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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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*}

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Mathematica [A]  time = 0.01, 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]

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fricas [B]  time = 0.45, size = 70, normalized size = 4.12 \[ \left [\frac {\sqrt {-a \cos \relax (x)^{2} + a} \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right )}{2 \, a \sin \relax (x)}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a \cos \relax (x)^{2} + a} \sqrt {-a} \cos \relax (x)}{a \sin \relax (x)}\right )}{a}\right ] \]

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]

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giac [A]  time = 0.16, size = 15, normalized size = 0.88 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\sin \relax (x)\right )} \]

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)))

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maple [B]  time = 0.90, size = 49, normalized size = 2.88 \[ -\frac {\sin \relax (x ) \sqrt {a \left (\cos ^{2}\relax (x )\right )}\, \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\relax (x )\right )}+2 a}{\sin \relax (x )}\right )}{\sqrt {a}\, \cos \relax (x ) \sqrt {a \left (\sin ^{2}\relax (x )\right )}} \]

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)

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maxima [A]  time = 0.63, size = 26, normalized size = 1.53 \[ \frac {\sqrt {-a} {\left (\arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )}}{a} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {a\,{\sin \relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*sin(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sin ^{2}{\relax (x )}}}\, dx \]

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)

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