Integrand size = 13, antiderivative size = 94 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=-\frac {(a-b \cosh (x)) \text {csch}^2(x)}{2 \left (a^2-b^2\right )}+\frac {(2 a+b) \log (1-\cosh (x))}{4 (a+b)^2}+\frac {(2 a-b) \log (1+\cosh (x))}{4 (a-b)^2}-\frac {a^3 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^2} \] Output:
-1/2*(a-b*cosh(x))*csch(x)^2/(a^2-b^2)+1/4*(2*a+b)*ln(1-cosh(x))/(a+b)^2+1 /4*(2*a-b)*ln(1+cosh(x))/(a-b)^2-a^3*ln(a+b*cosh(x))/(a^2-b^2)^2
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\frac {1}{8} \left (-\frac {\text {csch}^2\left (\frac {x}{2}\right )}{a+b}+\frac {4 (2 a-b) \log \left (\cosh \left (\frac {x}{2}\right )\right )}{(a-b)^2}-\frac {8 a^3 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^2}+\frac {4 (2 a+b) \log \left (\sinh \left (\frac {x}{2}\right )\right )}{(a+b)^2}+\frac {\text {sech}^2\left (\frac {x}{2}\right )}{a-b}\right ) \] Input:
Integrate[Coth[x]^3/(a + b*Cosh[x]),x]
Output:
(-(Csch[x/2]^2/(a + b)) + (4*(2*a - b)*Log[Cosh[x/2]])/(a - b)^2 - (8*a^3* Log[a + b*Cosh[x]])/(a^2 - b^2)^2 + (4*(2*a + b)*Log[Sinh[x/2]])/(a + b)^2 + Sech[x/2]^2/(a - b))/8
Time = 0.44 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.44, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 26, 3200, 601, 25, 27, 657, 2009}
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.
| \(\displaystyle \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \tan \left (-\frac {\pi }{2}+i x\right )^3}{a-b \sin \left (-\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\tan \left (i x-\frac {\pi }{2}\right )^3}{a-b \sin \left (i x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3200 |
| \(\displaystyle \int \frac {b^3 \cosh ^3(x)}{\left (b^2-b^2 \cosh ^2(x)\right )^2 (a+b \cosh (x))}d(b \cosh (x))\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle \frac {b^2 (a-b \cosh (x))}{2 \left (a^2-b^2\right ) \left (b^2-b^2 \cosh ^2(x)\right )}-\frac {\int -\frac {b^2 \left (\frac {a b^2}{a^2-b^2}-\frac {b \left (2 a^2-b^2\right ) \cosh (x)}{a^2-b^2}\right )}{(a+b \cosh (x)) \left (b^2-b^2 \cosh ^2(x)\right )}d(b \cosh (x))}{2 b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b^2 \left (a b^2-b \left (2 a^2-b^2\right ) \cosh (x)\right )}{\left (a^2-b^2\right ) (a+b \cosh (x)) \left (b^2-b^2 \cosh ^2(x)\right )}d(b \cosh (x))}{2 b^2}+\frac {b^2 (a-b \cosh (x))}{2 \left (a^2-b^2\right ) \left (b^2-b^2 \cosh ^2(x)\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a b^2-b \left (2 a^2-b^2\right ) \cosh (x)}{(a+b \cosh (x)) \left (b^2-b^2 \cosh ^2(x)\right )}d(b \cosh (x))}{2 \left (a^2-b^2\right )}+\frac {b^2 (a-b \cosh (x))}{2 \left (a^2-b^2\right ) \left (b^2-b^2 \cosh ^2(x)\right )}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {\int \left (-\frac {2 a^3}{(a-b) (a+b) (a+b \cosh (x))}+\frac {-2 a^2+b a+b^2}{2 (a+b) (b-b \cosh (x))}+\frac {(2 a-b) (a+b)}{2 (a-b) (\cosh (x) b+b)}\right )d(b \cosh (x))}{2 \left (a^2-b^2\right )}+\frac {b^2 (a-b \cosh (x))}{2 \left (a^2-b^2\right ) \left (b^2-b^2 \cosh ^2(x)\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^2 (a-b \cosh (x))}{2 \left (a^2-b^2\right ) \left (b^2-b^2 \cosh ^2(x)\right )}+\frac {-\frac {2 a^3 \log (a+b \cosh (x))}{a^2-b^2}+\frac {(a-b) (2 a+b) \log (b-b \cosh (x))}{2 (a+b)}+\frac {(2 a-b) (a+b) \log (b \cosh (x)+b)}{2 (a-b)}}{2 \left (a^2-b^2\right )}\) |
Input:
Int[Coth[x]^3/(a + b*Cosh[x]),x]
Output:
