Integrand size = 13, antiderivative size = 112 \[ \int \frac {\cosh ^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {b \left (a^2+2 b^2\right ) x}{2 a^4}+\frac {2 b^4 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^4 \sqrt {a-b} \sqrt {a+b}}+\frac {\left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh ^2(x) \sinh (x)}{3 a} \]
-1/2*b*(a^2+2*b^2)*x/a^4+1/3*(2*a^2+3*b^2)*sinh(x)/a^3-1/2*b*cosh(x)*sinh( x)/a^2+1/3*cosh(x)^2*sinh(x)/a+2*b^4*arctan((a-b)^(1/2)*tanh(1/2*x)/(a+b)^ (1/2))/a^4/(a-b)^(1/2)/(a+b)^(1/2)
Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {-6 b \left (a^2+2 b^2\right ) x-\frac {24 b^4 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+3 a \left (3 a^2+4 b^2\right ) \sinh (x)-3 a^2 b \sinh (2 x)+a^3 \sinh (3 x)}{12 a^4} \]
(-6*b*(a^2 + 2*b^2)*x - (24*b^4*ArcTan[((-a + b)*Tanh[x/2])/Sqrt[a^2 - b^2 ]])/Sqrt[a^2 - b^2] + 3*a*(3*a^2 + 4*b^2)*Sinh[x] - 3*a^2*b*Sinh[2*x] + a^ 3*Sinh[3*x])/(12*a^4)
Time = 1.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.15, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 4340, 25, 3042, 4592, 3042, 4592, 27, 3042, 4407, 3042, 4318, 3042, 3138, 218}
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.
| \(\displaystyle \int \frac {\cosh ^3(x)}{a+b \text {sech}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc \left (\frac {\pi }{2}+i x\right )^3 \left (a+b \csc \left (\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 4340 |
| \(\displaystyle \frac {\int -\frac {\cosh ^2(x) \left (-2 b \text {sech}^2(x)-2 a \text {sech}(x)+3 b\right )}{a+b \text {sech}(x)}dx}{3 a}+\frac {\sinh (x) \cosh ^2(x)}{3 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\int \frac {\cosh ^2(x) \left (-2 b \text {sech}^2(x)-2 a \text {sech}(x)+3 b\right )}{a+b \text {sech}(x)}dx}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\int \frac {-2 b \csc \left (i x+\frac {\pi }{2}\right )^2-2 a \csc \left (i x+\frac {\pi }{2}\right )+3 b}{\csc \left (i x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (i x+\frac {\pi }{2}\right )\right )}dx}{3 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\int \frac {\cosh (x) \left (-3 b^2 \text {sech}^2(x)+a b \text {sech}(x)+2 \left (2 a^2+3 b^2\right )\right )}{a+b \text {sech}(x)}dx}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\int \frac {-3 b^2 \csc \left (i x+\frac {\pi }{2}\right )^2+a b \csc \left (i x+\frac {\pi }{2}\right )+2 \left (2 a^2+3 b^2\right )}{\csc \left (i x+\frac {\pi }{2}\right ) \left (a+b \csc \left (i x+\frac {\pi }{2}\right )\right )}dx}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {\int \frac {3 \left (a \text {sech}(x) b^2+\left (a^2+2 b^2\right ) b\right )}{a+b \text {sech}(x)}dx}{a}}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \int \frac {a \text {sech}(x) b^2+\left (a^2+2 b^2\right ) b}{a+b \text {sech}(x)}dx}{a}}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \int \frac {a \csc \left (i x+\frac {\pi }{2}\right ) b^2+\left (a^2+2 b^2\right ) b}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx}{a}}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {b x \left (a^2+2 b^2\right )}{a}-\frac {2 b^4 \int \frac {\text {sech}(x)}{a+b \text {sech}(x)}dx}{a}\right )}{a}}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {b x \left (a^2+2 b^2\right )}{a}-\frac {2 b^4 \int \frac {\csc \left (i x+\frac {\pi }{2}\right )}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {b x \left (a^2+2 b^2\right )}{a}-\frac {2 b^3 \int \frac {1}{\frac {a \cosh (x)}{b}+1}dx}{a}\right )}{a}}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {b x \left (a^2+2 b^2\right )}{a}-\frac {2 b^3 \int \frac {1}{\frac {a \sin \left (i x+\frac {\pi }{2}\right )}{b}+1}dx}{a}\right )}{a}}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {b x \left (a^2+2 b^2\right )}{a}-\frac {4 b^3 \int \frac {1}{\frac {a+b}{b}-\left (1-\frac {a}{b}\right ) \tanh ^2\left (\frac {x}{2}\right )}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a}}{2 a}}{3 a}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\frac {2 \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {b x \left (a^2+2 b^2\right )}{a}-\frac {4 b^4 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{2 a}}{3 a}\) |
(Cosh[x]^2*Sinh[x])/(3*a) - ((3*b*Cosh[x]*Sinh[x])/(2*a) - ((-3*((b*(a^2 + 2*b^2)*x)/a - (4*b^4*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*Sqrt [a - b]*Sqrt[a + b])))/a + (2*(2*a^2 + 3*b^2)*Sinh[x])/a)/(2*a))/(3*a)
3.1.96.3.1 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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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[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. \(202\) vs. \(2(94)=188\).
