\(\int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 15 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=-\frac {\cot (x) \log (\cos (x))}{\sqrt {\cot ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4206, 3739, 3556} \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=-\frac {\cot (x) \log (\cos (x))}{\sqrt {\cot ^2(x)}} \]

[In]

Int[1/Sqrt[-1 + Csc[x]^2],x]

[Out]

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

Rule 3556

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

Rule 3739

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 4206

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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=-\frac {\cot (x) \log (\cos (x))}{\sqrt {\cot ^2(x)}} \]

[In]

Integrate[1/Sqrt[-1 + Csc[x]^2],x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.56 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60

method	result	size
default	\(-\frac {\operatorname {csgn}\left (\cot \left (x \right )\right ) \left (\ln \left (\csc \left (x \right )-\cot \left (x \right )+1\right )+\ln \left (-\cot \left (x \right )+\csc \left (x \right )-1\right )-\ln \left (\frac {2}{\cos \left (x \right )+1}\right )\right ) \sqrt {4}}{2}\)	\(39\)
risch	\(-\frac {\left ({\mathrm e}^{2 i x}+1\right ) x}{\sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i \left ({\mathrm e}^{2 i x}+1\right ) \ln \left ({\mathrm e}^{2 i x}+1\right )}{\sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}\)	\(92\)

[In]

int(1/(csc(x)^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*csgn(cot(x))*(ln(csc(x)-cot(x)+1)+ln(-cot(x)+csc(x)-1)-ln(2/(cos(x)+1)))*4^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=\log \left (-\cos \left (x\right )\right ) \]

[In]

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

[Out]

log(-cos(x))

Sympy [F]

\[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=\int \frac {1}{\sqrt {\csc ^{2}{\left (x \right )} - 1}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(csc(x)**2 - 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=\frac {1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) \]

[In]

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

[Out]

1/2*log(tan(x)^2 + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (13) = 26\).

Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=-\frac {1}{2} \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, x\right )^{2}} + 2\right ) + \frac {1}{2} \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, x\right )^{2}} - 2\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{{\sin \left (x\right )}^2}-1}} \,d x \]

[In]

int(1/(1/sin(x)^2 - 1)^(1/2),x)

[Out]

int(1/(1/sin(x)^2 - 1)^(1/2), x)