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

Optimal. Leaf size=17 \[ -\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{a \sin ^2(x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0113021, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3770} \[ -\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{a \sin ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Sin[x]^2],x]

[Out]

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

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

Mathematica [A]  time = 0.0148339, size = 30, normalized size = 1.76 \[ \frac{\sin (x) \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )}{\sqrt{a \sin ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Sin[x]^2],x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.599, size = 49, normalized size = 2.9 \begin{align*} -{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}+a}{\sin \left ( x \right ) }} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.59911, size = 35, normalized size = 2.06 \begin{align*} \frac{\sqrt{-a}{\left (\arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a)*(arctan2(sin(x), cos(x) + 1) - arctan2(sin(x), cos(x) - 1))/a

________________________________________________________________________________________

Fricas [B]  time = 1.71577, size = 193, normalized size = 11.35 \begin{align*} \left [\frac{\sqrt{-a \cos \left (x\right )^{2} + a} \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{2 \, a \sin \left (x\right )}, \frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-a \cos \left (x\right )^{2} + a} \sqrt{-a} \cos \left (x\right )}{a \sin \left (x\right )}\right )}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*sin(x)**2), x)

________________________________________________________________________________________

Giac [A]  time = 1.23031, size = 20, normalized size = 1.18 \begin{align*} \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{\sqrt{a} \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(tan(1/2*x)))/(sqrt(a)*sgn(sin(x)))