Integrand size = 13, antiderivative size = 97 \[ \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx=\frac {a x}{a^2-b^2}-\frac {\left (a^2+b^2\right ) \coth (x)}{a^3}+\frac {b \coth ^2(x)}{2 a^2}-\frac {\coth ^3(x)}{3 a}-\frac {b \left (a^2+b^2\right ) \log (\sinh (x))}{a^4}-\frac {b^5 \log (a \cosh (x)+b \sinh (x))}{a^4 \left (a^2-b^2\right )} \] Output:
a*x/(a^2-b^2)-(a^2+b^2)*coth(x)/a^3+1/2*b*coth(x)^2/a^2-1/3*coth(x)^3/a-b* (a^2+b^2)*ln(sinh(x))/a^4-b^5*ln(a*cosh(x)+b*sinh(x))/a^4/(a^2-b^2)
Time = 0.36 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx=\frac {1}{6} \left (-\frac {6 \left (a^2+b^2\right ) \coth (x)}{a^3}+\frac {3 b \coth ^2(x)}{a^2}-\frac {2 \coth ^3(x)}{a}-\frac {3 \log (1-\coth (x))}{a+b}+\frac {3 \log (1+\coth (x))}{a-b}+\frac {6 b^5 \log (b+a \coth (x))}{a^4 \left (-a^2+b^2\right )}\right ) \] Input:
Integrate[Coth[x]^4/(a + b*Tanh[x]),x]
Output:
((-6*(a^2 + b^2)*Coth[x])/a^3 + (3*b*Coth[x]^2)/a^2 - (2*Coth[x]^3)/a - (3 *Log[1 - Coth[x]])/(a + b) + (3*Log[1 + Coth[x]])/(a - b) + (6*b^5*Log[b + a*Coth[x]])/(a^4*(-a^2 + b^2)))/6
Result contains complex when optimal does not.
Time = 1.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.35, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.462, Rules used = {3042, 4052, 27, 3042, 26, 4132, 27, 3042, 25, 4133, 25, 3042, 26, 4134, 26, 3042, 26, 3956, 4013}
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.
| \(\displaystyle \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (i x)^4 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle -\frac {\int \frac {3 \coth ^3(x) \left (-b \tanh ^2(x)-a \tanh (x)+b\right )}{a+b \tanh (x)}dx}{3 a}-\frac {\coth ^3(x)}{3 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\coth ^3(x) \left (-b \tanh ^2(x)-a \tanh (x)+b\right )}{a+b \tanh (x)}dx}{a}-\frac {\coth ^3(x)}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}-\frac {\int -\frac {i \left (b \tan (i x)^2+i a \tan (i x)+b\right )}{\tan (i x)^3 (a-i b \tan (i x))}dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \int \frac {b \tan (i x)^2+i a \tan (i x)+b}{\tan (i x)^3 (a-i b \tan (i x))}dx}{a}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (-\frac {\int \frac {2 i \coth ^2(x) \left (a^2+b^2-b^2 \tanh ^2(x)\right )}{a+b \tanh (x)}dx}{2 a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (-\frac {i \int \frac {\coth ^2(x) \left (a^2+b^2-b^2 \tanh ^2(x)\right )}{a+b \tanh (x)}dx}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (-\frac {i \int -\frac {a^2+b^2+b^2 \tan (i x)^2}{\tan (i x)^2 (a-i b \tan (i x))}dx}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \int \frac {a^2+b^2+b^2 \tan (i x)^2}{\tan (i x)^2 (a-i b \tan (i x))}dx}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 4133 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}-\frac {\int -\frac {\coth (x) \left (-\tanh (x) a^3-b \left (a^2+b^2\right ) \tanh ^2(x)+b \left (a^2+b^2\right )\right )}{a+b \tanh (x)}dx}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\int \frac {\coth (x) \left (-\tanh (x) a^3-b \left (a^2+b^2\right ) \tanh ^2(x)+b \left (a^2+b^2\right )\right )}{a+b \tanh (x)}dx}{a}+\frac {\left (a^2+b^2\right ) \coth (x)}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}+\frac {\int \frac {i \left (i \tan (i x) a^3+b \left (a^2+b^2\right ) \tan (i x)^2+b \left (a^2+b^2\right )\right )}{\tan (i x) (a-i b \tan (i x))}dx}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}+\frac {i \int \frac {i \tan (i x) a^3+b \left (a^2+b^2\right ) \tan (i x)^2+b \left (a^2+b^2\right )}{\tan (i x) (a-i b \tan (i x))}dx}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 4134 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}+\frac {i \left (\frac {b \left (a^2+b^2\right ) \int -i \coth (x)dx}{a}+\frac {b^5 \int -\frac {i (b+a \tanh (x))}{a+b \tanh (x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^4 x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}+\frac {i \left (-\frac {i b \left (a^2+b^2\right ) \int \coth (x)dx}{a}-\frac {i b^5 \int \frac {b+a \tanh (x)}{a+b \tanh (x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^4 x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}+\frac {i \left (-\frac {i b \left (a^2+b^2\right ) \int -i \tan \left (i x+\frac {\pi }{2}\right )dx}{a}-\frac {i b^5 \int \frac {b-i a \tan (i x)}{a-i b \tan (i x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^4 x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}+\frac {i \left (-\frac {b \left (a^2+b^2\right ) \int \tan \left (i x+\frac {\pi }{2}\right )dx}{a}-\frac {i b^5 \int \frac {b-i a \tan (i x)}{a-i b \tan (i x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^4 x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}+\frac {i \left (-\frac {i b^5 \int \frac {b-i a \tan (i x)}{a-i b \tan (i x)}dx}{a \left (a^2-b^2\right )}-\frac {i b \left (a^2+b^2\right ) \log (\sinh (x))}{a}+\frac {i a^4 x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle -\frac {\coth ^3(x)}{3 a}+\frac {i \left (\frac {i \left (\frac {\left (a^2+b^2\right ) \coth (x)}{a}+\frac {i \left (-\frac {i b \left (a^2+b^2\right ) \log (\sinh (x))}{a}-\frac {i b^5 \log (a \cosh (x)+b \sinh (x))}{a \left (a^2-b^2\right )}+\frac {i a^4 x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i b \coth ^2(x)}{2 a}\right )}{a}\) |
Input:
Int[Coth[x]^4/(a + b*Tanh[x]),x]
Output:
-1/3*Coth[x]^3/a + (I*(((-1/2*I)*b*Coth[x]^2)/a + (I*(((a^2 + b^2)*Coth[x] )/a + (I*((I*a^4*x)/(a^2 - b^2) - (I*b*(a^2 + b^2)*Log[Sinh[x]])/a - (I*b^ 5*Log[a*Cosh[x] + b*Sinh[x]])/(a*(a^2 - b^2))))/a))/a))/a
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[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d) *(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Sim p[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*( m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m , -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^ 2)/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))*(x/ ((a^2 + b^2)*(c^2 + d^2))), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d) *(a^2 + b^2)) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Sim p[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)) Int[(d - c*Tan[e + f* x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Time = 0.58 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 b +2 a}+\frac {b}{2 a^{2} \tanh \left (x \right )^{2}}+\frac {-a^{2}-b^{2}}{a^{3} \tanh \left (x \right )}-\frac {\left (a^{2}+b^{2}\right ) b \ln \left (\tanh \left (x \right )\right )}{a^{4}}-\frac {1}{3 a \tanh \left (x \right )^{3}}-\frac {b^{5} \ln \left (a +b \tanh \left (x \right )\right )}{\left (a -b \right ) \left (a +b \right ) a^{4}}\) | \(114\) |