Time = 0.38 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.81
| method | result | size |
| default | \(\frac {2 b^{4} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \left (a^{2}+2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a^{4}}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a -b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b \left (a^{2}+2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a^{4}}\) | \(203\) |
| risch | \(-\frac {b x}{2 a^{2}}-\frac {b^{3} x}{a^{4}}+\frac {{\mathrm e}^{3 x}}{24 a}-\frac {b \,{\mathrm e}^{2 x}}{8 a^{2}}+\frac {3 \,{\mathrm e}^{x}}{8 a}+\frac {{\mathrm e}^{x} b^{2}}{2 a^{3}}-\frac {3 \,{\mathrm e}^{-x}}{8 a}-\frac {{\mathrm e}^{-x} b^{2}}{2 a^{3}}+\frac {b \,{\mathrm e}^{-2 x}}{8 a^{2}}-\frac {{\mathrm e}^{-3 x}}{24 a}-\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{4}}+\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{4}}\) | \(213\) |
2*b^4/a^4/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2) )-1/3/a/(tanh(1/2*x)-1)^3-1/2*(a+b)/a^2/(tanh(1/2*x)-1)^2-1/2*(2*a^2+a*b+2 *b^2)/a^3/(tanh(1/2*x)-1)+1/2*b*(a^2+2*b^2)/a^4*ln(tanh(1/2*x)-1)-1/3/a/(t anh(1/2*x)+1)^3-1/2*(-a-b)/a^2/(tanh(1/2*x)+1)^2-1/2*(2*a^2+a*b+2*b^2)/a^3 /(tanh(1/2*x)+1)-1/2*b*(a^2+2*b^2)/a^4*ln(tanh(1/2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (94) = 188\).
Time = 0.28 (sec) , antiderivative size = 1562, normalized size of antiderivative = 13.95 \[ \int \frac {\cosh ^3(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \]
[1/24*((a^5 - a^3*b^2)*cosh(x)^6 + (a^5 - a^3*b^2)*sinh(x)^6 - 3*(a^4*b - a^2*b^3)*cosh(x)^5 - 3*(a^4*b - a^2*b^3 - 2*(a^5 - a^3*b^2)*cosh(x))*sinh( x)^5 - a^5 + a^3*b^2 - 12*(a^4*b + a^2*b^3 - 2*b^5)*x*cosh(x)^3 + 3*(3*a^5 + a^3*b^2 - 4*a*b^4)*cosh(x)^4 + 3*(3*a^5 + a^3*b^2 - 4*a*b^4 + 5*(a^5 - a^3*b^2)*cosh(x)^2 - 5*(a^4*b - a^2*b^3)*cosh(x))*sinh(x)^4 + 2*(10*(a^5 - a^3*b^2)*cosh(x)^3 - 15*(a^4*b - a^2*b^3)*cosh(x)^2 - 6*(a^4*b + a^2*b^3 - 2*b^5)*x + 6*(3*a^5 + a^3*b^2 - 4*a*b^4)*cosh(x))*sinh(x)^3 - 3*(3*a^5 + a^3*b^2 - 4*a*b^4)*cosh(x)^2 - 3*(3*a^5 + a^3*b^2 - 4*a*b^4 - 5*(a^5 - a^ 3*b^2)*cosh(x)^4 + 10*(a^4*b - a^2*b^3)*cosh(x)^3 + 12*(a^4*b + a^2*b^3 - 2*b^5)*x*cosh(x) - 6*(3*a^5 + a^3*b^2 - 4*a*b^4)*cosh(x)^2)*sinh(x)^2 - 24 *(b^4*cosh(x)^3 + 3*b^4*cosh(x)^2*sinh(x) + 3*b^4*cosh(x)*sinh(x)^2 + b^4* sinh(x)^3)*sqrt(-a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cos h(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)) + 3*(a^4*b - a^2*b^3)*cosh(x) + 3*(2*(a^5 - a^ 3*b^2)*cosh(x)^5 + a^4*b - a^2*b^3 - 5*(a^4*b - a^2*b^3)*cosh(x)^4 - 12*(a ^4*b + a^2*b^3 - 2*b^5)*x*cosh(x)^2 + 4*(3*a^5 + a^3*b^2 - 4*a*b^4)*cosh(x )^3 - 2*(3*a^5 + a^3*b^2 - 4*a*b^4)*cosh(x))*sinh(x))/((a^6 - a^4*b^2)*cos h(x)^3 + 3*(a^6 - a^4*b^2)*cosh(x)^2*sinh(x) + 3*(a^6 - a^4*b^2)*cosh(x)*s inh(x)^2 + (a^6 - a^4*b^2)*sinh(x)^3), 1/24*((a^5 - a^3*b^2)*cosh(x)^6 ...
