Integrand size = 13, antiderivative size = 80 \[ \int \frac {\sinh ^2(x)}{a+b \text {csch}(x)} \, dx=-\frac {\left (a^2-2 b^2\right ) x}{2 a^3}+\frac {2 b^3 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2}}-\frac {b \cosh (x)}{a^2}+\frac {\cosh (x) \sinh (x)}{2 a} \]
-1/2*(a^2-2*b^2)*x/a^3-b*cosh(x)/a^2+1/2*cosh(x)*sinh(x)/a+2*b^3*arctanh(( a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/a^3/(a^2+b^2)^(1/2)
Time = 0.82 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int \frac {\sinh ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {-2 a^2 x+4 b^2 x-\frac {8 b^3 \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a b \cosh (x)+a^2 \sinh (2 x)}{4 a^3} \]
(-2*a^2*x + 4*b^2*x - (8*b^3*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a^2 - b^2]])/S qrt[-a^2 - b^2] - 4*a*b*Cosh[x] + a^2*Sinh[2*x])/(4*a^3)
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.31, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.308, Rules used = {3042, 25, 4340, 3042, 26, 4592, 26, 3042, 4407, 26, 3042, 26, 4318, 3042, 3139, 1083, 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.
| \(\displaystyle \int \frac {\sinh ^2(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\csc (i x)^2 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\csc (i x)^2 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 4340 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}-\frac {\int \frac {\left (b \text {csch}^2(x)+a \text {csch}(x)+2 b\right ) \sinh (x)}{a+b \text {csch}(x)}dx}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}-\frac {\int -\frac {i \left (-b \csc (i x)^2+i a \csc (i x)+2 b\right )}{\csc (i x) (a+i b \csc (i x))}dx}{2 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \int \frac {-b \csc (i x)^2+i a \csc (i x)+2 b}{\csc (i x) (a+i b \csc (i x))}dx}{2 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i b \cosh (x)}{a}-\frac {\int -\frac {i \left (a^2+b \text {csch}(x) a-2 b^2\right )}{a+b \text {csch}(x)}dx}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \int \frac {a^2+b \text {csch}(x) a-2 b^2}{a+b \text {csch}(x)}dx}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \int \frac {a^2+i b \csc (i x) a-2 b^2}{a+i b \csc (i x)}dx}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {x \left (a^2-2 b^2\right )}{a}+\frac {2 i b^3 \int -\frac {i \text {csch}(x)}{a+b \text {csch}(x)}dx}{a}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {2 b^3 \int \frac {\text {csch}(x)}{a+b \text {csch}(x)}dx}{a}+\frac {x \left (a^2-2 b^2\right )}{a}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {x \left (a^2-2 b^2\right )}{a}+\frac {2 b^3 \int \frac {i \csc (i x)}{a+i b \csc (i x)}dx}{a}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {x \left (a^2-2 b^2\right )}{a}+\frac {2 i b^3 \int \frac {\csc (i x)}{a+i b \csc (i x)}dx}{a}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {2 b^2 \int \frac {1}{\frac {a \sinh (x)}{b}+1}dx}{a}+\frac {x \left (a^2-2 b^2\right )}{a}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {x \left (a^2-2 b^2\right )}{a}+\frac {2 b^2 \int \frac {1}{1-\frac {i a \sin (i x)}{b}}dx}{a}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {4 b^2 \int \frac {1}{-\tanh ^2\left (\frac {x}{2}\right )+\frac {2 a \tanh \left (\frac {x}{2}\right )}{b}+1}d\tanh \left (\frac {x}{2}\right )}{a}+\frac {x \left (a^2-2 b^2\right )}{a}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {x \left (a^2-2 b^2\right )}{a}-\frac {8 b^2 \int \frac {1}{4 \left (\frac {a^2}{b^2}+1\right )-\left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )^2}d\left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{a}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {i \left (\frac {x \left (a^2-2 b^2\right )}{a}-\frac {4 b^3 \text {arctanh}\left (\frac {b \left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )}{a}+\frac {2 i b \cosh (x)}{a}\right )}{2 a}\) |
((I/2)*((I*(((a^2 - 2*b^2)*x)/a - (4*b^3*ArcTanh[(b*((2*a)/b - 2*Tanh[x/2] ))/(2*Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2])))/a + ((2*I)*b*Cosh[x])/a))/a + (Cosh[x]*Sinh[x])/(2*a)
3.1.78.3.1 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_)^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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *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[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[Cot[e + f*x]*((d*Csc[e + f*x])^n/(a*f*n)), x] - Sim p[1/(a*d*n) Int[((d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x]))*Simp[b*n - a*(n + 1)*Csc[e + f*x] - b*(n + 1)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(151\) vs. \(2(70)=140\).
