Integrand size = 10, antiderivative size = 42 \[ \int \sqrt {-1+\cot ^2(x)} \, dx=-\text {arctanh}\left (\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {-1+\cot ^2(x)}}\right ) \] Output:
-arctanh(cot(x)/(-1+cot(x)^2)^(1/2))+2^(1/2)*arctanh(2^(1/2)*cot(x)/(-1+co t(x)^2)^(1/2))
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.43 \[ \int \sqrt {-1+\cot ^2(x)} \, dx=\frac {\sqrt {-1+\cot ^2(x)} \left (-\text {arctanh}\left (\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right )+\sqrt {2} \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )\right ) \sin (x)}{\sqrt {\cos (2 x)}} \] Input:
Integrate[Sqrt[-1 + Cot[x]^2],x]
Output:
(Sqrt[-1 + Cot[x]^2]*(-ArcTanh[Cos[x]/Sqrt[Cos[2*x]]] + Sqrt[2]*Log[Sqrt[2 ]*Cos[x] + Sqrt[Cos[2*x]]])*Sin[x])/Sqrt[Cos[2*x]]
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 4144, 301, 224, 219, 291, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {\cot ^2(x)-1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {\tan \left (x+\frac {\pi }{2}\right )^2-1}dx\) |
\(\Big \downarrow \) 4144 |
\(\displaystyle -\int \frac {\sqrt {\cot ^2(x)-1}}{\cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 301 |
\(\displaystyle 2 \int \frac {1}{\sqrt {\cot ^2(x)-1} \left (\cot ^2(x)+1\right )}d\cot (x)-\int \frac {1}{\sqrt {\cot ^2(x)-1}}d\cot (x)\) |
\(\Big \downarrow \) 224 |
\(\displaystyle 2 \int \frac {1}{\sqrt {\cot ^2(x)-1} \left (\cot ^2(x)+1\right )}d\cot (x)-\int \frac {1}{1-\frac {\cot ^2(x)}{\cot ^2(x)-1}}d\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \int \frac {1}{\sqrt {\cot ^2(x)-1} \left (\cot ^2(x)+1\right )}d\cot (x)-\text {arctanh}\left (\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle 2 \int \frac {1}{1-\frac {2 \cot ^2(x)}{\cot ^2(x)-1}}d\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}-\text {arctanh}\left (\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {\cot ^2(x)-1}}\right )-\text {arctanh}\left (\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\right )\) |
Input:
Int[Sqrt[-1 + Cot[x]^2],x]
Output:
-ArcTanh[Cot[x]/Sqrt[-1 + Cot[x]^2]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Cot[x])/Sq rt[-1 + Cot[x]^2]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ d Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b*x^2)^ (p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E qQ[b*c + 3*a*d, 0]))
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(a + b* (ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\ln \left (\cot \left (x \right )+\sqrt {-1+\cot \left (x \right )^{2}}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \cot \left (x \right )}{\sqrt {-1+\cot \left (x \right )^{2}}}\right )\) | \(35\) |
default | \(-\ln \left (\cot \left (x \right )+\sqrt {-1+\cot \left (x \right )^{2}}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \cot \left (x \right )}{\sqrt {-1+\cot \left (x \right )^{2}}}\right )\) | \(35\) |
Input:
int((-1+cot(x)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-ln(cot(x)+(-1+cot(x)^2)^(1/2))+2^(1/2)*arctanh(2^(1/2)*cot(x)/(-1+cot(x)^ 2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (34) = 68\).
