Integrand size = 23, antiderivative size = 337 \[ \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx=\sqrt {2} \left (\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}} \left (-\sqrt {2}-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )}{2 \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}\right )-\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2+2 \sqrt {2}} \left (-\sqrt {2}-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )}{2 \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {-2+2 \sqrt {2}} \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt {2}-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}\right )+\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+2 \sqrt {2}} \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt {2}-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}\right )\right ) \cot (x) \sqrt {-1+\sec (x)} \sqrt {1+\sec (x)} \]
cot(x)*2^(1/2)*(-1+sec(x))^(1/2)*(1+sec(x))^(1/2)*(arctan(1/2*(-2^(1/2)-(- 1+sec(x))^(1/2)+(1+sec(x))^(1/2))*(-2+2*2^(1/2))^(1/2)/(-(-1+sec(x))^(1/2) +(1+sec(x))^(1/2))^(1/2))*(2^(1/2)-1)^(1/2)+arctanh((2+2*2^(1/2))^(1/2)*(- (-1+sec(x))^(1/2)+(1+sec(x))^(1/2))^(1/2)/(2^(1/2)-(-1+sec(x))^(1/2)+(1+se c(x))^(1/2)))*(2^(1/2)-1)^(1/2)-arctan(1/2*(-2^(1/2)-(-1+sec(x))^(1/2)+(1+ sec(x))^(1/2))*(2+2*2^(1/2))^(1/2)/(-(-1+sec(x))^(1/2)+(1+sec(x))^(1/2))^( 1/2))*(1+2^(1/2))^(1/2)-arctanh((-2+2*2^(1/2))^(1/2)*(-(-1+sec(x))^(1/2)+( 1+sec(x))^(1/2))^(1/2)/(2^(1/2)-(-1+sec(x))^(1/2)+(1+sec(x))^(1/2)))*(1+2^ (1/2))^(1/2))
Time = 2.85 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.64 \[ \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx=\frac {\sqrt [4]{2} \cos (x) \left (\sqrt {-1+\sec (x)}-\sqrt {1+\sec (x)}\right )^2 \left (2 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\csc \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt [4]{2}}\right ) \cos \left (\frac {\pi }{8}\right )-2 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\csc \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt [4]{2}}\right ) \cos \left (\frac {\pi }{8}\right )+\cos \left (\frac {\pi }{8}\right ) \log \left (2+\sqrt {2} \left (-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )-2\ 2^{3/4} \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \sin \left (\frac {\pi }{8}\right )\right )-\cos \left (\frac {\pi }{8}\right ) \log \left (2+\sqrt {2} \left (-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )+2\ 2^{3/4} \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \sin \left (\frac {\pi }{8}\right )\right )+2 \arctan \left (\frac {\sec \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt [4]{2}}-\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+2 \arctan \left (\frac {\sec \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt [4]{2}}+\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\log \left (2-2\ 2^{3/4} \cos \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}+\sqrt {2} \left (-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\log \left (2+\sqrt [4]{2} \csc \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}+\sqrt {2} \left (-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )\right ) \sin \left (\frac {\pi }{8}\right )\right ) \sin (x)}{-1+\cos (2 x)+2 \cos (x) \sqrt {-1+\sec (x)} \sqrt {1+\sec (x)}} \]
(2^(1/4)*Cos[x]*(Sqrt[-1 + Sec[x]] - Sqrt[1 + Sec[x]])^2*(2*ArcTan[Cot[Pi/ 8] - (Csc[Pi/8]*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]])/2^(1/4)]*Cos[ Pi/8] - 2*ArcTan[Cot[Pi/8] + (Csc[Pi/8]*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]])/2^(1/4)]*Cos[Pi/8] + Cos[Pi/8]*Log[2 + Sqrt[2]*(-Sqrt[-1 + Sec[ x]] + Sqrt[1 + Sec[x]]) - 2*2^(3/4)*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec [x]]]*Sin[Pi/8]] - Cos[Pi/8]*Log[2 + Sqrt[2]*(-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]) + 2*2^(3/4)*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]]*Sin[Pi/ 8]] + 2*ArcTan[(Sec[Pi/8]*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]])/2^( 1/4) - Tan[Pi/8]]*Sin[Pi/8] + 2*ArcTan[(Sec[Pi/8]*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]])/2^(1/4) + Tan[Pi/8]]*Sin[Pi/8] - Log[2 - 2*2^(3/4)*Co s[Pi/8]*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]] + Sqrt[2]*(-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]])]*Sin[Pi/8] + Log[2 + 2^(1/4)*Csc[Pi/8]*Sqrt[-S qrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]] + Sqrt[2]*(-Sqrt[-1 + Sec[x]] + Sqrt[ 1 + Sec[x]])]*Sin[Pi/8])*Sin[x])/(-1 + Cos[2*x] + 2*Cos[x]*Sqrt[-1 + Sec[x ]]*Sqrt[1 + Sec[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.