Time = 0.32 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.90
| method | result | size |
| default | \(-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-a +2 b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\left (-a^{2}+2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a^{3}}+\frac {2 b^{3} \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3} \sqrt {a^{2}+b^{2}}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\left (a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a^{3}}-\frac {-a -2 b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(152\) |
| risch | \(-\frac {x}{2 a}+\frac {x \,b^{2}}{a^{3}}+\frac {{\mathrm e}^{2 x}}{8 a}-\frac {b \,{\mathrm e}^{x}}{2 a^{2}}-\frac {b \,{\mathrm e}^{-x}}{2 a^{2}}-\frac {{\mathrm e}^{-2 x}}{8 a}+\frac {b^{3} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, a^{3}}-\frac {b^{3} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, a^{3}}\) | \(159\) |
-1/2/a/(tanh(1/2*x)+1)^2-1/2*(-a+2*b)/a^2/(tanh(1/2*x)+1)+1/2/a^3*(-a^2+2* b^2)*ln(tanh(1/2*x)+1)+2*b^3/a^3/(a^2+b^2)^(1/2)*arctanh(1/2*(-2*b*tanh(1/ 2*x)+2*a)/(a^2+b^2)^(1/2))+1/2/a/(tanh(1/2*x)-1)^2+1/2/a^3*(a^2-2*b^2)*ln( tanh(1/2*x)-1)-1/2*(-a-2*b)/a^2/(tanh(1/2*x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (72) = 144\).
Time = 0.27 (sec) , antiderivative size = 456, normalized size of antiderivative = 5.70 \[ \int \frac {\sinh ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {{\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{4} + {\left (a^{4} + a^{2} b^{2}\right )} \sinh \left (x\right )^{4} - a^{4} - a^{2} b^{2} - 4 \, {\left (a^{4} - a^{2} b^{2} - 2 \, b^{4}\right )} x \cosh \left (x\right )^{2} - 4 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{3} - 4 \, {\left (a^{3} b + a b^{3} - {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (3 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{4} - a^{2} b^{2} - 2 \, b^{4}\right )} x - 6 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 8 \, {\left (b^{3} \cosh \left (x\right )^{2} + 2 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right ) + b^{3} \sinh \left (x\right )^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) - 4 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right ) - 4 \, {\left (a^{3} b + a b^{3} - {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (a^{4} - a^{2} b^{2} - 2 \, b^{4}\right )} x \cosh \left (x\right ) + 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )}{8 \, {\left ({\left (a^{5} + a^{3} b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{5} + a^{3} b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{5} + a^{3} b^{2}\right )} \sinh \left (x\right )^{2}\right )}} \]