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.93 \[ \int \sqrt {-1+\cot ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-2 \, \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - 1\right ) - \frac {1}{2} \, \log \left (\frac {\sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\cos \left (2 \, x\right ) + 1}\right ) + \frac {1}{2} \, \log \left (\frac {\sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1}{\cos \left (2 \, x\right ) + 1}\right ) \] Input:
integrate((-1+cot(x)^2)^(1/2),x, algorithm="fricas")
Output:
1/2*sqrt(2)*log(-2*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - 2*cos(2*x) - 1) - 1/2*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) + cos(2*x) + 1)/(cos(2*x) + 1)) + 1/2*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin( 2*x) - cos(2*x) - 1)/(cos(2*x) + 1))
\[ \int \sqrt {-1+\cot ^2(x)} \, dx=\int \sqrt {\cot ^{2}{\left (x \right )} - 1}\, dx \] Input:
integrate((-1+cot(x)**2)**(1/2),x)
Output:
Integral(sqrt(cot(x)**2 - 1), x)
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 941, normalized size of antiderivative = 22.40 \[ \int \sqrt {-1+\cot ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate((-1+cot(x)^2)^(1/2),x, algorithm="maxima")
Output:
1/2*sqrt(2)*arcsinh(1) + 1/4*sqrt(2)*log(cos(2*x)^2 + sin(2*x)^2 + sqrt(co s(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*(cos(1/2*arctan2(sin(4*x), cos(4*x ) + 1))^2 + sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2) + 2*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(cos(2*x)*cos(1/2*arctan2(sin(4*x), cos (4*x) + 1)) + sin(2*x)*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1)))) - 1/2*lo g((sqrt(abs(2*e^(2*I*x) - 2)^4 + 16*cos(2*x)^4 + 16*sin(2*x)^4 + 8*(cos(2* x)^2 - sin(2*x)^2 + 2*cos(2*x) + 1)*abs(2*e^(2*I*x) - 2)^2 + 64*cos(2*x)^3 + 32*(cos(2*x)^2 + 2*cos(2*x) + 1)*sin(2*x)^2 + 96*cos(2*x)^2 + 64*cos(2* x) + 16)*cos(1/2*arctan2(8*(cos(2*x) + 1)*sin(2*x)/abs(2*e^(2*I*x) - 2)^2, (abs(2*e^(2*I*x) - 2)^2 + 4*cos(2*x)^2 - 4*sin(2*x)^2 + 8*cos(2*x) + 4)/a bs(2*e^(2*I*x) - 2)^2))^2 + sqrt(abs(2*e^(2*I*x) - 2)^4 + 16*cos(2*x)^4 + 16*sin(2*x)^4 + 8*(cos(2*x)^2 - sin(2*x)^2 + 2*cos(2*x) + 1)*abs(2*e^(2*I* x) - 2)^2 + 64*cos(2*x)^3 + 32*(cos(2*x)^2 + 2*cos(2*x) + 1)*sin(2*x)^2 + 96*cos(2*x)^2 + 64*cos(2*x) + 16)*sin(1/2*arctan2(8*(cos(2*x) + 1)*sin(2*x )/abs(2*e^(2*I*x) - 2)^2, (abs(2*e^(2*I*x) - 2)^2 + 4*cos(2*x)^2 - 4*sin(2 *x)^2 + 8*cos(2*x) + 4)/abs(2*e^(2*I*x) - 2)^2))^2 + 4*(abs(2*e^(2*I*x) - 2)^4 + 16*cos(2*x)^4 + 16*sin(2*x)^4 + 8*(cos(2*x)^2 - sin(2*x)^2 + 2*cos( 2*x) + 1)*abs(2*e^(2*I*x) - 2)^2 + 64*cos(2*x)^3 + 32*(cos(2*x)^2 + 2*cos( 2*x) + 1)*sin(2*x)^2 + 96*cos(2*x)^2 + 64*cos(2*x) + 16)^(1/4)*(cos(2*x) + 1)*cos(1/2*arctan2(8*(cos(2*x) + 1)*sin(2*x)/abs(2*e^(2*I*x) - 2)^2, (...
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (34) = 68\).
Time = 5.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.24 \[ \int \sqrt {-1+\cot ^2(x)} \, dx=-\frac {1}{2} \, {\left (\sqrt {2} \log \left ({\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{2}\right ) - \log \left ({\left | {\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} + 2 \, \sqrt {2} - 3 \right |}\right ) + \log \left ({\left | {\left (\sqrt {2} \cos \left (x\right ) - \sqrt {2 \, \cos \left (x\right )^{2} - 1}\right )}^{2} - 2 \, \sqrt {2} - 3 \right |}\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate((-1+cot(x)^2)^(1/2),x, algorithm="giac")
Output:
-1/2*(sqrt(2)*log((sqrt(2)*cos(x) - sqrt(2*cos(x)^2 - 1))^2) - log(abs((sq rt(2)*cos(x) - sqrt(2*cos(x)^2 - 1))^2 + 2*sqrt(2) - 3)) + log(abs((sqrt(2 )*cos(x) - sqrt(2*cos(x)^2 - 1))^2 - 2*sqrt(2) - 3)))*sgn(sin(x))
Time = 9.78 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \sqrt {-1+\cot ^2(x)} \, dx=\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\mathrm {cot}\left (x\right )}{\sqrt {{\mathrm {cot}\left (x\right )}^2-1}}\right )-\ln \left (\mathrm {cot}\left (x\right )+\sqrt {{\mathrm {cot}\left (x\right )}^2-1}\right ) \] Input:
int((cot(x)^2 - 1)^(1/2),x)
Output:
2^(1/2)*atanh((2^(1/2)*cot(x))/(cot(x)^2 - 1)^(1/2)) - log(cot(x) + (cot(x )^2 - 1)^(1/2))
\[ \int \sqrt {-1+\cot ^2(x)} \, dx=\int \sqrt {\cot \left (x \right )^{2}-1}d x \] Input:
int((-1+cot(x)^2)^(1/2),x)
Output:
int(sqrt(cot(x)**2 - 1),x)