(b^2*(a - b*Cosh[x]))/(2*(a^2 - b^2)*(b^2 - b^2*Cosh[x]^2)) + (((a - b)*(2 *a + b)*Log[b - b*Cosh[x]])/(2*(a + b)) - (2*a^3*Log[a + b*Cosh[x]])/(a^2 - b^2) + ((2*a - b)*(a + b)*Log[b + b*Cosh[x]])/(2*(a - b)))/(2*(a^2 - b^2 ))
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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Time = 1.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8 \left (a -b \right )}-\frac {a^{3} \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -b \tanh \left (\frac {x}{2}\right )^{2}-a -b \right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {1}{8 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a +2 b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 \left (a +b \right )^{2}}\) | \(91\) |
| risch | \(-\frac {x a}{a^{2}+2 a b +b^{2}}-\frac {x b}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {x a}{a^{2}-2 a b +b^{2}}+\frac {b x}{2 a^{2}-4 a b +2 b^{2}}+\frac {2 x \,a^{3}}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {{\mathrm e}^{x} \left (-{\mathrm e}^{2 x} b +2 a \,{\mathrm e}^{x}-b \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} \left (a^{2}-b^{2}\right )}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) a}{a^{2}+2 a b +b^{2}}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) b}{2 a^{2}+4 a b +2 b^{2}}+\frac {\ln \left ({\mathrm e}^{x}+1\right ) a}{a^{2}-2 a b +b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+1\right ) b}{2 \left (a^{2}-2 a b +b^{2}\right )}-\frac {a^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}\) | \(251\) |
Input:
int(coth(x)^3/(a+b*cosh(x)),x,method=_RETURNVERBOSE)
Output:
-1/8*tanh(1/2*x)^2/(a-b)-1/(a-b)^2*a^3/(a+b)^2*ln(tanh(1/2*x)^2*a-b*tanh(1 /2*x)^2-a-b)-1/8/(a+b)/tanh(1/2*x)^2+1/4/(a+b)^2*(4*a+2*b)*ln(tanh(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (89) = 178\).
Time = 0.10 (sec) , antiderivative size = 839, normalized size of antiderivative = 8.93 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^3/(a+b*cosh(x)),x, algorithm="fricas")
Output:
1/2*(2*(a^2*b - b^3)*cosh(x)^3 + 2*(a^2*b - b^3)*sinh(x)^3 - 4*(a^3 - a*b^ 2)*cosh(x)^2 - 2*(2*a^3 - 2*a*b^2 - 3*(a^2*b - b^3)*cosh(x))*sinh(x)^2 + 2 *(a^2*b - b^3)*cosh(x) - 2*(a^3*cosh(x)^4 + 4*a^3*cosh(x)*sinh(x)^3 + a^3* sinh(x)^4 - 2*a^3*cosh(x)^2 + a^3 + 2*(3*a^3*cosh(x)^2 - a^3)*sinh(x)^2 + 4*(a^3*cosh(x)^3 - a^3*cosh(x))*sinh(x))*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))) + ((2*a^3 + 3*a^2*b - b^3)*cosh(x)^4 + 4*(2*a^3 + 3*a^2*b - b^3) *cosh(x)*sinh(x)^3 + (2*a^3 + 3*a^2*b - b^3)*sinh(x)^4 + 2*a^3 + 3*a^2*b - b^3 - 2*(2*a^3 + 3*a^2*b - b^3)*cosh(x)^2 - 2*(2*a^3 + 3*a^2*b - b^3 - 3* (2*a^3 + 3*a^2*b - b^3)*cosh(x)^2)*sinh(x)^2 + 4*((2*a^3 + 3*a^2*b - b^3)* cosh(x)^3 - (2*a^3 + 3*a^2*b - b^3)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x ) + 1) + ((2*a^3 - 3*a^2*b + b^3)*cosh(x)^4 + 4*(2*a^3 - 3*a^2*b + b^3)*co sh(x)*sinh(x)^3 + (2*a^3 - 3*a^2*b + b^3)*sinh(x)^4 + 2*a^3 - 3*a^2*b + b^ 3 - 2*(2*a^3 - 3*a^2*b + b^3)*cosh(x)^2 - 2*(2*a^3 - 3*a^2*b + b^3 - 3*(2* a^3 - 3*a^2*b + b^3)*cosh(x)^2)*sinh(x)^2 + 4*((2*a^3 - 3*a^2*b + b^3)*cos h(x)^3 - (2*a^3 - 3*a^2*b + b^3)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(a^2*b - b^3 + 3*(a^2*b - b^3)*cosh(x)^2 - 4*(a^3 - a*b^2)*cosh(x) )*sinh(x))/((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 4*(a^4 - 2*a^2*b^2 + b^4)* cosh(x)*sinh(x)^3 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 - 2*(a^4 - 2*a^2*b^2 + b^4 - 3*( a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 4*((a^4 - 2*a^2*b^2 + b^4...
