Integrand size = 13, antiderivative size = 76 \[ \int \frac {\coth ^4(x)}{a+b \coth (x)} \, dx=\frac {a x}{a^2-b^2}+\frac {a \coth (x)}{b^2}-\frac {\coth ^2(x)}{2 b}-\frac {a^4 \log (a+b \coth (x))}{b^3 \left (a^2-b^2\right )}-\frac {b \log (\sinh (x))}{a^2-b^2} \]
a*x/(a^2-b^2)+a*coth(x)/b^2-1/2*coth(x)^2/b-a^4*ln(a+b*coth(x))/b^3/(a^2-b ^2)-b*ln(sinh(x))/(a^2-b^2)
Time = 0.44 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int \frac {\coth ^4(x)}{a+b \coth (x)} \, dx=\frac {a \coth (x)}{b^2}-\frac {\coth ^2(x)}{2 b}-\frac {\log (1-\coth (x))}{2 (a+b)}+\frac {\log (1+\coth (x))}{2 (a-b)}-\frac {a^4 \log (a+b \coth (x))}{b^3 \left (a^2-b^2\right )} \]
(a*Coth[x])/b^2 - Coth[x]^2/(2*b) - Log[1 - Coth[x]]/(2*(a + b)) + Log[1 + Coth[x]]/(2*(a - b)) - (a^4*Log[a + b*Coth[x]])/(b^3*(a^2 - b^2))
Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.30, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 4049, 27, 3042, 26, 4130, 25, 3042, 4110, 26, 3042, 26, 3956, 4100, 16}
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.
\(\displaystyle \int \frac {\coth ^4(x)}{a+b \coth (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan \left (\frac {\pi }{2}+i x\right )^4}{a-i b \tan \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 4049 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}+\frac {i \int -\frac {2 i \coth (x) \left (-a \coth ^2(x)+b \coth (x)+a\right )}{a+b \coth (x)}dx}{2 b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\coth (x) \left (-a \coth ^2(x)+b \coth (x)+a\right )}{a+b \coth (x)}dx}{b}-\frac {\coth ^2(x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}+\frac {\int -\frac {i \tan \left (i x+\frac {\pi }{2}\right ) \left (a \tan \left (i x+\frac {\pi }{2}\right )^2-i b \tan \left (i x+\frac {\pi }{2}\right )+a\right )}{a-i b \tan \left (i x+\frac {\pi }{2}\right )}dx}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \int \frac {\tan \left (i x+\frac {\pi }{2}\right ) \left (a \tan \left (i x+\frac {\pi }{2}\right )^2-i b \tan \left (i x+\frac {\pi }{2}\right )+a\right )}{a-i b \tan \left (i x+\frac {\pi }{2}\right )}dx}{b}\) |
\(\Big \downarrow \) 4130 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i \int -\frac {a^2-\left (a^2+b^2\right ) \coth ^2(x)}{a+b \coth (x)}dx}{b}+\frac {i a \coth (x)}{b}\right )}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \int \frac {a^2-\left (a^2+b^2\right ) \coth ^2(x)}{a+b \coth (x)}dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \int \frac {a^2+\left (a^2+b^2\right ) \tan \left (i x+\frac {\pi }{2}\right )^2}{a-i b \tan \left (i x+\frac {\pi }{2}\right )}dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 4110 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \left (-\frac {i b^3 \int i \coth (x)dx}{a^2-b^2}+\frac {a^4 \int \frac {1-\coth ^2(x)}{a+b \coth (x)}dx}{a^2-b^2}-\frac {a b^2 x}{a^2-b^2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \left (\frac {b^3 \int \coth (x)dx}{a^2-b^2}+\frac {a^4 \int \frac {1-\coth ^2(x)}{a+b \coth (x)}dx}{a^2-b^2}-\frac {a b^2 x}{a^2-b^2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \left (\frac {b^3 \int -i \tan \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {a^4 \int \frac {\tan \left (i x+\frac {\pi }{2}\right )^2+1}{a-i b \tan \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a b^2 x}{a^2-b^2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \left (-\frac {i b^3 \int \tan \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {a^4 \int \frac {\tan \left (i x+\frac {\pi }{2}\right )^2+1}{a-i b \tan \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a b^2 