left (2 \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right )^{3} + {\left (2 \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right ) \sin \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right )^{2} - {\left (\cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) + 1\right )} \sin \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right )^{3} - {\left (\cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) - 1\right )} \cos \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right ) - {\left ({\left (\cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) + 1\right )} \cos \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right )^{2} + \cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) - 1\right )} \sin \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right )\right )}}{{\left (4 \, {\left (\cos \left (2 \, x\right ) - \sin \left (2 \, x\right )\right )} \cos \left (4 \, x\right ) + 2 \, \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + 4 \, {\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \sin \left (4 \, x\right ) + 2 \, \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right ) + 2\right )}^{\frac {1}{4}} {\left (\cos \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (-\cos \left (4 \, x\right ) + \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right ) + 1, \cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right ) + 1\right )\right )^{2}\right )}} \]
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))
\[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\int { \frac {\sqrt {\cos \left (x\right ) + \sin \left (x\right )} \csc \left (x\right )}{\cos \left (x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc (x) \sqrt {\cos (x)+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx=\int \frac {\sqrt {\cos \left (x\right )+\sin \left (x\right )}}{{\cos \left (x\right )}^{3/2}\,\sin \left (x\right )} \,d x \]