\[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \] Input:
integrate(coth(x)**3/(a+b*cosh(x)),x)
Output:
Integral(coth(x)**3/(a + b*cosh(x)), x)
Time = 0.04 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.66 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=-\frac {a^{3} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (2 \, a - b\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {{\left (2 \, a + b\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {b e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} + b e^{\left (-3 \, x\right )}}{a^{2} - b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} - b^{2}\right )} e^{\left (-4 \, x\right )}} \] Input:
integrate(coth(x)^3/(a+b*cosh(x)),x, algorithm="maxima")
Output:
-a^3*log(2*a*e^(-x) + b*e^(-2*x) + b)/(a^4 - 2*a^2*b^2 + b^4) + 1/2*(2*a - b)*log(e^(-x) + 1)/(a^2 - 2*a*b + b^2) + 1/2*(2*a + b)*log(e^(-x) - 1)/(a ^2 + 2*a*b + b^2) + (b*e^(-x) - 2*a*e^(-2*x) + b*e^(-3*x))/(a^2 - b^2 - 2* (a^2 - b^2)*e^(-2*x) + (a^2 - b^2)*e^(-4*x))
Time = 0.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.89 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=-\frac {a^{3} b \log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} + \frac {{\left (2 \, a - b\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {{\left (2 \, a + b\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} - 4 \, a b^{2}}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}} \] Input:
integrate(coth(x)^3/(a+b*cosh(x)),x, algorithm="giac")
Output:
-a^3*b*log(abs(b*(e^(-x) + e^x) + 2*a))/(a^4*b - 2*a^2*b^3 + b^5) + 1/4*(2 *a - b)*log(e^(-x) + e^x + 2)/(a^2 - 2*a*b + b^2) + 1/4*(2*a + b)*log(e^(- x) + e^x - 2)/(a^2 + 2*a*b + b^2) - 1/2*(a^3*(e^(-x) + e^x)^2 - 2*a^2*b*(e ^(-x) + e^x) + 2*b^3*(e^(-x) + e^x) - 4*a*b^2)/((a^4 - 2*a^2*b^2 + b^4)*(( e^(-x) + e^x)^2 - 4))
Time = 2.51 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.10 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\frac {\frac {2\,\left (a\,b^2-a^3\right )}{{\left (a^2-b^2\right )}^2}+\frac {{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {2\,a}{a^2-b^2}-\frac {2\,b\,{\mathrm {e}}^x}{a^2-b^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (2\,a-b\right )}{2\,a^2-4\,a\,b+2\,b^2}-\frac {a^3\,\ln \left (b^7\,{\mathrm {e}}^{2\,x}-16\,a^6\,b+b^7-6\,a^2\,b^5+9\,a^4\,b^3-32\,a^7\,{\mathrm {e}}^x-6\,a^2\,b^5\,{\mathrm {e}}^{2\,x}+9\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+2\,a\,b^6\,{\mathrm {e}}^x-16\,a^6\,b\,{\mathrm {e}}^{2\,x}-12\,a^3\,b^4\,{\mathrm {e}}^x+18\,a^5\,b^2\,{\mathrm {e}}^x\right )}{a^4-2\,a^2\,b^2+b^4}+\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (2\,a+b\right )}{2\,a^2+4\,a\,b+2\,b^2} \] Input:
int(coth(x)^3/(a + b*cosh(x)),x)
Output:
