Integrand size = 12, antiderivative size = 45 \[ \int \sqrt {-1-\tanh ^2(x)} \, dx=\arctan \left (\frac {\tanh (x)}{\sqrt {-1-\tanh ^2(x)}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} \tanh (x)}{\sqrt {-1-\tanh ^2(x)}}\right ) \]
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18 \[ \int \sqrt {-1-\tanh ^2(x)} \, dx=\frac {\left (\sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh (x)\right )-\text {arctanh}\left (\frac {\sinh (x)}{\sqrt {\cosh (2 x)}}\right )\right ) \cosh (x) \sqrt {-1-\tanh ^2(x)}}{\sqrt {\cosh (2 x)}} \]
((Sqrt[2]*ArcSinh[Sqrt[2]*Sinh[x]] - ArcTanh[Sinh[x]/Sqrt[Cosh[2*x]]])*Cos h[x]*Sqrt[-1 - Tanh[x]^2])/Sqrt[Cosh[2*x]]
Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3042, 4144, 301, 224, 216, 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.
\(\displaystyle \int \sqrt {-\tanh ^2(x)-1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {-1+\tan (i x)^2}dx\) |
\(\Big \downarrow \) 4144 |
\(\displaystyle \int \frac {\sqrt {-\tanh ^2(x)-1}}{1-\tanh ^2(x)}d\tanh (x)\) |
\(\Big \downarrow \) 301 |
\(\displaystyle \int \frac {1}{\sqrt {-\tanh ^2(x)-1}}d\tanh (x)-2 \int \frac {1}{\sqrt {-\tanh ^2(x)-1} \left (1-\tanh ^2(x)\right )}d\tanh (x)\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \int \frac {1}{\frac {\tanh ^2(x)}{-\tanh ^2(x)-1}+1}d\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}-2 \int \frac {1}{\sqrt {-\tanh ^2(x)-1} \left (1-\tanh ^2(x)\right )}d\tanh (x)\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \arctan \left (\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )-2 \int \frac {1}{\sqrt {-\tanh ^2(x)-1} \left (1-\tanh ^2(x)\right )}d\tanh (x)\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \arctan \left (\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )-2 \int \frac {1}{\frac {2 \tanh ^2(x)}{-\tanh ^2(x)-1}+1}d\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \arctan \left (\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} \tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )\) |
ArcTan[Tanh[x]/Sqrt[-1 - Tanh[x]^2]] - Sqrt[2]*ArcTan[(Sqrt[2]*Tanh[x])/Sq rt[-1 - Tanh[x]^2]]
3.3.26.3.1 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[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])
Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(37)=74\).
Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.16
method | result | size |
derivativedivides | \(\frac {\sqrt {-\left (1+\tanh \left (x \right )\right )^{2}+2 \tanh \left (x \right )}}{2}+\frac {\arctan \left (\frac {\tanh \left (x \right )}{\sqrt {-\left (1+\tanh \left (x \right )\right )^{2}+2 \tanh \left (x \right )}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\tanh \left (x \right )\right )^{2}+2 \tanh \left (x \right )}}\right )}{2}-\frac {\sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}{2}+\frac {\arctan \left (\frac {\tanh \left (x \right )}{\sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}\right )}{2}\) | \(142\) |
