3.51 \(\int \frac {1}{\sqrt {1-\cos ^2(x)}} \, dx\)

Optimal. Leaf size=15 \[ -\frac {\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt {\sin ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3176, 3207, 3770} \[ -\frac {\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt {\sin ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - Cos[x]^2],x]

[Out]

-((ArcTanh[Cos[x]]*Sin[x])/Sqrt[Sin[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 {1-\cos ^2(x)}} \, dx &=\int \frac {1}{\sqrt {\sin ^2(x)}} \, dx\\ &=\frac {\sin (x) \int \csc (x) \, dx}{\sqrt {\sin ^2(x)}}\\ &=-\frac {\tanh ^{-1}(\cos (x)) \sin (x)}{\sqrt {\sin ^2(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.87 \[ \frac {\sin (x) \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )\right )}{\sqrt {\sin ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - Cos[x]^2],x]

[Out]

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

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fricas [A]  time = 1.35, size = 19, normalized size = 1.27 \[ -\frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(1/2*cos(x) + 1/2) + 1/2*log(-1/2*cos(x) + 1/2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\cos \relax (x)^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-cos(x)^2 + 1), x)

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maple [A]  time = 0.71, size = 14, normalized size = 0.93 \[ -\frac {2 \arctanh \left (\cos \relax (x )\right ) \sin \relax (x )}{\sqrt {2-2 \cos \left (2 x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.13, size = 35, normalized size = 2.33 \[ \frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {1-{\cos \relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - cos(x)^2)^(1/2),x)

[Out]

int(1/(1 - cos(x)^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(1 - cos(x)**2), x)

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