| default | \(\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 b +2 a}+\frac {b}{2 a^{2} \tanh \left (x \right )^{2}}+\frac {-a^{2}-b^{2}}{a^{3} \tanh \left (x \right )}-\frac {\left (a^{2}+b^{2}\right ) b \ln \left (\tanh \left (x \right )\right )}{a^{4}}-\frac {1}{3 a \tanh \left (x \right )^{3}}-\frac {b^{5} \ln \left (a +b \tanh \left (x \right )\right )}{\left (a -b \right ) \left (a +b \right ) a^{4}}\) | \(114\) |
| parallelrisch | \(\frac {-6 b^{5} \ln \left (a +b \tanh \left (x \right )\right )+6 \ln \left (1-\tanh \left (x \right )\right ) a^{4} b +\left (-6 a^{4} b +6 b^{5}\right ) \ln \left (\tanh \left (x \right )\right )+\left (-2 a^{5}+2 a^{3} b^{2}\right ) \coth \left (x \right )^{3}+\left (3 a^{4} b -3 a^{2} b^{3}\right ) \coth \left (x \right )^{2}+\left (-6 a^{5}+6 a \,b^{4}\right ) \coth \left (x \right )+6 a^{4} x \left (a +b \right )}{6 a^{6}-6 a^{4} b^{2}}\) | \(123\) |
| risch | \(\frac {x}{a +b}+\frac {2 x b}{a^{2}}+\frac {2 x \,b^{3}}{a^{4}}+\frac {2 x \,b^{5}}{a^{4} \left (a^{2}-b^{2}\right )}-\frac {2 \left (6 a^{2} {\mathrm e}^{4 x}-3 \,{\mathrm e}^{4 x} a b +3 b^{2} {\mathrm e}^{4 x}-6 \,{\mathrm e}^{2 x} a^{2}+3 \,{\mathrm e}^{2 x} a b -6 b^{2} {\mathrm e}^{2 x}+4 a^{2}+3 b^{2}\right )}{3 a^{3} \left ({\mathrm e}^{2 x}-1\right )^{3}}-\frac {b \ln \left ({\mathrm e}^{2 x}-1\right )}{a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 x}-1\right )}{a^{4}}-\frac {b^{5} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{4} \left (a^{2}-b^{2}\right )}\) | \(185\) |
Input:
int(coth(x)^4/(a+b*tanh(x)),x,method=_RETURNVERBOSE)
Output:
1/(2*a-2*b)*ln(1+tanh(x))-1/(2*b+2*a)*ln(tanh(x)-1)+1/2*b/a^2/tanh(x)^2+(- a^2-b^2)/a^3/tanh(x)-(a^2+b^2)/a^4*b*ln(tanh(x))-1/3/a/tanh(x)^3-b^5/(a-b) /(a+b)/a^4*ln(a+b*tanh(x))
Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (93) = 186\).
Time = 0.12 (sec) , antiderivative size = 1299, normalized size of antiderivative = 13.39 \[ \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^4/(a+b*tanh(x)),x, algorithm="fricas")
Output:
1/3*(3*(a^5 + a^4*b)*x*cosh(x)^6 + 18*(a^5 + a^4*b)*x*cosh(x)*sinh(x)^5 + 3*(a^5 + a^4*b)*x*sinh(x)^6 - 8*a^5 + 2*a^3*b^2 + 6*a*b^4 - 3*(4*a^5 - 2*a ^4*b - 2*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 + 3*(a^5 + a^4*b)*x)*cosh(x)^4 - 3* (4*a^5 - 2*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - 15*(a^5 + a^4*b)*x*co sh(x)^2 + 3*(a^5 + a^4*b)*x)*sinh(x)^4 + 12*(5*(a^5 + a^4*b)*x*cosh(x)^3 - (4*a^5 - 2*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 + 3*(a^5 + a^4*b)*x)*c osh(x))*sinh(x)^3 + 3*(4*a^5 - 2*a^4*b + 2*a^2*b^3 - 4*a*b^4 + 3*(a^5 + a^ 4*b)*x)*cosh(x)^2 + 3*(15*(a^5 + a^4*b)*x*cosh(x)^4 + 4*a^5 - 2*a^4*b + 2* a^2*b^3 - 4*a*b^4 - 6*(4*a^5 - 2*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 + 3*(a^5 + a^4*b)*x)*cosh(x)^2 + 3*(a^5 + a^4*b)*x)*sinh(x)^2 - 3*(a^5 + a^ 4*b)*x - 3*(b^5*cosh(x)^6 + 6*b^5*cosh(x)*sinh(x)^5 + b^5*sinh(x)^6 - 3*b^ 5*cosh(x)^4 + 3*b^5*cosh(x)^2 - b^5 + 3*(5*b^5*cosh(x)^2 - b^5)*sinh(x)^4 + 4*(5*b^5*cosh(x)^3 - 3*b^5*cosh(x))*sinh(x)^3 + 3*(5*b^5*cosh(x)^4 - 6*b ^5*cosh(x)^2 + b^5)*sinh(x)^2 + 6*(b^5*cosh(x)^5 - 2*b^5*cosh(x)^3 + b^5*c osh(x))*sinh(x))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) - 3*(( a^4*b - b^5)*cosh(x)^6 + 6*(a^4*b - b^5)*cosh(x)*sinh(x)^5 + (a^4*b - b^5) *sinh(x)^6 - a^4*b + b^5 - 3*(a^4*b - b^5)*cosh(x)^4 - 3*(a^4*b - b^5 - 5* (a^4*b - b^5)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^4*b - b^5)*cosh(x)^3 - 3*(a^4 *b - b^5)*cosh(x))*sinh(x)^3 + 3*(a^4*b - b^5)*cosh(x)^2 + 3*(a^4*b - b^5 + 5*(a^4*b - b^5)*cosh(x)^4 - 6*(a^4*b - b^5)*cosh(x)^2)*sinh(x)^2 + 6*...