default | \(\frac {\sqrt {-\left (1+\tanh \left (x \right )\right )^{2}+2 \tanh \left (x \right )}}{2}+\frac {\arctan \left (\frac {\tanh \left (x \right )}{\sqrt {-\left (1+\tanh \left (x \right )\right )^{2}+2 \tanh \left (x \right )}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\tanh \left (x \right )\right )^{2}+2 \tanh \left (x \right )}}\right )}{2}-\frac {\sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}{2}+\frac {\arctan \left (\frac {\tanh \left (x \right )}{\sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}\right )}{2}\) | \(142\) |
1/2*(-(1+tanh(x))^2+2*tanh(x))^(1/2)+1/2*arctan(tanh(x)/(-(1+tanh(x))^2+2* tanh(x))^(1/2))-1/2*2^(1/2)*arctan(1/4*(-2+2*tanh(x))*2^(1/2)/(-(1+tanh(x) )^2+2*tanh(x))^(1/2))-1/2*(-(tanh(x)-1)^2-2*tanh(x))^(1/2)+1/2*arctan(tanh (x)/(-(tanh(x)-1)^2-2*tanh(x))^(1/2))+1/2*2^(1/2)*arctan(1/4*(-2-2*tanh(x) )*2^(1/2)/(-(tanh(x)-1)^2-2*tanh(x))^(1/2))
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 226, normalized size of antiderivative = 5.02 \[ \int \sqrt {-1-\tanh ^2(x)} \, dx=-\frac {1}{4} \, \sqrt {-2} \log \left (-{\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )}\right ) + \frac {1}{4} \, \sqrt {-2} \log \left ({\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} - 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) + \frac {1}{4} \, \sqrt {-2} \log \left (-2 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} - 2\right )} + \sqrt {-2} e^{\left (4 \, x\right )} - \sqrt {-2} e^{\left (2 \, x\right )} + 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) - \frac {1}{4} \, \sqrt {-2} \log \left (-2 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} - 2\right )} - \sqrt {-2} e^{\left (4 \, x\right )} + \sqrt {-2} e^{\left (2 \, x\right )} - 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) + \frac {1}{2} i \, \log \left (-4 \, {\left (i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{2} i \, \log \left (-4 \, {\left (-i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}\right ) \]
-1/4*sqrt(-2)*log(-(sqrt(-2)*sqrt(-2*e^(4*x) - 2) + 2*e^(2*x) + 2)*e^(-2*x )) + 1/4*sqrt(-2)*log((sqrt(-2)*sqrt(-2*e^(4*x) - 2) - 2*e^(2*x) - 2)*e^(- 2*x)) + 1/4*sqrt(-2)*log(-2*(sqrt(-2*e^(4*x) - 2)*(e^(2*x) - 2) + sqrt(-2) *e^(4*x) - sqrt(-2)*e^(2*x) + 2*sqrt(-2))*e^(-4*x)) - 1/4*sqrt(-2)*log(-2* (sqrt(-2*e^(4*x) - 2)*(e^(2*x) - 2) - sqrt(-2)*e^(4*x) + sqrt(-2)*e^(2*x) - 2*sqrt(-2))*e^(-4*x)) + 1/2*I*log(-4*(I*sqrt(-2*e^(4*x) - 2) + e^(2*x) - 1)*e^(-2*x)) - 1/2*I*log(-4*(-I*sqrt(-2*e^(4*x) - 2) + e^(2*x) - 1)*e^(-2 *x))
\[ \int \sqrt {-1-\tanh ^2(x)} \, dx=\int \sqrt {- \tanh ^{2}{\left (x \right )} - 1}\, dx \]
\[ \int \sqrt {-1-\tanh ^2(x)} \, dx=\int { \sqrt {-\tanh \left (x\right )^{2} - 1} \,d x } \]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.31 \[ \int \sqrt {-1-\tanh ^2(x)} \, dx=-\frac {1}{2} i \, \sqrt {2} {\left (\sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \]
-1/2*I*sqrt(2)*(sqrt(2)*log((sqrt(2) - sqrt(e^(4*x) + 1) + e^(2*x) + 1)/(s qrt(2) + sqrt(e^(4*x) + 1) - e^(2*x) - 1)) + log(sqrt(e^(4*x) + 1) - e^(2* x) + 1) + log(sqrt(e^(4*x) + 1) - e^(2*x)) - log(-sqrt(e^(4*x) + 1) + e^(2 *x) + 1))
Time = 1.94 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \sqrt {-1-\tanh ^2(x)} \, dx=-\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tanh}\left (x\right )}{\sqrt {-{\mathrm {tanh}\left (x\right )}^2-1}}\right )-\ln \left (\mathrm {tanh}\left (x\right )-\sqrt {-{\mathrm {tanh}\left (x\right )}^2-1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]