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 = 0.18356, 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, Rules used = {4148, 6722, 1988, 2008, 206} \[ -\frac{\sqrt{1-\cos ^4(x)} \sec ^2(x) \tanh ^{-1}\left (\frac{\sqrt{2} \sin (x)}{\sqrt{2 \sin ^2(x)-\sin ^4(x)}}\right )}{\sqrt{2} \sqrt{\sec ^4(x)-1}} \]

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]))

Rule 4148

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x
], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] &&  !IntegerQ
[p]

Rule 6722

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[(a + b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b + a/
v^n)^FracPart[p]), Int[u*v^(n*p)*(b + a/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[p] && ILtQ[n, 0] &
& BinomialQ[v, x] &&  !LinearQ[v, x]

Rule 1988

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && TrinomialQ[u, x] &&  !TrinomialMatch
Q[u, x]

Rule 2008

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sec (x)}{\sqrt{-1+\sec ^4(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{-1+\frac{1}{\left (1-x^2\right )^2}}} \, dx,x,\sin (x)\right )\\ &=\frac{\left (\sqrt{1-\cos ^4(x)} \sec ^2(x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\left (1-x^2\right )^2}} \, dx,x,\sin (x)\right )}{\sqrt{-1+\sec ^4(x)}}\\ &=\frac{\left (\sqrt{1-\cos ^4(x)} \sec ^2(x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 x^2-x^4}} \, dx,x,\sin (x)\right )}{\sqrt{-1+\sec ^4(x)}}\\ &=-\frac{\left (\sqrt{1-\cos ^4(x)} \sec ^2(x)\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\sin (x)}{\sqrt{2 \sin ^2(x)-\sin ^4(x)}}\right )}{\sqrt{-1+\sec ^4(x)}}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sin (x)}{\sqrt{2 \sin ^2(x)-\sin ^4(x)}}\right ) \sqrt{1-\cos ^4(x)} \sec ^2(x)}{\sqrt{2} \sqrt{-1+\sec ^4(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0346768, size = 45, normalized size = 1.61 \[ -\frac{\sqrt{\cos (2 x)+3} \tan (x) \sec (x) \tanh ^{-1}\left (\frac{1}{2} \sqrt{4-2 \sin ^2(x)}\right )}{2 \sqrt{\sec ^4(x)-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.137, size = 91, normalized size = 3.3 \begin{align*} -{\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}{\cos \left ( x \right ) +1}} \right ) -{\it Artanh} \left ({\frac{\sqrt{2}\sqrt{4}}{4}{\frac{1}{\sqrt{{\frac{1+ \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}}}} \right ) \right ) \sqrt{{\frac{1+ \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}{\frac{1}{\sqrt{-2\,{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}-1}{ \left ( \cos \left ( x \right ) \right ) ^{4}}}}}}} \end{align*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(-1+sec(x)^4)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.37648, size = 163, normalized size = 5.82 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(-1+sec(x)^4)^(1/2),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. \begin{align*} \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 \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(-1+sec(x)^4)^(1/2),x, algorithm="giac")

[Out]

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