Integrand size = 10, antiderivative size = 33 \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=-\arctan \left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )-\text {arctanh}\left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(33)=66\).
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.06 \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=\frac {\sqrt {2} \sqrt {-1+\text {csch}^2(x)} \left (\arctan \left (\frac {\sqrt {2} \cosh (x)}{\sqrt {-3+\cosh (2 x)}}\right )+\log \left (\sqrt {2} \cosh (x)+\sqrt {-3+\cosh (2 x)}\right )\right ) \sinh (x)}{\sqrt {-3+\cosh (2 x)}} \]
(Sqrt[2]*Sqrt[-1 + Csch[x]^2]*(ArcTan[(Sqrt[2]*Cosh[x])/Sqrt[-3 + Cosh[2*x ]]] + Log[Sqrt[2]*Cosh[x] + Sqrt[-3 + Cosh[2*x]]])*Sinh[x])/Sqrt[-3 + Cosh [2*x]]
Time = 0.22 (sec) , antiderivative size = 33, 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, 4616, 301, 224, 219, 291, 216}
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.
\(\displaystyle \int \sqrt {\text {csch}^2(x)-1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {-1-\sec \left (\frac {\pi }{2}+i x\right )^2}dx\) |
\(\Big \downarrow \) 4616 |
\(\displaystyle \int \frac {\sqrt {\coth ^2(x)-2}}{1-\coth ^2(x)}d\coth (x)\) |
\(\Big \downarrow \) 301 |
\(\displaystyle -\int \frac {1}{\sqrt {\coth ^2(x)-2}}d\coth (x)-\int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {\coth ^2(x)-2}}d\coth (x)\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {\coth ^2(x)-2}}d\coth (x)-\int \frac {1}{1-\frac {\coth ^2(x)}{\coth ^2(x)-2}}d\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {\coth ^2(x)-2}}d\coth (x)-\text {arctanh}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\int \frac {1}{\frac {\coth ^2(x)}{\coth ^2(x)-2}+1}d\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}-\text {arctanh}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\arctan \left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )-\text {arctanh}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )\) |
3.1.23.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b + b*ff^2*x^2)^p /(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
\[\int \sqrt {-1+\operatorname {csch}\left (x \right )^{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 358, normalized size of antiderivative = 10.85 \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=\frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right ) - \frac {1}{2} \, \log \left (\frac {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt {2} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac {1}{2} \, \log \left (\frac {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt {2} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) \]
1/2*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-( cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(c osh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 2)*sinh(x)^2 + 4*cosh(x)^2 + 4*(cosh(x)^3 + 2*cosh(x))*sinh(x) - 1)) + 1/2*arctan(sqrt (2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-(cosh(x)^2 + sin h(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 + 4*co sh(x)*sinh(x)^3 + sinh(x)^4 + 6*(cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x))*sinh(x) + 1)) - 1/2*log((cosh(x)^2 + 2*cosh(x)*s inh(x) + sinh(x)^2 + sqrt(2)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + si nh(x)^2)) + 1/2*log((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(2)*s qrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^ 2)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2))
\[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=\int \sqrt {\operatorname {csch}^{2}{\left (x \right )} - 1}\, dx \]
\[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=\int { \sqrt {\operatorname {csch}\left (x\right )^{2} - 1} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (29) = 58\).
Time = 0.30 (sec) , antiderivative size = 157, normalized size of antiderivative = 4.76 \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=\frac {1}{2} \, {\left (\arcsin \left (\frac {1}{4} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 3\right )}\right ) + 2 \, \arctan \left (-2 \, \sqrt {2} - \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3}\right ) - 2 \, \log \left ({\left | -\sqrt {2} - \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3} + 1 \right |}\right ) + 2 \, \log \left ({\left | -\sqrt {2} - \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3} - 1 \right |}\right )\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) \]
1/2*(arcsin(1/4*sqrt(2)*(e^(2*x) - 3)) + 2*arctan(-2*sqrt(2) - 3*(2*sqrt(2 ) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/(e^(2*x) - 3)) - 2*log(abs(-sqrt(2) - (2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/(e^(2*x) - 3) + 1)) + 2*log(a bs(-sqrt(2) - (2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/(e^(2*x) - 3) - 1)))*sgn(-e^(2*x) + 1)
Timed out. \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=\int \sqrt {\frac {1}{{\mathrm {sinh}\left (x\right )}^2}-1} \,d x \]