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

Optimal. Leaf size=14 \[ \tan (x) \sqrt {\cot ^2(x)} \log (\sin (x)) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 14, 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]

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 7, normalized size = 0.50 \[ -\log \left (\frac {1}{2} \, \sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(1/2*sin(x))

________________________________________________________________________________________

giac [B]  time = 0.58, size = 44, normalized size = 3.14 \[ \frac {1}{2} \, {\left (2 \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) - \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) \]

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

________________________________________________________________________________________

maple [B]  time = 0.68, size = 51, normalized size = 3.64 \[ \frac {\left (\ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (\frac {2}{\cos \relax (x )+1}\right )\right ) \sin \relax (x ) \sqrt {-\frac {\cos ^{2}\relax (x )}{-1+\cos ^{2}\relax (x )}}\, \sqrt {4}}{2 \cos \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 13, normalized size = 0.93 \[ -\frac {1}{2} \, \log \left (\tan \relax (x)^{2} + 1\right ) + \log \left (\tan \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(tan(x)^2 + 1) + log(tan(x))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(x)^2 - 1)^(1/2),x)

[Out]

int((1/sin(x)^2 - 1)^(1/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(csc(x)**2 - 1), x)

________________________________________________________________________________________