3.20 \(\int \sqrt {1-\csc ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ \tan (x) \sqrt {-\cot ^2(x)} \log (\sin (x)) \]

[Out]

ln(sin(x))*(-cot(x)^2)^(1/2)*tan(x)

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Rubi [A]  time = 0.03, antiderivative size = 16, 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 = {4121, 3658, 3475} \[ \tan (x) \sqrt {-\cot ^2(x)} \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Csc[x]^2],x]

[Out]

Sqrt[-Cot[x]^2]*Log[Sin[x]]*Tan[x]

Rule 3475

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

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[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 4121

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

Rubi steps

\begin {align*} \int \sqrt {1-\csc ^2(x)} \, dx &=\int \sqrt {-\cot ^2(x)} \, dx\\ &=\left (\sqrt {-\cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=\sqrt {-\cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \tan (x) \sqrt {-\cot ^2(x)} \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Csc[x]^2],x]

[Out]

Sqrt[-Cot[x]^2]*Log[Sin[x]]*Tan[x]

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fricas [A]  time = 0.45, size = 10, normalized size = 0.62 \[ x + \arctan \left (\frac {\cos \relax (x)}{\sin \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + arctan(cos(x)/sin(x))

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giac [C]  time = 1.26, size = 48, normalized size = 3.00 \[ -\frac {1}{2} \, {\left (2 i \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{4} + 1\right ) - i \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{4} + 1\right )\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*I*log(tan(1/2*x)^2 + 1)*sgn(-tan(1/2*x)^4 + 1) - I*log(tan(1/2*x)^2)*sgn(-tan(1/2*x)^4 + 1))*sgn(sin(x
))

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maple [B]  time = 0.54, size = 46, normalized size = 2.88 \[ \frac {\left (\ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (\frac {2}{\cos \relax (x )+1}\right )\right ) \sin \relax (x ) \sqrt {\frac {\cos ^{2}\relax (x )}{-1+\cos ^{2}\relax (x )}}}{\cos \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.41, size = 15, normalized size = 0.94 \[ -\frac {1}{2} i \, \log \left (\tan \relax (x)^{2} + 1\right ) + i \, \log \left (\tan \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*log(tan(x)^2 + 1) + I*log(tan(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 1/sin(x)^2)^(1/2),x)

[Out]

int((1 - 1/sin(x)^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 - csc(x)**2), x)

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