3.337 \(\int \frac{1}{\sqrt{-\cos (x)+\sec (x)}} \, dx\)

Optimal. Leaf size=52 \[ \frac{\sin (x) \tan ^{-1}\left (\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\sin (x) \tanh ^{-1}\left (\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}} \]

[Out]

(ArcTan[Sqrt[Cos[x]]]*Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) - (ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(Sqrt[Cos[x]
]*Sqrt[Sin[x]*Tan[x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0780825, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {4397, 4400, 2601, 2565, 329, 298, 203, 206} \[ \frac{\sin (x) \tan ^{-1}\left (\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\sin (x) \tanh ^{-1}\left (\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-Cos[x] + Sec[x]],x]

[Out]

(ArcTan[Sqrt[Cos[x]]]*Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) - (ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(Sqrt[Cos[x]
]*Sqrt[Sin[x]*Tan[x]])

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 4400

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rule 2601

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(Cos[e + f*x
]^n*(b*Tan[e + f*x])^n)/(a*Sin[e + f*x])^n, Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(-1)]) || IntegersQ[m - 1/2, n -
1/2])

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

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

Mathematica [A]  time = 0.245932, size = 43, normalized size = 0.83 \[ \frac{\cos (x) \cot (x) \sqrt{\sin (x) \tan (x)} \left (\tan ^{-1}\left (\sqrt [4]{\cos ^2(x)}\right )-\tanh ^{-1}\left (\sqrt [4]{\cos ^2(x)}\right )\right )}{\cos ^2(x)^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-Cos[x] + Sec[x]],x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.108, size = 105, normalized size = 2. \begin{align*} -{\frac{1+\cos \left ( x \right ) }{2\,\sin \left ( x \right ) } \left ( \arctan \left ({\frac{1}{2}{\frac{1}{\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}}}} \right ) +\ln \left ( -{\frac{1}{ \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}- \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\cos \left ( x \right ) -2\,\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}-1 \right ) } \right ) \right ) \sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}\sqrt{{\frac{1- \left ( \cos \left ( x \right ) \right ) ^{2}}{\cos \left ( x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(arctan(1/2/(-cos(x)/(1+cos(x))^2)^(1/2))+ln(-(2*cos(x)^2*(-cos(x)/(1+cos(x))^2)^(1/2)-cos(x)^2+2*cos(x)-
2*(-cos(x)/(1+cos(x))^2)^(1/2)-1)/sin(x)^2))*(1+cos(x))*(-cos(x)/(1+cos(x))^2)^(1/2)*((1-cos(x)^2)/cos(x))^(1/
2)/sin(x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.32496, size = 228, normalized size = 4.38 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) + \frac{1}{2} \, \log \left (\frac{{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(2*sqrt(-(cos(x)^2 - 1)/cos(x))*cos(x)/((cos(x) - 1)*sin(x))) + 1/2*log(((cos(x) + 1)*sin(x) - 2*sq
rt(-(cos(x)^2 - 1)/cos(x))*cos(x))/((cos(x) - 1)*sin(x)))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-cos(x) + sec(x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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