((2*(a*b^2 - a^3))/(a^2 - b^2)^2 + (exp(x)*(a^2*b - b^3))/(a^2 - b^2)^2)/( exp(2*x) - 1) - ((2*a)/(a^2 - b^2) - (2*b*exp(x))/(a^2 - b^2))/(exp(4*x) - 2*exp(2*x) + 1) + (log(exp(x) + 1)*(2*a - b))/(2*a^2 - 4*a*b + 2*b^2) - ( a^3*log(b^7*exp(2*x) - 16*a^6*b + b^7 - 6*a^2*b^5 + 9*a^4*b^3 - 32*a^7*exp (x) - 6*a^2*b^5*exp(2*x) + 9*a^4*b^3*exp(2*x) + 2*a*b^6*exp(x) - 16*a^6*b* exp(2*x) - 12*a^3*b^4*exp(x) + 18*a^5*b^2*exp(x)))/(a^4 + b^4 - 2*a^2*b^2) + (log(exp(x) - 1)*(2*a + b))/(4*a*b + 2*a^2 + 2*b^2)
Time = 0.33 (sec) , antiderivative size = 494, normalized size of antiderivative = 5.26 \[ \int \frac {\coth ^3(x)}{a+b \cosh (x)} \, dx=\frac {2 a \,b^{2}+2 e^{x} a^{2} b -3 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b +3 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b +6 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b -6 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b +2 \,\mathrm {log}\left (e^{x}-1\right ) a^{3}+2 \,\mathrm {log}\left (e^{x}+1\right ) a^{3}+\mathrm {log}\left (e^{x}-1\right ) b^{3}-\mathrm {log}\left (e^{x}+1\right ) b^{3}-2 a^{3}-2 e^{x} b^{3}+2 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{3}+2 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{3}-2 e^{4 x} a^{3}-2 e^{3 x} b^{3}-2 \,\mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) a^{3}+e^{4 x} \mathrm {log}\left (e^{x}-1\right ) b^{3}-e^{4 x} \mathrm {log}\left (e^{x}+1\right ) b^{3}-4 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{3}-2 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) b^{3}-4 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{3}+2 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) b^{3}-2 e^{4 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) a^{3}+2 e^{4 x} a \,b^{2}+2 e^{3 x} a^{2} b +4 e^{2 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) a^{3}-3 \,\mathrm {log}\left (e^{x}-1\right ) a^{2} b +3 \,\mathrm {log}\left (e^{x}+1\right ) a^{2} b}{2 e^{4 x} a^{4}-4 e^{4 x} a^{2} b^{2}+2 e^{4 x} b^{4}-4 e^{2 x} a^{4}+8 e^{2 x} a^{2} b^{2}-4 e^{2 x} b^{4}+2 a^{4}-4 a^{2} b^{2}+2 b^{4}} \] Input:
int(coth(x)^3/(a+b*cosh(x)),x)
Output:
(2*e**(4*x)*log(e**x - 1)*a**3 - 3*e**(4*x)*log(e**x - 1)*a**2*b + e**(4*x )*log(e**x - 1)*b**3 + 2*e**(4*x)*log(e**x + 1)*a**3 + 3*e**(4*x)*log(e**x + 1)*a**2*b - e**(4*x)*log(e**x + 1)*b**3 - 2*e**(4*x)*log(e**(2*x)*b + 2 *e**x*a + b)*a**3 - 2*e**(4*x)*a**3 + 2*e**(4*x)*a*b**2 + 2*e**(3*x)*a**2* b - 2*e**(3*x)*b**3 - 4*e**(2*x)*log(e**x - 1)*a**3 + 6*e**(2*x)*log(e**x - 1)*a**2*b - 2*e**(2*x)*log(e**x - 1)*b**3 - 4*e**(2*x)*log(e**x + 1)*a** 3 - 6*e**(2*x)*log(e**x + 1)*a**2*b + 2*e**(2*x)*log(e**x + 1)*b**3 + 4*e* *(2*x)*log(e**(2*x)*b + 2*e**x*a + b)*a**3 + 2*e**x*a**2*b - 2*e**x*b**3 + 2*log(e**x - 1)*a**3 - 3*log(e**x - 1)*a**2*b + log(e**x - 1)*b**3 + 2*lo g(e**x + 1)*a**3 + 3*log(e**x + 1)*a**2*b - log(e**x + 1)*b**3 - 2*log(e** (2*x)*b + 2*e**x*a + b)*a**3 - 2*a**3 + 2*a*b**2)/(2*(e**(4*x)*a**4 - 2*e* *(4*x)*a**2*b**2 + e**(4*x)*b**4 - 2*e**(2*x)*a**4 + 4*e**(2*x)*a**2*b**2 - 2*e**(2*x)*b**4 + a**4 - 2*a**2*b**2 + b**4))