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

Optimal. Leaf size=17 \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {-\cot ^2(x)}} \]

[Out]

-cot(x)*ln(cos(x))/(-cot(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 17, 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} \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {-\cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - Csc[x]^2],x]

[Out]

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

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 \frac {1}{\sqrt {1-\csc ^2(x)}} \, dx &=\int \frac {1}{\sqrt {-\cot ^2(x)}} \, dx\\ &=\frac {\cot (x) \int \tan (x) \, dx}{\sqrt {-\cot ^2(x)}}\\ &=-\frac {\cot (x) \log (\cos (x))}{\sqrt {-\cot ^2(x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {-\cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - Csc[x]^2],x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 12, normalized size = 0.71 \[ x - \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - arctan(sin(x)/cos(x))

________________________________________________________________________________________

giac [C]  time = 0.64, size = 24, normalized size = 1.41 \[ -i \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + i \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right )^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.69, size = 67, normalized size = 3.94 \[ -\frac {\left (\ln \left (-\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}\right )+\ln \left (-\frac {-\sin \relax (x )-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (\frac {2}{\cos \relax (x )+1}\right )\right ) \cos \relax (x ) \sqrt {4}}{2 \sqrt {\frac {\cos ^{2}\relax (x )}{-1+\cos ^{2}\relax (x )}}\, \sin \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(ln(-(-1+cos(x)+sin(x))/sin(x))+ln(-(-sin(x)-1+cos(x))/sin(x))-ln(2/(cos(x)+1)))*cos(x)*4^(1/2)/(cos(x)^2
/(-1+cos(x)^2))^(1/2)/sin(x)

________________________________________________________________________________________

maxima [C]  time = 0.41, size = 9, normalized size = 0.53 \[ -\frac {1}{2} i \, \log \left (\tan \relax (x)^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*log(tan(x)^2 + 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - 1/sin(x)^2)^(1/2),x)

[Out]

int(1/(1 - 1/sin(x)^2)^(1/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(1 - csc(x)**2), x)

________________________________________________________________________________________