1/8*((a^4 + a^2*b^2)*cosh(x)^4 + (a^4 + a^2*b^2)*sinh(x)^4 - a^4 - a^2*b^2 - 4*(a^4 - a^2*b^2 - 2*b^4)*x*cosh(x)^2 - 4*(a^3*b + a*b^3)*cosh(x)^3 - 4 *(a^3*b + a*b^3 - (a^4 + a^2*b^2)*cosh(x))*sinh(x)^3 + 2*(3*(a^4 + a^2*b^2 )*cosh(x)^2 - 2*(a^4 - a^2*b^2 - 2*b^4)*x - 6*(a^3*b + a*b^3)*cosh(x))*sin h(x)^2 + 8*(b^3*cosh(x)^2 + 2*b^3*cosh(x)*sinh(x) + b^3*sinh(x)^2)*sqrt(a^ 2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) + a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(a^2 + b^2)*(a*cosh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x ) - a)) - 4*(a^3*b + a*b^3)*cosh(x) - 4*(a^3*b + a*b^3 - (a^4 + a^2*b^2)*c osh(x)^3 + 2*(a^4 - a^2*b^2 - 2*b^4)*x*cosh(x) + 3*(a^3*b + a*b^3)*cosh(x) ^2)*sinh(x))/((a^5 + a^3*b^2)*cosh(x)^2 + 2*(a^5 + a^3*b^2)*cosh(x)*sinh(x ) + (a^5 + a^3*b^2)*sinh(x)^2)
\[ \int \frac {\sinh ^2(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\sinh ^{2}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.45 \[ \int \frac {\sinh ^2(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^{3} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} - \frac {{\left (4 \, b e^{\left (-x\right )} - a\right )} e^{\left (2 \, x\right )}}{8 \, a^{2}} - \frac {4 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )}}{8 \, a^{2}} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} x}{2 \, a^{3}} \]
-b^3*log((a*e^(-x) - b - sqrt(a^2 + b^2))/(a*e^(-x) - b + sqrt(a^2 + b^2)) )/(sqrt(a^2 + b^2)*a^3) - 1/8*(4*b*e^(-x) - a)*e^(2*x)/a^2 - 1/8*(4*b*e^(- x) + a*e^(-2*x))/a^2 - 1/2*(a^2 - 2*b^2)*x/a^3
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.44 \[ \int \frac {\sinh ^2(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^{3} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} + \frac {a e^{\left (2 \, x\right )} - 4 \, b e^{x}}{8 \, a^{2}} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} x}{2 \, a^{3}} - \frac {{\left (4 \, a b e^{x} + a^{2}\right )} e^{\left (-2 \, x\right )}}{8 \, a^{3}} \]
-b^3*log(abs(2*a*e^x + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^x + 2*b + 2*sqrt (a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3) + 1/8*(a*e^(2*x) - 4*b*e^x)/a^2 - 1/2* (a^2 - 2*b^2)*x/a^3 - 1/8*(4*a*b*e^x + a^2)*e^(-2*x)/a^3
Time = 2.40 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.96 \[ \int \frac {\sinh ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {b\,{\mathrm {e}}^x}{2\,a^2}-\frac {b\,{\mathrm {e}}^{-x}}{2\,a^2}-\frac {x\,\left (a^2-2\,b^2\right )}{2\,a^3}-\frac {b^3\,\ln \left (\frac {2\,b^3\,{\mathrm {e}}^x}{a^4}-\frac {2\,b^3\,\left (a-b\,{\mathrm {e}}^x\right )}{a^4\,\sqrt {a^2+b^2}}\right )}{a^3\,\sqrt {a^2+b^2}}+\frac {b^3\,\ln \left (\frac {2\,b^3\,{\mathrm {e}}^x}{a^4}+\frac {2\,b^3\,\left (a-b\,{\mathrm {e}}^x\right )}{a^4\,\sqrt {a^2+b^2}}\right )}{a^3\,\sqrt {a^2+b^2}} \]
exp(2*x)/(8*a) - exp(-2*x)/(8*a) - (b*exp(x))/(2*a^2) - (b*exp(-x))/(2*a^2 ) - (x*(a^2 - 2*b^2))/(2*a^3) - (b^3*log((2*b^3*exp(x))/a^4 - (2*b^3*(a - b*exp(x)))/(a^4*(a^2 + b^2)^(1/2))))/(a^3*(a^2 + b^2)^(1/2)) + (b^3*log((2 *b^3*exp(x))/a^4 + (2*b^3*(a - b*exp(x)))/(a^4*(a^2 + b^2)^(1/2))))/(a^3*( a^2 + b^2)^(1/2))