x}{a^2-b^2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \left (\frac {a^4 \int \frac {\tan \left (i x+\frac {\pi }{2}\right )^2+1}{a-i b \tan \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a b^2 x}{a^2-b^2}+\frac {b^3 \log (\sinh (x))}{a^2-b^2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 4100 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \left (\frac {a^4 \int \frac {1}{a+b \coth (x)}d(b \coth (x))}{b \left (a^2-b^2\right )}-\frac {a b^2 x}{a^2-b^2}+\frac {b^3 \log (\sinh (x))}{a^2-b^2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\frac {\coth ^2(x)}{2 b}-\frac {i \left (\frac {i a \coth (x)}{b}-\frac {i \left (-\frac {a b^2 x}{a^2-b^2}+\frac {b^3 \log (\sinh (x))}{a^2-b^2}+\frac {a^4 \log (a+b \coth (x))}{b \left (a^2-b^2\right )}\right )}{b}\right )}{b}\) |
-1/2*Coth[x]^2/b - (I*((I*a*Coth[x])/b - (I*(-((a*b^2*x)/(a^2 - b^2)) + (a ^4*Log[a + b*Coth[x]])/(b*(a^2 - b^2)) + (b^3*Log[Sinh[x]])/(a^2 - b^2)))/ b))/b
3.2.48.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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[((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 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*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] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/(b*f) Subst[Int[(a + x)^m, x], x, b* Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
Int[((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/((a_) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[a*(A - C)*(x/(a^2 + b^2)), x] + (Simp[(a^2*C + A*b^2)/(a^2 + b^2) Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[b*((A - C)/(a^2 + b^2)) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2*C + A*b^2, 0] && NeQ[a^2 + b^2, 0] && NeQ[A, C]
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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[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] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\frac {\coth \left (x \right )^{2}}{2 b}+\frac {a \coth \left (x \right )}{b^{2}}+\frac {\ln \left (1+\coth \left (x \right )\right )}{2 a -2 b}-\frac {a^{4} \ln \left (a +b \coth \left (x \right )\right )}{b^{3} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (\coth \left (x \right )-1\right )}{2 a +2 b}\) | \(76\) |
default | \(-\frac {\coth \left (x \right )^{2}}{2 b}+\frac {a \coth \left (x \right )}{b^{2}}+\frac {\ln \left (1+\coth \left (x \right )\right )}{2 a -2 b}-\frac {a^{4} \ln \left (a +b \coth \left (x \right )\right )}{b^{3} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (\coth \left (x \right )-1\right )}{2 a +2 b}\) | \(76\) |
parallelrisch | \(\frac {-2 \ln \left (b +a \tanh \left (x \right )\right ) a^{4}+2 \ln \left (1-\tanh \left (x \right )\right ) b^{4}+\left (2 a^{4}-2 b^{4}\right ) \ln \left (\tanh \left (x \right )\right )+2 \left (-\frac {b \coth \left (x \right )^{2} \left (a -b \right )}{2}+a \coth \left (x \right ) \left (a -b \right )+b^{2} x \right ) b \left (a +b \right )}{2 a^{2} b^{3}-2 b^{5}}\) | \(91\) |
risch | \(\frac {x}{a +b}-\frac {2 x \,a^{2}}{b^{3}}-\frac {2 x}{b}+\frac {2 x \,a^{4}}{b^{3} \left (a^{2}-b^{2}\right )}+\frac {2 a \,{\mathrm e}^{2 x}-2 b \,{\mathrm e}^{2 x}-2 a}{\left ({\mathrm e}^{2 x}-1\right )^{2} b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{b}-\frac {a^{4} \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{b^{3} \left (a^{2}-b^{2}\right )}\) | \(136\) |
-1/2*coth(x)^2/b+a*coth(x)/b^2+1/(2*a-2*b)*ln(1+coth(x))-1/b^3*a^4/(a+b)/( a-b)*ln(a+b*coth(x))-1/(2*a+2*b)*ln(coth(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (74) = 148\).
