3.45 \(\int \sqrt{-\sqrt{-1+\sec (x)}+\sqrt{1+\sec (x)}} \, dx\)

Optimal. Leaf size=337 \[ \sqrt{2} \cot (x) \sqrt{\sec (x)-1} \sqrt{\sec (x)+1} \left (\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{2}-2} \left (-\sqrt{\sec (x)-1}+\sqrt{\sec (x)+1}-\sqrt{2}\right )}{2 \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}\right )-\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+2 \sqrt{2}} \left (-\sqrt{\sec (x)-1}+\sqrt{\sec (x)+1}-\sqrt{2}\right )}{2 \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}\right )-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{2}-2} \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{-\sqrt{\sec (x)-1}+\sqrt{\sec (x)+1}+\sqrt{2}}\right )+\sqrt{\sqrt{2}-1} \tanh ^{-1}\left (\frac{\sqrt{2+2 \sqrt{2}} \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{-\sqrt{\sec (x)-1}+\sqrt{\sec (x)+1}+\sqrt{2}}\right )\right ) \]

[Out]

Sqrt[2]*(Sqrt[-1 + Sqrt[2]]*ArcTan[(Sqrt[-2 + 2*Sqrt[2]]*(-Sqrt[2] - Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]))/(2
*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]])] - Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2 + 2*Sqrt[2]]*(-Sqrt[2] - Sqr
t[-1 + Sec[x]] + Sqrt[1 + Sec[x]]))/(2*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]])] - Sqrt[1 + Sqrt[2]]*ArcTa
nh[(Sqrt[-2 + 2*Sqrt[2]]*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]])/(Sqrt[2] - Sqrt[-1 + Sec[x]] + Sqrt[1 +
Sec[x]])] + Sqrt[-1 + Sqrt[2]]*ArcTanh[(Sqrt[2 + 2*Sqrt[2]]*Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]])/(Sqrt
[2] - Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]])])*Cot[x]*Sqrt[-1 + Sec[x]]*Sqrt[1 + Sec[x]]

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Rubi [F]  time = 0.794238, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \sqrt{-\sqrt{-1+\sec (x)}+\sqrt{1+\sec (x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]],x]

[Out]

Defer[Int][Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]], x]

Rubi steps

\begin{align*} \int \sqrt{-\sqrt{-1+\sec (x)}+\sqrt{1+\sec (x)}} \, dx &=\int \sqrt{-\sqrt{-1+\sec (x)}+\sqrt{1+\sec (x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 1.99538, size = 552, normalized size = 1.64 \[ \frac{\sqrt [4]{2} \sin (x) \cos (x) \left (\sqrt{\sec (x)-1}-\sqrt{\sec (x)+1}\right )^2 \left (2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{\sqrt [4]{2}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{\sqrt [4]{2}}+\tan \left (\frac{\pi }{8}\right )\right )+\cos \left (\frac{\pi }{8}\right ) \log \left (\sqrt{2} \left (\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}\right )-2\ 2^{3/4} \sin \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}+2\right )-\cos \left (\frac{\pi }{8}\right ) \log \left (\sqrt{2} \left (\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}\right )+2\ 2^{3/4} \sin \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}+2\right )-\sin \left (\frac{\pi }{8}\right ) \log \left (\sqrt{2} \left (\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}\right )-2\ 2^{3/4} \cos \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}+2\right )+\sin \left (\frac{\pi }{8}\right ) \log \left (\sqrt{2} \left (\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}\right )+\sqrt [4]{2} \csc \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}+2\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\csc \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{\sqrt [4]{2}}\right )-2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\csc \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{\sqrt [4]{2}}+\cot \left (\frac{\pi }{8}\right )\right )\right )}{\cos (2 x)+2 \cos (x) \sqrt{\sec (x)-1} \sqrt{\sec (x)+1}-1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]],x]

[Out]

(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]] + Sqr
t[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 + Se
c[x]] + Sqrt[1 + Sec[x]]])/2^(1/4) + Tan[Pi/8]]*Sin[Pi/8] - Log[2 - 2*2^(3/4)*Cos[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[-Sqrt[-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]])

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Maple [F]  time = 0.16, size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\sqrt{-1+\sec \left ( x \right ) }+\sqrt{1+\sec \left ( x \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(-1+sec(x))^(1/2)+(1+sec(x))^(1/2))^(1/2),x)

[Out]

int((-(-1+sec(x))^(1/2)+(1+sec(x))^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{\sec \left (x\right ) + 1} - \sqrt{\sec \left (x\right ) - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sqrt(sec(x) + 1) - sqrt(sec(x) - 1)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- \sqrt{\sec{\left (x \right )} - 1} + \sqrt{\sec{\left (x \right )} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(-1+sec(x))**(1/2)+(1+sec(x))**(1/2))**(1/2),x)

[Out]

Integral(sqrt(-sqrt(sec(x) - 1) + sqrt(sec(x) + 1)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out