Integrand size = 13, antiderivative size = 110 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh (x)} \, dx=\frac {2 b^4 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}+\frac {\left (3 b^3+a \left (2 a^2-5 b^2\right ) \cosh (x)\right ) \text {csch}(x)}{3 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^3(x)}{3 \left (a^2-b^2\right )} \]
2*b^4*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b)^(5/2)/(a+b)^(5/2) +1/3*(3*b^3+a*(2*a^2-5*b^2)*cosh(x))*csch(x)/(a^2-b^2)^2+1/3*(b-a*cosh(x)) *csch(x)^3/(a^2-b^2)
Time = 0.43 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.28 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh (x)} \, dx=\frac {1}{24} \left (-\frac {48 b^4 \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {2 (4 a+7 b) \coth \left (\frac {x}{2}\right )}{(a+b)^2}+\frac {8 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )}{a-b}-\frac {\text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)}{2 (a+b)}+\frac {8 a \tanh \left (\frac {x}{2}\right )}{(a-b)^2}-\frac {14 b \tanh \left (\frac {x}{2}\right )}{(a-b)^2}\right ) \]
((-48*b^4*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (2*(4*a + 7*b)*Coth[x/2])/(a + b)^2 + (8*Csch[x]^3*Sinh[x/2]^4)/(a - b) - (Csch[x/2]^4*Sinh[x])/(2*(a + b)) + (8*a*Tanh[x/2])/(a - b)^2 - (14*b*T anh[x/2])/(a - b)^2)/24
Time = 0.60 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3042, 3175, 3042, 25, 3345, 27, 3042, 3138, 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.
| \(\displaystyle \int \frac {\text {csch}^4(x)}{a+b \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (-\frac {\pi }{2}+i x\right )^4 \left (a-b \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 3175 |
| \(\displaystyle \frac {\text {csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}-\frac {\int \frac {\left (2 a^2+2 b \cosh (x) a-3 b^2\right ) \text {csch}^2(x)}{a+b \cosh (x)}dx}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}-\frac {\int -\frac {2 a^2-2 b \sin \left (i x-\frac {\pi }{2}\right ) a-3 b^2}{\cos \left (i x-\frac {\pi }{2}\right )^2 \left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\text {csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}+\frac {\int \frac {2 a^2-2 b \sin \left (i x-\frac {\pi }{2}\right ) a-3 b^2}{\cos \left (i x-\frac {\pi }{2}\right )^2 \left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\text {csch}(x) \left (a \left (2 a^2-5 b^2\right ) \cosh (x)+3 b^3\right )}{a^2-b^2}-\frac {\int -\frac {3 b^4}{a+b \cosh (x)}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}+\frac {\text {csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 b^4 \int \frac {1}{a+b \cosh (x)}dx}{a^2-b^2}+\frac {\text {csch}(x) \left (a \left (2 a^2-5 b^2\right ) \cosh (x)+3 b^3\right )}{a^2-b^2}}{3 \left (a^2-b^2\right )}+\frac {\text {csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}+\frac {\frac {\text {csch}(x) \left (a \left (2 a^2-5 b^2\right ) \cosh (x)+3 b^3\right )}{a^2-b^2}+\frac {3 b^4 \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {6 b^4 \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2}+\frac {\text {csch}(x) \left (a \left (2 a^2-5 b^2\right ) \cosh (x)+3 b^3\right )}{a^2-b^2}}{3 \left (a^2-b^2\right )}+\frac {\text {csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {6 b^4 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}+\frac {\text {csch}(x) \left (a \left (2 a^2-5 b^2\right ) \cosh (x)+3 b^3\right )}{a^2-b^2}}{3 \left (a^2-b^2\right )}+\frac {\text {csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}\) |
((b - a*Cosh[x])*Csch[x]^3)/(3*(a^2 - b^2)) + ((6*b^4*ArcTanh[(Sqrt[a - b] *Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)) + ((3*b^3 + a*(2*a^2 - 5*b^2)*Cosh[x])*Csch[x])/(a^2 - b^2))/(3*(a^2 - b^2))
3.2.75.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ (m + 1)*((b - a*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2* (a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m* (a^2*(p + 2) - b^2*(m + p + 2) + a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; F reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegersQ [2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Time = 3.52 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.15
| method | result | size |
| default | \(-\frac {\frac {a \tanh \left (\frac {x}{2}\right )^{3}}{3}-\frac {b \tanh \left (\frac {x}{2}\right )^{3}}{3}-3 a \tanh \left (\frac {x}{2}\right )+5 b \tanh \left (\frac {x}{2}\right )}{8 \left (a -b \right )^{2}}-\frac {1}{24 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-3 a -5 b}{8 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )}+\frac {2 b^{4} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(127\) |
| risch | \(-\frac {2 \left (-3 b^{3} {\mathrm e}^{5 x}+3 a \,b^{2} {\mathrm e}^{4 x}-4 a^{2} b \,{\mathrm e}^{3 x}+10 b^{3} {\mathrm e}^{3 x}+6 a^{3} {\mathrm e}^{2 x}-12 a \,b^{2} {\mathrm e}^{2 x}-3 b^{3} {\mathrm e}^{x}-2 a^{3}+5 a \,b^{2}\right )}{3 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 x}-1\right )^{3}}+\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(242\) |
-1/8/(a-b)^2*(1/3*a*tanh(1/2*x)^3-1/3*b*tanh(1/2*x)^3-3*a*tanh(1/2*x)+5*b* tanh(1/2*x))-1/24/(a+b)/tanh(1/2*x)^3-1/8/(a+b)^2*(-3*a-5*b)/tanh(1/2*x)+2 /(a-b)^2/(a+b)^2*b^4/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)* (a-b))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (97) = 194\).