Time = 0.29 (sec) , antiderivative size = 648, normalized size of antiderivative = 8.53 \[ \int \frac {\coth ^4(x)}{a+b \coth (x)} \, dx=\frac {{\left (a b^{3} + b^{4}\right )} x \cosh \left (x\right )^{4} + 4 \, {\left (a b^{3} + b^{4}\right )} x \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a b^{3} + b^{4}\right )} x \sinh \left (x\right )^{4} - 2 \, a^{3} b + 2 \, a b^{3} + 2 \, {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a b^{3} + b^{4}\right )} x\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} + 3 \, {\left (a b^{3} + b^{4}\right )} x \cosh \left (x\right )^{2} - {\left (a b^{3} + b^{4}\right )} x\right )} \sinh \left (x\right )^{2} + {\left (a b^{3} + b^{4}\right )} x - {\left (a^{4} \cosh \left (x\right )^{4} + 4 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{4} \sinh \left (x\right )^{4} - 2 \, a^{4} \cosh \left (x\right )^{2} + a^{4} + 2 \, {\left (3 \, a^{4} \cosh \left (x\right )^{2} - a^{4}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{4} \cosh \left (x\right )^{3} - a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left ({\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{4} - b^{4}\right )} \sinh \left (x\right )^{4} + a^{4} - b^{4} - 2 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{4} - b^{4} - 3 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{3} - {\left (a^{4} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left ({\left (a b^{3} + b^{4}\right )} x \cosh \left (x\right )^{3} + {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a b^{3} + b^{4}\right )} x\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} b^{3} - b^{5}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b^{3} - b^{5} - 3 \, {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{3} - {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
((a*b^3 + b^4)*x*cosh(x)^4 + 4*(a*b^3 + b^4)*x*cosh(x)*sinh(x)^3 + (a*b^3 + b^4)*x*sinh(x)^4 - 2*a^3*b + 2*a*b^3 + 2*(a^3*b - a^2*b^2 - a*b^3 + b^4 - (a*b^3 + b^4)*x)*cosh(x)^2 + 2*(a^3*b - a^2*b^2 - a*b^3 + b^4 + 3*(a*b^3 + b^4)*x*cosh(x)^2 - (a*b^3 + b^4)*x)*sinh(x)^2 + (a*b^3 + b^4)*x - (a^4* cosh(x)^4 + 4*a^4*cosh(x)*sinh(x)^3 + a^4*sinh(x)^4 - 2*a^4*cosh(x)^2 + a^ 4 + 2*(3*a^4*cosh(x)^2 - a^4)*sinh(x)^2 + 4*(a^4*cosh(x)^3 - a^4*cosh(x))* sinh(x))*log(2*(b*cosh(x) + a*sinh(x))/(cosh(x) - sinh(x))) + ((a^4 - b^4) *cosh(x)^4 + 4*(a^4 - b^4)*cosh(x)*sinh(x)^3 + (a^4 - b^4)*sinh(x)^4 + a^4 - b^4 - 2*(a^4 - b^4)*cosh(x)^2 - 2*(a^4 - b^4 - 3*(a^4 - b^4)*cosh(x)^2) *sinh(x)^2 + 4*((a^4 - b^4)*cosh(x)^3 - (a^4 - b^4)*cosh(x))*sinh(x))*log( 2*sinh(x)/(cosh(x) - sinh(x))) + 4*((a*b^3 + b^4)*x*cosh(x)^3 + (a^3*b - a ^2*b^2 - a*b^3 + b^4 - (a*b^3 + b^4)*x)*cosh(x))*sinh(x))/(a^2*b^3 - b^5 + (a^2*b^3 - b^5)*cosh(x)^4 + 4*(a^2*b^3 - b^5)*cosh(x)*sinh(x)^3 + (a^2*b^ 3 - b^5)*sinh(x)^4 - 2*(a^2*b^3 - b^5)*cosh(x)^2 - 2*(a^2*b^3 - b^5 - 3*(a ^2*b^3 - b^5)*cosh(x)^2)*sinh(x)^2 + 4*((a^2*b^3 - b^5)*cosh(x)^3 - (a^2*b ^3 - b^5)*cosh(x))*sinh(x))
Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (61) = 122\).
