3.119 \(\int \frac{1}{\sqrt{a-a \sin ^2(x)}} \, dx\)

Optimal. Leaf size=16 \[ \frac{\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \cos ^2(x)}} \]

[Out]

(ArcTanh[Sin[x]]*Cos[x])/Sqrt[a*Cos[x]^2]

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Rubi [A]  time = 0.0306739, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3176, 3207, 3770} \[ \frac{\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \cos ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a - a*Sin[x]^2],x]

[Out]

(ArcTanh[Sin[x]]*Cos[x])/Sqrt[a*Cos[x]^2]

Rule 3176

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a-a \sin ^2(x)}} \, dx &=\int \frac{1}{\sqrt{a \cos ^2(x)}} \, dx\\ &=\frac{\cos (x) \int \sec (x) \, dx}{\sqrt{a \cos ^2(x)}}\\ &=\frac{\tanh ^{-1}(\sin (x)) \cos (x)}{\sqrt{a \cos ^2(x)}}\\ \end{align*}

Mathematica [B]  time = 0.0194808, size = 46, normalized size = 2.88 \[ \frac{\cos (x) \left (\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right )}{\sqrt{a \cos ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a - a*Sin[x]^2],x]

[Out]

(Cos[x]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]]))/Sqrt[a*Cos[x]^2]

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Maple [C]  time = 0.11, size = 20, normalized size = 1.3 \begin{align*}{\frac{\cos \left ( x \right ){\it InverseJacobiAM} \left ( x,1 \right ) }{{\it csgn} \left ( \cos \left ( x \right ) \right ) }{\frac{1}{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-a*sin(x)^2)^(1/2),x)

[Out]

1/(a*cos(x)^2)^(1/2)/csgn(cos(x))*cos(x)*InverseJacobiAM(x,1)

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Maxima [B]  time = 1.58033, size = 51, normalized size = 3.19 \begin{align*} \frac{\log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right )}{2 \, \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(x)^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.65344, size = 182, normalized size = 11.38 \begin{align*} \left [-\frac{\sqrt{a \cos \left (x\right )^{2}} \log \left (-\frac{\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right )}{2 \, a \cos \left (x\right )}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{a \cos \left (x\right )^{2}} \sqrt{-a} \sin \left (x\right )}{a \cos \left (x\right )}\right )}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(x)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/2*sqrt(a*cos(x)^2)*log(-(sin(x) - 1)/(sin(x) + 1))/(a*cos(x)), -sqrt(-a)*arctan(sqrt(a*cos(x)^2)*sqrt(-a)*
sin(x)/(a*cos(x)))/a]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(x)**2)**(1/2),x)

[Out]

Integral(1/sqrt(-a*sin(x)**2 + a), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a \sin \left (x\right )^{2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-a*sin(x)^2 + a), x)