\[ \int \frac {\cosh ^3(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\cosh ^{3}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\cosh ^3(x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \]
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*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.19 \[ \int \frac {\cosh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \, b^{4} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{4}} + \frac {a^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} + 9 \, a^{2} e^{x} + 12 \, b^{2} e^{x}}{24 \, a^{3}} - \frac {{\left (a^{2} b + 2 \, b^{3}\right )} x}{2 \, a^{4}} + \frac {{\left (3 \, a^{2} b e^{x} - a^{3} - 3 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, a^{4}} \]
2*b^4*arctan((a*e^x + b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a^4) + 1/24*(a^ 2*e^(3*x) - 3*a*b*e^(2*x) + 9*a^2*e^x + 12*b^2*e^x)/a^3 - 1/2*(a^2*b + 2*b ^3)*x/a^4 + 1/24*(3*a^2*b*e^x - a^3 - 3*(3*a^3 + 4*a*b^2)*e^(2*x))*e^(-3*x )/a^4
Time = 2.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.87 \[ \int \frac {\cosh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a}-\frac {x\,\left (a^2\,b+2\,b^3\right )}{2\,a^4}+\frac {{\mathrm {e}}^x\,\left (3\,a^2+4\,b^2\right )}{8\,a^3}+\frac {b\,{\mathrm {e}}^{-2\,x}}{8\,a^2}-\frac {b\,{\mathrm {e}}^{2\,x}}{8\,a^2}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a^2+4\,b^2\right )}{8\,a^3}+\frac {b^4\,\ln \left (-\frac {2\,b^4\,{\mathrm {e}}^x}{a^5}-\frac {2\,b^4\,\left (a+b\,{\mathrm {e}}^x\right )}{a^5\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^4\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {b^4\,\ln \left (\frac {2\,b^4\,\left (a+b\,{\mathrm {e}}^x\right )}{a^5\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {2\,b^4\,{\mathrm {e}}^x}{a^5}\right )}{a^4\,\sqrt {a+b}\,\sqrt {b-a}} \]
exp(3*x)/(24*a) - exp(-3*x)/(24*a) - (x*(a^2*b + 2*b^3))/(2*a^4) + (exp(x) *(3*a^2 + 4*b^2))/(8*a^3) + (b*exp(-2*x))/(8*a^2) - (b*exp(2*x))/(8*a^2) - (exp(-x)*(3*a^2 + 4*b^2))/(8*a^3) + (b^4*log(- (2*b^4*exp(x))/a^5 - (2*b^ 4*(a + b*exp(x)))/(a^5*(a + b)^(1/2)*(b - a)^(1/2))))/(a^4*(a + b)^(1/2)*( b - a)^(1/2)) - (b^4*log((2*b^4*(a + b*exp(x)))/(a^5*(a + b)^(1/2)*(b - a) ^(1/2)) - (2*b^4*exp(x))/a^5))/(a^4*(a + b)^(1/2)*(b - a)^(1/2))