Time = 1.42 (sec) , antiderivative size = 882, normalized size of antiderivative = 11.61 \[ \int \frac {\coth ^4(x)}{a+b \coth (x)} \, dx=\text {Too large to display} \]
Piecewise((zoo*(x - log(tanh(x) + 1) + log(tanh(x)) - 1/(2*tanh(x)**2)), E q(a, 0) & Eq(b, 0)), ((x - log(tanh(x) + 1) + log(tanh(x)) - 1/(2*tanh(x)* *2))/b, Eq(a, 0)), (7*x*tanh(x)**3/(2*b*tanh(x)**3 - 2*b*tanh(x)**2) - 7*x *tanh(x)**2/(2*b*tanh(x)**3 - 2*b*tanh(x)**2) - 4*log(tanh(x) + 1)*tanh(x) **3/(2*b*tanh(x)**3 - 2*b*tanh(x)**2) + 4*log(tanh(x) + 1)*tanh(x)**2/(2*b *tanh(x)**3 - 2*b*tanh(x)**2) + 4*log(tanh(x))*tanh(x)**3/(2*b*tanh(x)**3 - 2*b*tanh(x)**2) - 4*log(tanh(x))*tanh(x)**2/(2*b*tanh(x)**3 - 2*b*tanh(x )**2) - 3*tanh(x)**2/(2*b*tanh(x)**3 - 2*b*tanh(x)**2) + tanh(x)/(2*b*tanh (x)**3 - 2*b*tanh(x)**2) + 1/(2*b*tanh(x)**3 - 2*b*tanh(x)**2), Eq(a, -b)) , (x*tanh(x)**3/(2*b*tanh(x)**3 + 2*b*tanh(x)**2) + x*tanh(x)**2/(2*b*tanh (x)**3 + 2*b*tanh(x)**2) - 4*log(tanh(x) + 1)*tanh(x)**3/(2*b*tanh(x)**3 + 2*b*tanh(x)**2) - 4*log(tanh(x) + 1)*tanh(x)**2/(2*b*tanh(x)**3 + 2*b*tan h(x)**2) + 4*log(tanh(x))*tanh(x)**3/(2*b*tanh(x)**3 + 2*b*tanh(x)**2) + 4 *log(tanh(x))*tanh(x)**2/(2*b*tanh(x)**3 + 2*b*tanh(x)**2) + 3*tanh(x)**2/ (2*b*tanh(x)**3 + 2*b*tanh(x)**2) + tanh(x)/(2*b*tanh(x)**3 + 2*b*tanh(x)* *2) - 1/(2*b*tanh(x)**3 + 2*b*tanh(x)**2), Eq(a, b)), ((x - 1/tanh(x) - 1/ (3*tanh(x)**3))/a, Eq(b, 0)), (-2*a**4*log(tanh(x) + b/a)*tanh(x)**2/(2*a* *2*b**3*tanh(x)**2 - 2*b**5*tanh(x)**2) + 2*a**4*log(tanh(x))*tanh(x)**2/( 2*a**2*b**3*tanh(x)**2 - 2*b**5*tanh(x)**2) + 2*a**3*b*tanh(x)/(2*a**2*b** 3*tanh(x)**2 - 2*b**5*tanh(x)**2) - a**2*b**2/(2*a**2*b**3*tanh(x)**2 -...
Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.57 \[ \int \frac {\coth ^4(x)}{a+b \coth (x)} \, dx=-\frac {a^{4} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{2} b^{3} - b^{5}} + \frac {2 \, {\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} - a\right )}}{2 \, b^{2} e^{\left (-2 \, x\right )} - b^{2} e^{\left (-4 \, x\right )} - b^{2}} + \frac {x}{a + b} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{b^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{b^{3}} \]
-a^4*log(-(a - b)*e^(-2*x) + a + b)/(a^2*b^3 - b^5) + 2*((a + b)*e^(-2*x) - a)/(2*b^2*e^(-2*x) - b^2*e^(-4*x) - b^2) + x/(a + b) + (a^2 + b^2)*log(e ^(-x) + 1)/b^3 + (a^2 + b^2)*log(e^(-x) - 1)/b^3
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.32 \[ \int \frac {\coth ^4(x)}{a+b \coth (x)} \, dx=-\frac {a^{4} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a^{2} b^{3} - b^{5}} + \frac {x}{a - b} + \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{b^{3}} - \frac {2 \, {\left (a b - {\left (a b - b^{2}\right )} e^{\left (2 \, x\right )}\right )}}{b^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \]
-a^4*log(abs(a*e^(2*x) + b*e^(2*x) - a + b))/(a^2*b^3 - b^5) + x/(a - b) + (a^2 + b^2)*log(abs(e^(2*x) - 1))/b^3 - 2*(a*b - (a*b - b^2)*e^(2*x))/(b^ 3*(e^(2*x) - 1)^2)
Time = 2.24 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.45 \[ \int \frac {\coth ^4(x)}{a+b \coth (x)} \, dx=\frac {x}{a-b}-\frac {2}{b\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}+\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^2+b^2\right )}{b^3}+\frac {a^4\,\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{b^5-a^2\,b^3}+\frac {2\,\left (a^2-b^2\right )}{b^2\,\left (a+b\right )\,\left ({\mathrm {e}}^{2\,x}-1\right )} \]