Time = 0.30 (sec) , antiderivative size = 2339, normalized size of antiderivative = 21.26 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh (x)} \, dx=\text {Too large to display} \]
[1/3*(6*(a^2*b^3 - b^5)*cosh(x)^5 + 6*(a^2*b^3 - b^5)*sinh(x)^5 + 4*a^5 - 14*a^3*b^2 + 10*a*b^4 - 6*(a^3*b^2 - a*b^4)*cosh(x)^4 - 6*(a^3*b^2 - a*b^4 - 5*(a^2*b^3 - b^5)*cosh(x))*sinh(x)^4 + 4*(2*a^4*b - 7*a^2*b^3 + 5*b^5)* cosh(x)^3 + 4*(2*a^4*b - 7*a^2*b^3 + 5*b^5 + 15*(a^2*b^3 - b^5)*cosh(x)^2 - 6*(a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^3 - 12*(a^5 - 3*a^3*b^2 + 2*a*b^4)* cosh(x)^2 - 12*(a^5 - 3*a^3*b^2 + 2*a*b^4 - 5*(a^2*b^3 - b^5)*cosh(x)^3 + 3*(a^3*b^2 - a*b^4)*cosh(x)^2 - (2*a^4*b - 7*a^2*b^3 + 5*b^5)*cosh(x))*sin h(x)^2 + 3*(b^4*cosh(x)^6 + 6*b^4*cosh(x)*sinh(x)^5 + b^4*sinh(x)^6 - 3*b^ 4*cosh(x)^4 + 3*b^4*cosh(x)^2 + 3*(5*b^4*cosh(x)^2 - b^4)*sinh(x)^4 - b^4 + 4*(5*b^4*cosh(x)^3 - 3*b^4*cosh(x))*sinh(x)^3 + 3*(5*b^4*cosh(x)^4 - 6*b ^4*cosh(x)^2 + b^4)*sinh(x)^2 + 6*(b^4*cosh(x)^5 - 2*b^4*cosh(x)^3 + b^4*c osh(x))*sinh(x))*sqrt(a^2 - b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a* b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 - b^2 )*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) + b)) + 6*(a^2*b^3 - b^5)*cosh(x) + 6*(a^2*b^3 - b^5 + 5*(a^2*b^3 - b^5)*cosh(x)^4 - 4*(a^3*b^2 - a*b^4)*cosh(x)^3 + 2*(2* a^4*b - 7*a^2*b^3 + 5*b^5)*cosh(x)^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*cosh( x))*sinh(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 + 6*(a^6 - 3*a ^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)*sinh(x)^5 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)^6 - a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6 - 3*(a^6 - 3*a^4*...