| \(\displaystyle \int \sqrt {\sqrt {\sec (x)+1}-\sqrt {\sec (x)-1}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sqrt {\sec (x)+1}-\sqrt {\sec (x)-1}}dx\) |
| \(\Big \downarrow \) 4902 |
| \(\displaystyle 2 \int \frac {\sqrt [4]{2} \sqrt {\sqrt {\frac {1}{1-\tan ^2\left (\frac {x}{2}\right )}}-\sqrt {\frac {\tan ^2\left (\frac {x}{2}\right )}{1-\tan ^2\left (\frac {x}{2}\right )}}}}{\tan ^2\left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \sqrt [4]{2} \int \frac {\sqrt {\sqrt {\frac {1}{1-\tan ^2\left (\frac {x}{2}\right )}}-\sqrt {\frac {\tan ^2\left (\frac {x}{2}\right )}{1-\tan ^2\left (\frac {x}{2}\right )}}}}{\tan ^2\left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \sqrt [4]{2} \int \left (\frac {i \sqrt {\sqrt {\frac {1}{1-\tan ^2\left (\frac {x}{2}\right )}}-\sqrt {\frac {\tan ^2\left (\frac {x}{2}\right )}{1-\tan ^2\left (\frac {x}{2}\right )}}}}{2 \left (i-\tan \left (\frac {x}{2}\right )\right )}+\frac {i \sqrt {\sqrt {\frac {1}{1-\tan ^2\left (\frac {x}{2}\right )}}-\sqrt {\frac {\tan ^2\left (\frac {x}{2}\right )}{1-\tan ^2\left (\frac {x}{2}\right )}}}}{2 \left (\tan \left (\frac {x}{2}\right )+i\right )}\right )d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \sqrt [4]{2} \left (\frac {1}{2} i \int \frac {\sqrt {\sqrt {\frac {1}{1-\tan ^2\left (\frac {x}{2}\right )}}-\sqrt {\frac {\tan ^2\left (\frac {x}{2}\right )}{1-\tan ^2\left (\frac {x}{2}\right )}}}}{i-\tan \left (\frac {x}{2}\right )}d\tan \left (\frac {x}{2}\right )+\frac {1}{2} i \int \frac {\sqrt {\sqrt {\frac {1}{1-\tan ^2\left (\frac {x}{2}\right )}}-\sqrt {\frac {\tan ^2\left (\frac {x}{2}\right )}{1-\tan ^2\left (\frac {x}{2}\right )}}}}{\tan \left (\frac {x}{2}\right )+i}d\tan \left (\frac {x}{2}\right )\right )\) |
3.1.45.3.1 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_, 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_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
\[\int \sqrt {-\sqrt {-1+\sec \left (x \right )}+\sqrt {1+\sec \left (x \right )}}d x\]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00 \[ \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx=\frac {1}{2} \, \sqrt {4 i + 4} \log \left (i \, \sqrt {4 i + 4} + 2 \, \sqrt {\frac {\sqrt {\frac {\cos \left (x\right ) + 1}{\cos \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + 1}}\right ) - \frac {1}{2} \, \sqrt {4 i + 4} \log \left (-i \, \sqrt {4 i + 4} + 2 \, \sqrt {\frac {\sqrt {\frac {\cos \left (x\right ) + 1}{\cos \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + 1}}\right ) - \frac {1}{2} \, \sqrt {-4 i + 4} \log \left (i \, \sqrt {-4 i + 4} + 2 \, \sqrt {\frac {\sqrt {\frac {\cos \left (x\right ) + 1}{\cos \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + 1}}\right ) + \frac {1}{2} \, \sqrt {-4 i + 4} \log \left (-i \, \sqrt {-4 i + 4} + 2 \, \sqrt {\frac {\sqrt {\frac {\cos \left (x\right ) + 1}{\cos \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + 1}}\right ) - \frac {1}{2} \, \sqrt {4 i - 4} \log \left (i \, \sqrt {4 i - 4} + 2 \, \sqrt {\frac {\sqrt {\frac {\cos \left (x\right ) + 1}{\cos \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + 1}}\right ) + \frac {1}{2} \, \sqrt {4 i - 4} \log \left (-i \, \sqrt {4 i - 4} + 2 \, \sqrt {\frac {\sqrt {\frac {\cos \left (x\right ) + 1}{\cos \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + 1}}\right ) + \frac {1}{2} \, \sqrt {-4 i - 4} \log \left (i \, \sqrt {-4 i - 4} + 2 \, \sqrt {\frac {\sqrt {\frac {\cos \left (x\right ) + 1}{\cos \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + 1}}\right ) - \frac {1}{2} \, \sqrt {-4 i - 4} \log \left (-i \, \sqrt {-4 i - 4} + 2 \, \sqrt {\frac {\sqrt {\frac {\cos \left (x\right ) + 1}{\cos \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + 1}}\right ) \]
1/2*sqrt(4*I + 4)*log(I*sqrt(4*I + 4) + 2*sqrt(sqrt((cos(x) + 1)/cos(x))*( cos(x) - sin(x) + 1)/(cos(x) + 1))) - 1/2*sqrt(4*I + 4)*log(-I*sqrt(4*I + 4) + 2*sqrt(sqrt((cos(x) + 1)/cos(x))*(cos(x) - sin(x) + 1)/(cos(x) + 1))) - 1/2*sqrt(-4*I + 4)*log(I*sqrt(-4*I + 4) + 2*sqrt(sqrt((cos(x) + 1)/cos( x))*(cos(x) - sin(x) + 1)/(cos(x) + 1))) + 1/2*sqrt(-4*I + 4)*log(-I*sqrt( -4*I + 4) + 2*sqrt(sqrt((cos(x) + 1)/cos(x))*(cos(x) - sin(x) + 1)/(cos(x) + 1))) - 1/2*sqrt(4*I - 4)*log(I*sqrt(4*I - 4) + 2*sqrt(sqrt((cos(x) + 1) /cos(x))*(cos(x) - sin(x) + 1)/(cos(x) + 1))) + 1/2*sqrt(4*I - 4)*log(-I*s qrt(4*I - 4) + 2*sqrt(sqrt((cos(x) + 1)/cos(x))*(cos(x) - sin(x) + 1)/(cos (x) + 1))) + 1/2*sqrt(-4*I - 4)*log(I*sqrt(-4*I - 4) + 2*sqrt(sqrt((cos(x) + 1)/cos(x))*(cos(x) - sin(x) + 1)/(cos(x) + 1))) - 1/2*sqrt(-4*I - 4)*lo g(-I*sqrt(-4*I - 4) + 2*sqrt(sqrt((cos(x) + 1)/cos(x))*(cos(x) - sin(x) + 1)/(cos(x) + 1)))
\[ \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx=\int \sqrt {- \sqrt {\sec {\left (x \right )} - 1} + \sqrt {\sec {\left (x \right )} + 1}}\, dx \]
\[ \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx=\int { \sqrt {\sqrt {\sec \left (x\right ) + 1} - \sqrt {\sec \left (x\right ) - 1}} \,d x } \]
\[ \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx=\int { \sqrt {\sqrt {\sec \left (x\right ) + 1} - \sqrt {\sec \left (x\right ) - 1}} \,d x } \]
Timed out. \[ \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx=\int \sqrt {\sqrt {\frac {1}{\cos \left (x\right )}+1}-\sqrt {\frac {1}{\cos \left (x\right )}-1}} \,d x \]