Integrand size = 15, antiderivative size = 44 \[ \int \coth (x) \sqrt {a+b \coth ^2(x)} \, dx=\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )-\sqrt {a+b \coth ^2(x)} \]
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \coth (x) \sqrt {a+b \coth ^2(x)} \, dx=\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )-\sqrt {a+b \coth ^2(x)} \]
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 26, 4153, 26, 353, 60, 73, 221}
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.
\(\displaystyle \int \coth (x) \sqrt {a+b \coth ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \tan \left (\frac {\pi }{2}+i x\right ) \sqrt {a-b \tan \left (\frac {\pi }{2}+i x\right )^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \tan \left (i x+\frac {\pi }{2}\right ) \sqrt {a-b \tan \left (i x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle -i \int \frac {i \coth (x) \sqrt {b \coth ^2(x)+a}}{1-\coth ^2(x)}d\coth (x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\coth (x) \sqrt {a+b \coth ^2(x)}}{1-\coth ^2(x)}d\coth (x)\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {b \coth ^2(x)+a}}{1-\coth ^2(x)}d\coth ^2(x)\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left ((a+b) \int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth ^2(x)-2 \sqrt {a+b \coth ^2(x)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {2 (a+b) \int \frac {1}{\frac {a+b}{b}-\frac {\coth ^4(x)}{b}}d\sqrt {b \coth ^2(x)+a}}{b}-2 \sqrt {a+b \coth ^2(x)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (2 \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )-2 \sqrt {a+b \coth ^2(x)}\right )\) |
3.1.18.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(237\) vs. \(2(36)=72\).
Time = 0.09 (sec) , antiderivative size = 238, normalized size of antiderivative = 5.41
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2}-\frac {\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2}\) | \(238\) |
default | \(-\frac {\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2}-\frac {\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2}\) | \(238\) |
-1/2*(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)+1/2*b^(1/2)*ln((b*(1+coth (x))-b)/b^(1/2)+(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))+1/2*(a+b)^(1/ 2)*ln((2*a+2*b-2*b*(1+coth(x))+2*(a+b)^(1/2)*(b*(1+coth(x))^2-2*b*(1+coth( x))+a+b)^(1/2))/(1+coth(x)))-1/2*(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/ 2)-1/2*b^(1/2)*ln((b*(coth(x)-1)+b)/b^(1/2)+(b*(coth(x)-1)^2+2*b*(coth(x)- 1)+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b+2*b*(coth(x)-1)+2*(a+b)^(1/2)*( b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))/(coth(x)-1))
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (36) = 72\).
Time = 0.31 (sec) , antiderivative size = 1551, normalized size of antiderivative = 35.25 \[ \int \coth (x) \sqrt {a+b \coth ^2(x)} \, dx=\text {Too large to display} \]
[1/4*((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(a + b)*log(-((a ^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*b)* sinh(x)^8 - 2*(2*a^3 + a^2*b)*cosh(x)^6 - 2*(2*a^3 + a^2*b - 14*(a^3 + a^2 *b)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + a^2*b)*cosh(x)^3 - 3*(2*a^3 + a^2* b)*cosh(x))*sinh(x)^5 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^4 + (70*(a ^3 + a^2*b)*cosh(x)^4 + 6*a^3 + 4*a^2*b - a*b^2 + b^3 - 30*(2*a^3 + a^2*b) *cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + a^2*b)*cosh(x)^5 - 10*(2*a^3 + a^2*b) *cosh(x)^3 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 + a^3 + 3* a^2*b + 3*a*b^2 + b^3 - 2*(2*a^3 + 3*a^2*b - b^3)*cosh(x)^2 + 2*(14*(a^3 + a^2*b)*cosh(x)^6 - 15*(2*a^3 + a^2*b)*cosh(x)^4 - 2*a^3 - 3*a^2*b + b^3 + 3*(6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(a^2*cos h(x)^6 + 6*a^2*cosh(x)*sinh(x)^5 + a^2*sinh(x)^6 - 3*a^2*cosh(x)^4 + 3*(5* a^2*cosh(x)^2 - a^2)*sinh(x)^4 + 4*(5*a^2*cosh(x)^3 - 3*a^2*cosh(x))*sinh( x)^3 + (3*a^2 + 2*a*b - b^2)*cosh(x)^2 + (15*a^2*cosh(x)^4 - 18*a^2*cosh(x )^2 + 3*a^2 + 2*a*b - b^2)*sinh(x)^2 - a^2 - 2*a*b - b^2 + 2*(3*a^2*cosh(x )^5 - 6*a^2*cosh(x)^3 + (3*a^2 + 2*a*b - b^2)*cosh(x))*sinh(x))*sqrt(a + b )*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh (x)*sinh(x) + sinh(x)^2)) + 4*(2*(a^3 + a^2*b)*cosh(x)^7 - 3*(2*a^3 + a^2* b)*cosh(x)^5 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^3 - (2*a^3 + 3*a^2* b - b^3)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x...
\[ \int \coth (x) \sqrt {a+b \coth ^2(x)} \, dx=\int \sqrt {a + b \coth ^{2}{\left (x \right )}} \coth {\left (x \right )}\, dx \]
\[ \int \coth (x) \sqrt {a+b \coth ^2(x)} \, dx=\int { \sqrt {b \coth \left (x\right )^{2} + a} \coth \left (x\right ) \,d x } \]
Exception generated. \[ \int \coth (x) \sqrt {a+b \coth ^2(x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 2.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \coth (x) \sqrt {a+b \coth ^2(x)} \, dx=-\sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a}-2\,\mathrm {atan}\left (\frac {2\,\sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a}\,\sqrt {-\frac {a}{4}-\frac {b}{4}}}{a+b}\right )\,\sqrt {-\frac {a}{4}-\frac {b}{4}} \]