\[ \int \frac {\text {csch}^4(x)}{a+b \cosh (x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\text {csch}^4(x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.42 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh (x)} \, dx=\frac {2 \, b^{4} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, b^{3} e^{\left (5 \, x\right )} - 3 \, a b^{2} e^{\left (4 \, x\right )} + 4 \, a^{2} b e^{\left (3 \, x\right )} - 10 \, b^{3} e^{\left (3 \, x\right )} - 6 \, a^{3} e^{\left (2 \, x\right )} + 12 \, a b^{2} e^{\left (2 \, x\right )} + 3 \, b^{3} e^{x} + 2 \, a^{3} - 5 \, a b^{2}\right )}}{3 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]
2*b^4*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/((a^4 - 2*a^2*b^2 + b^4)*sqrt(- a^2 + b^2)) + 2/3*(3*b^3*e^(5*x) - 3*a*b^2*e^(4*x) + 4*a^2*b*e^(3*x) - 10* b^3*e^(3*x) - 6*a^3*e^(2*x) + 12*a*b^2*e^(2*x) + 3*b^3*e^x + 2*a^3 - 5*a*b ^2)/((a^4 - 2*a^2*b^2 + b^4)*(e^(2*x) - 1)^3)
Time = 2.76 (sec) , antiderivative size = 642, normalized size of antiderivative = 5.84 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh (x)} \, dx=\frac {\frac {4\,\left (a\,b^2-a^3\right )}{{\left (a^2-b^2\right )}^2}+\frac {8\,{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{3\,{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,a\,b^2}{{\left (a^2-b^2\right )}^2}-\frac {2\,b^3\,{\mathrm {e}}^x}{{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {8\,a}{3\,\left (a^2-b^2\right )}-\frac {8\,b\,{\mathrm {e}}^x}{3\,\left (a^2-b^2\right )}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}+\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,b^2}{{\left (a^2-b^2\right )}^2\,\sqrt {b^8}\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,a\,\left (a^5\,\sqrt {b^8}-2\,a^3\,b^2\,\sqrt {b^8}+a\,b^4\,\sqrt {b^8}\right )}{b^6\,\sqrt {-{\left (a^2-b^2\right )}^5}\,\left (a^4-2\,a^2\,b^2+b^4\right )\,\sqrt {-a^{10}+5\,a^8\,b^2-10\,a^6\,b^4+10\,a^4\,b^6-5\,a^2\,b^8+b^{10}}}\right )+\frac {2\,a\,\left (b^5\,\sqrt {b^8}-2\,a^2\,b^3\,\sqrt {b^8}+a^4\,b\,\sqrt {b^8}\right )}{b^6\,\sqrt {-{\left (a^2-b^2\right )}^5}\,\left (a^4-2\,a^2\,b^2+b^4\right )\,\sqrt {-a^{10}+5\,a^8\,b^2-10\,a^6\,b^4+10\,a^4\,b^6-5\,a^2\,b^8+b^{10}}}\right )\,\left (\frac {b^5\,\sqrt {-a^{10}+5\,a^8\,b^2-10\,a^6\,b^4+10\,a^4\,b^6-5\,a^2\,b^8+b^{10}}}{2}-a^2\,b^3\,\sqrt {-a^{10}+5\,a^8\,b^2-10\,a^6\,b^4+10\,a^4\,b^6-5\,a^2\,b^8+b^{10}}+\frac {a^4\,b\,\sqrt {-a^{10}+5\,a^8\,b^2-10\,a^6\,b^4+10\,a^4\,b^6-5\,a^2\,b^8+b^{10}}}{2}\right )\right )\,\sqrt {b^8}}{\sqrt {-a^{10}+5\,a^8\,b^2-10\,a^6\,b^4+10\,a^4\,b^6-5\,a^2\,b^8+b^{10}}} \]
((4*(a*b^2 - a^3))/(a^2 - b^2)^2 + (8*exp(x)*(a^2*b - b^3))/(3*(a^2 - b^2) ^2))/(exp(4*x) - 2*exp(2*x) + 1) - ((2*a*b^2)/(a^2 - b^2)^2 - (2*b^3*exp(x ))/(a^2 - b^2)^2)/(exp(2*x) - 1) - ((8*a)/(3*(a^2 - b^2)) - (8*b*exp(x))/( 3*(a^2 - b^2)))/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1) + (2*atan((exp(x) *((2*b^2)/((a^2 - b^2)^2*(b^8)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)) + (2*a*(a^5* (b^8)^(1/2) - 2*a^3*b^2*(b^8)^(1/2) + a*b^4*(b^8)^(1/2)))/(b^6*(-(a^2 - b^ 2)^5)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)*(b^10 - a^10 - 5*a^2*b^8 + 10*a^4*b^6 - 10*a^6*b^4 + 5*a^8*b^2)^(1/2))) + (2*a*(b^5*(b^8)^(1/2) - 2*a^2*b^3*(b^8 )^(1/2) + a^4*b*(b^8)^(1/2)))/(b^6*(-(a^2 - b^2)^5)^(1/2)*(a^4 + b^4 - 2*a ^2*b^2)*(b^10 - a^10 - 5*a^2*b^8 + 10*a^4*b^6 - 10*a^6*b^4 + 5*a^8*b^2)^(1 /2)))*((b^5*(b^10 - a^10 - 5*a^2*b^8 + 10*a^4*b^6 - 10*a^6*b^4 + 5*a^8*b^2 )^(1/2))/2 - a^2*b^3*(b^10 - a^10 - 5*a^2*b^8 + 10*a^4*b^6 - 10*a^6*b^4 + 5*a^8*b^2)^(1/2) + (a^4*b*(b^10 - a^10 - 5*a^2*b^8 + 10*a^4*b^6 - 10*a^6*b ^4 + 5*a^8*b^2)^(1/2))/2))*(b^8)^(1/2))/(b^10 - a^10 - 5*a^2*b^8 + 10*a^4* b^6 - 10*a^6*b^4 + 5*a^8*b^2)^(1/2)