3.42 \(\int \frac{\sec (x)}{\sqrt{-1+\sec ^4(x)}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{\tanh ^{-1}\left (\frac{\cos (x) \cot (x) \sqrt{\sec ^4(x)-1}}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

-(ArcTanh[(Cos[x]*Cot[x]*Sqrt[-1 + Sec[x]^4])/Sqrt[2]]/Sqrt[2])

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Rubi [B]  time = 0.296654, antiderivative size = 59, normalized size of antiderivative = 2.11, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{\sqrt{1-\cos ^4(x)} \sec ^2(x) \tanh ^{-1}\left (\frac{\sqrt{2} \sin (x)}{\sqrt{2 \sin ^2(x)-\sin ^4(x)}}\right )}{\sqrt{2} \sqrt{\sec ^4(x)-1}} \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]/Sqrt[-1 + Sec[x]^4],x]

[Out]

-((ArcTanh[(Sqrt[2]*Sin[x])/Sqrt[2*Sin[x]^2 - Sin[x]^4]]*Sqrt[1 - Cos[x]^4]*Sec[
x]^2)/(Sqrt[2]*Sqrt[-1 + Sec[x]^4]))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int ^{\sin{\left (x \right )}} \frac{1}{\sqrt{-1 + \frac{1}{x^{4} - 2 x^{2} + 1}} \left (- 2 x + 2\right )}\, dx + \int ^{\sin{\left (x \right )}} \frac{1}{\sqrt{-1 + \frac{1}{x^{4} - 2 x^{2} + 1}} \left (2 x + 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sec(x)/(-1+sec(x)**4)**(1/2),x)

[Out]

Integral(1/(sqrt(-1 + 1/(x**4 - 2*x**2 + 1))*(-2*x + 2)), (x, sin(x))) + Integra
l(1/(sqrt(-1 + 1/(x**4 - 2*x**2 + 1))*(2*x + 2)), (x, sin(x)))

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Mathematica [A]  time = 0.10674, size = 43, normalized size = 1.54 \[ -\frac{\sqrt{\cos (2 x)+3} \tan (x) \sec (x) \tanh ^{-1}\left (\frac{1}{2} \sqrt{\cos (2 x)+3}\right )}{2 \sqrt{\sec ^4(x)-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]/Sqrt[-1 + Sec[x]^4],x]

[Out]

-(ArcTanh[Sqrt[3 + Cos[2*x]]/2]*Sqrt[3 + Cos[2*x]]*Sec[x]*Tan[x])/(2*Sqrt[-1 + S
ec[x]^4])

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Maple [B]  time = 0.161, size = 91, normalized size = 3.3 \[ -{\frac{\sqrt{8}\sqrt{2} \left ( \sin \left ( x \right ) \right ) ^{3}}{ \left ( 8\,\cos \left ( x \right ) -8 \right ) \left ( \cos \left ( x \right ) \right ) ^{2}} \left ({\it Arcsinh} \left ({\frac{\cos \left ( x \right ) -1}{1+\cos \left ( x \right ) }} \right ) -{\it Artanh} \left ({\frac{\sqrt{2}\sqrt{4}}{4}{\frac{1}{\sqrt{{\frac{1+ \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}}}} \right ) \right ) \sqrt{{\frac{1+ \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}{\frac{1}{\sqrt{-2\,{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}-1}{ \left ( \cos \left ( x \right ) \right ) ^{4}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)/(-1+sec(x)^4)^(1/2),x)

[Out]

-1/8*8^(1/2)*2^(1/2)*(arcsinh((cos(x)-1)/(1+cos(x)))-arctanh(1/4*2^(1/2)*4^(1/2)
/((1+cos(x)^2)/(1+cos(x))^2)^(1/2)))*sin(x)^3*((1+cos(x)^2)/(1+cos(x))^2)^(1/2)/
(cos(x)-1)/(-2*(cos(x)^4-1)/cos(x)^4)^(1/2)/cos(x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sec \left (x\right )}{\sqrt{\sec \left (x\right )^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)/sqrt(sec(x)^4 - 1),x, algorithm="maxima")

[Out]

integrate(sec(x)/sqrt(sec(x)^4 - 1), x)

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Fricas [A]  time = 0.250748, size = 73, normalized size = 2.61 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{2 \,{\left (2 \, \sqrt{2} \sqrt{-\frac{\cos \left (x\right )^{4} - 1}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )\right )}}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)/sqrt(sec(x)^4 - 1),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sec{\left (x \right )}}{\sqrt{\left (\sec{\left (x \right )} - 1\right ) \left (\sec{\left (x \right )} + 1\right ) \left (\sec ^{2}{\left (x \right )} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)/(-1+sec(x)**4)**(1/2),x)

[Out]

Integral(sec(x)/sqrt((sec(x) - 1)*(sec(x) + 1)*(sec(x)**2 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sec \left (x\right )}{\sqrt{\sec \left (x\right )^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)/sqrt(sec(x)^4 - 1),x, algorithm="giac")

[Out]

integrate(sec(x)/sqrt(sec(x)^4 - 1), x)