\[ \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx=\int \frac {\coth ^{4}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \] Input:
integrate(coth(x)**4/(a+b*tanh(x)),x)
Output:
Integral(coth(x)**4/(a + b*tanh(x)), x)
Time = 0.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.78 \[ \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx=-\frac {b^{5} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - a^{4} b^{2}} + \frac {2 \, {\left (4 \, a^{2} + 3 \, b^{2} - 3 \, {\left (2 \, a^{2} + a b + 2 \, b^{2}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (2 \, a^{2} + a b + b^{2}\right )} e^{\left (-4 \, x\right )}\right )}}{3 \, {\left (3 \, a^{3} e^{\left (-2 \, x\right )} - 3 \, a^{3} e^{\left (-4 \, x\right )} + a^{3} e^{\left (-6 \, x\right )} - a^{3}\right )}} + \frac {x}{a + b} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{4}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{4}} \] Input:
integrate(coth(x)^4/(a+b*tanh(x)),x, algorithm="maxima")
Output:
-b^5*log(-(a - b)*e^(-2*x) - a - b)/(a^6 - a^4*b^2) + 2/3*(4*a^2 + 3*b^2 - 3*(2*a^2 + a*b + 2*b^2)*e^(-2*x) + 3*(2*a^2 + a*b + b^2)*e^(-4*x))/(3*a^3 *e^(-2*x) - 3*a^3*e^(-4*x) + a^3*e^(-6*x) - a^3) + x/(a + b) - (a^2*b + b^ 3)*log(e^(-x) + 1)/a^4 - (a^2*b + b^3)*log(e^(-x) - 1)/a^4
Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.46 \[ \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx=-\frac {b^{5} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - a^{4} b^{2}} + \frac {x}{a - b} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{a^{4}} - \frac {2 \, {\left (4 \, a^{3} + 3 \, a b^{2} + 3 \, {\left (2 \, a^{3} - a^{2} b + a b^{2}\right )} e^{\left (4 \, x\right )} - 3 \, {\left (2 \, a^{3} - a^{2} b + 2 \, a b^{2}\right )} e^{\left (2 \, x\right )}\right )}}{3 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \] Input:
integrate(coth(x)^4/(a+b*tanh(x)),x, algorithm="giac")
Output:
-b^5*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^6 - a^4*b^2) + x/(a - b) - (a^2*b + b^3)*log(abs(e^(2*x) - 1))/a^4 - 2/3*(4*a^3 + 3*a*b^2 + 3*(2*a^3 - a^2*b + a*b^2)*e^(4*x) - 3*(2*a^3 - a^2*b + 2*a*b^2)*e^(2*x))/(a^4*(e^( 2*x) - 1)^3)
Time = 2.66 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.68 \[ \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx=\frac {x}{a-b}-\frac {8}{3\,a\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {b^5\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^6-a^4\,b^2}-\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^2\,b+b^3\right )}{a^4}-\frac {2\,\left (2\,a^3+a^2\,b+b^3\right )}{a^3\,\left (a+b\right )\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2\,\left (2\,a^2+a\,b-b^2\right )}{a^2\,\left (a+b\right )\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )} \] Input:
int(coth(x)^4/(a + b*tanh(x)),x)
Output:
x/(a - b) - 8/(3*a*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (b^5*log(a - b + a*exp(2*x) + b*exp(2*x)))/(a^6 - a^4*b^2) - (log(exp(2*x) - 1)*(a^2* b + b^3))/a^4 - (2*(a^2*b + 2*a^3 + b^3))/(a^3*(a + b)*(exp(2*x) - 1)) - ( 2*(a*b + 2*a^2 - b^2))/(a^2*(a + b)*(exp(4*x) - 2*exp(2*x) + 1))
\[ \int \frac {\coth ^4(x)}{a+b \tanh (x)} \, dx=\int \frac {\coth \left (x \right )^{4}}{\tanh \left (x \right ) b +a}d x \] Input:
int(coth(x)^4/(a+b*tanh(x)),x)
Output:
int(coth(x)^4/(a+b*tanh(x)),x)