Integrand size = 10, antiderivative size = 31 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=-\text {arcsinh}(\tanh (x))+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right ) \] Output:
-arcsinh(tanh(x))+2^(1/2)*arctanh(2^(1/2)*tanh(x)/(1+tanh(x)^2)^(1/2))
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=\frac {\left (\sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh (x)\right )-\text {arctanh}\left (\frac {\sinh (x)}{\sqrt {\cosh (2 x)}}\right )\right ) \cosh (x) \sqrt {1+\tanh ^2(x)}}{\sqrt {\cosh (2 x)}} \] Input:
Integrate[Sqrt[1 + Tanh[x]^2],x]
Output:
((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 = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4144, 301, 222, 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.
\(\displaystyle \int \sqrt {\tanh ^2(x)+1} \, dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle 2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {\tanh ^2(x)+1}}d\tanh (x)-\int \frac {1}{\sqrt {\tanh ^2(x)+1}}d\tanh (x)\) |
\(\Big \downarrow \) 222 |
\(\displaystyle 2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {\tanh ^2(x)+1}}d\tanh (x)-\text {arcsinh}(\tanh (x))\) |
\(\Big \downarrow \) 291 |
\(\displaystyle 2 \int \frac {1}{1-\frac {2 \tanh ^2(x)}{\tanh ^2(x)+1}}d\frac {\tanh (x)}{\sqrt {\tanh ^2(x)+1}}-\text {arcsinh}(\tanh (x))\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {\tanh ^2(x)+1}}\right )-\text {arcsinh}(\tanh (x))\) |
Input:
Int[Sqrt[1 + Tanh[x]^2],x]
Output:
-ArcSinh[Tanh[x]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Tanh[x])/Sqrt[1 + Tanh[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] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
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. \(96\) vs. \(2(25)=50\).
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.13
method | result | size |
derivativedivides | \(-\frac {\sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}{2}-\operatorname {arcsinh}\left (\tanh \left (x \right )\right )+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}\right )}{2}+\frac {\sqrt {\left (\tanh \left (x \right )+1\right )^{2}-2 \tanh \left (x \right )}}{2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )+1\right )^{2}-2 \tanh \left (x \right )}}\right )}{2}\) | \(97\) |
default | \(-\frac {\sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}{2}-\operatorname {arcsinh}\left (\tanh \left (x \right )\right )+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}\right )}{2}+\frac {\sqrt {\left (\tanh \left (x \right )+1\right )^{2}-2 \tanh \left (x \right )}}{2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )+1\right )^{2}-2 \tanh \left (x \right )}}\right )}{2}\) | \(97\) |
Input:
int((1+tanh(x)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*((tanh(x)-1)^2+2*tanh(x))^(1/2)-arcsinh(tanh(x))+1/2*2^(1/2)*arctanh( 1/4*(2+2*tanh(x))*2^(1/2)/((tanh(x)-1)^2+2*tanh(x))^(1/2))+1/2*((tanh(x)+1 )^2-2*tanh(x))^(1/2)-1/2*2^(1/2)*arctanh(1/4*(2-2*tanh(x))*2^(1/2)/((tanh( x)+1)^2-2*tanh(x))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 679, normalized size of antiderivative = 21.90 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate((1+tanh(x)^2)^(1/2),x, algorithm="fricas")
Output:
1/4*sqrt(2)*log(-2*(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + (28*cosh (x)^2 - 3)*sinh(x)^6 - 3*cosh(x)^6 + 2*(28*cosh(x)^3 - 9*cosh(x))*sinh(x)^ 5 + 5*(14*cosh(x)^4 - 9*cosh(x)^2 + 1)*sinh(x)^4 + 5*cosh(x)^4 + 4*(14*cos h(x)^5 - 15*cosh(x)^3 + 5*cosh(x))*sinh(x)^3 + (28*cosh(x)^6 - 45*cosh(x)^ 4 + 30*cosh(x)^2 - 4)*sinh(x)^2 - 4*cosh(x)^2 + 2*(4*cosh(x)^7 - 9*cosh(x) ^5 + 10*cosh(x)^3 - 4*cosh(x))*sinh(x) + (sqrt(2)*cosh(x)^6 + 6*sqrt(2)*co sh(x)*sinh(x)^5 + sqrt(2)*sinh(x)^6 + 3*(5*sqrt(2)*cosh(x)^2 - sqrt(2))*si nh(x)^4 - 3*sqrt(2)*cosh(x)^4 + 4*(5*sqrt(2)*cosh(x)^3 - 3*sqrt(2)*cosh(x) )*sinh(x)^3 + (15*sqrt(2)*cosh(x)^4 - 18*sqrt(2)*cosh(x)^2 + 4*sqrt(2))*si nh(x)^2 + 4*sqrt(2)*cosh(x)^2 + 2*(3*sqrt(2)*cosh(x)^5 - 6*sqrt(2)*cosh(x) ^3 + 4*sqrt(2)*cosh(x))*sinh(x) - 4*sqrt(2))*sqrt((cosh(x)^2 + sinh(x)^2)/ (cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4)/(cosh(x)^6 + 6*cosh(x)^5 *sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2* sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 1/4*sqrt(2)*log(2*(cosh(x) ^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 + 1)*sinh(x)^2 + cosh( x)^2 + 2*(2*cosh(x)^3 + cosh(x))*sinh(x) + (sqrt(2)*cosh(x)^2 + 2*sqrt(2)* cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*sqrt((cosh(x)^2 + sinh(x)^2 )/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1)/(cosh(x)^2 + 2*cosh(x) *sinh(x) + sinh(x)^2)) - 1/2*log((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^ 2 + 2*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sin...
\[ \int \sqrt {1+\tanh ^2(x)} \, dx=\int \sqrt {\tanh ^{2}{\left (x \right )} + 1}\, dx \] Input:
integrate((1+tanh(x)**2)**(1/2),x)
Output:
Integral(sqrt(tanh(x)**2 + 1), x)
\[ \int \sqrt {1+\tanh ^2(x)} \, dx=\int { \sqrt {\tanh \left (x\right )^{2} + 1} \,d x } \] Input:
integrate((1+tanh(x)^2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(tanh(x)^2 + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.35 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \] Input:
integrate((1+tanh(x)^2)^(1/2),x, algorithm="giac")
Output:
-1/2*sqrt(2)*(sqrt(2)*log((sqrt(2) - sqrt(e^(4*x) + 1) + e^(2*x) + 1)/(sqr t(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 = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \sqrt {1+\tanh ^2(x)} \, dx=\frac {\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\left (x\right )+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}-\mathrm {tanh}\left (x\right )+1\right )\right )}{2}-\mathrm {asinh}\left (\mathrm {tanh}\left (x\right )\right )+\frac {\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\left (x\right )+\sqrt {2}\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}+1\right )-\ln \left (\mathrm {tanh}\left (x\right )-1\right )\right )}{2} \] Input:
int((tanh(x)^2 + 1)^(1/2),x)
Output:
(2^(1/2)*(log(tanh(x) + 1) - log(2^(1/2)*(tanh(x)^2 + 1)^(1/2) - tanh(x) + 1)))/2 - asinh(tanh(x)) + (2^(1/2)*(log(tanh(x) + 2^(1/2)*(tanh(x)^2 + 1) ^(1/2) + 1) - log(tanh(x) - 1)))/2
\[ \int \sqrt {1+\tanh ^2(x)} \, dx=\int \sqrt {\tanh \left (x \right )^{2}+1}d x \] Input:
int((1+tanh(x)^2)^(1/2),x)
Output:
int(sqrt(tanh(x)**2 + 1),x)