\(\int \frac {1}{(a+b \coth (c+d x))^4} \, dx\) [84]

Optimal result

Integrand size = 12, antiderivative size = 169 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\frac {\left (a^4+6 a^2 b^2+b^4\right ) x}{\left (a^2-b^2\right )^4}+\frac {b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac {a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac {b \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \coth (c+d x))}-\frac {4 a b \left (a^2+b^2\right ) \log (b \cosh (c+d x)+a \sinh (c+d x))}{\left (a^2-b^2\right )^4 d} \]

[Out]

(a^4+6*a^2*b^2+b^4)*x/(a^2-b^2)^4+1/3*b/(a^2-b^2)/d/(a+b*coth(d*x+c))^3+a*b/(a^2-b^2)^2/d/(a+b*coth(d*x+c))^2+
b*(3*a^2+b^2)/(a^2-b^2)^3/d/(a+b*coth(d*x+c))-4*a*b*(a^2+b^2)*ln(b*cosh(d*x+c)+a*sinh(d*x+c))/(a^2-b^2)^4/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3564, 3610, 3612, 3611} \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\frac {b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a+b \coth (c+d x))}+\frac {a b}{d \left (a^2-b^2\right )^2 (a+b \coth (c+d x))^2}+\frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}-\frac {4 a b \left (a^2+b^2\right ) \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )^4}+\frac {x \left (a^4+6 a^2 b^2+b^4\right )}{\left (a^2-b^2\right )^4} \]

[In]

Int[(a + b*Coth[c + d*x])^(-4),x]

[Out]

((a^4 + 6*a^2*b^2 + b^4)*x)/(a^2 - b^2)^4 + b/(3*(a^2 - b^2)*d*(a + b*Coth[c + d*x])^3) + (a*b)/((a^2 - b^2)^2
*d*(a + b*Coth[c + d*x])^2) + (b*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(a + b*Coth[c + d*x])) - (4*a*b*(a^2 + b^2)*L
og[b*Cosh[c + d*x] + a*Sinh[c + d*x]])/((a^2 - b^2)^4*d)

Rule 3564

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*
(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3611

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*Sin[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]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[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] && NeQ[a*c + b*d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac {\int \frac {a-b \coth (c+d x)}{(a+b \coth (c+d x))^3} \, dx}{a^2-b^2} \\ & = \frac {b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac {a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac {\int \frac {a^2+b^2-2 a b \coth (c+d x)}{(a+b \coth (c+d x))^2} \, dx}{\left (a^2-b^2\right )^2} \\ & = \frac {b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac {a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac {b \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \coth (c+d x))}+\frac {\int \frac {a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{\left (a^2-b^2\right )^3} \\ & = \frac {\left (a^4+6 a^2 b^2+b^4\right ) x}{\left (a^2-b^2\right )^4}+\frac {b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac {a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac {b \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \coth (c+d x))}-\frac {\left (4 i a b \left (a^2+b^2\right )\right ) \int \frac {-i b-i a \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{\left (a^2-b^2\right )^4} \\ & = \frac {\left (a^4+6 a^2 b^2+b^4\right ) x}{\left (a^2-b^2\right )^4}+\frac {b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac {a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac {b \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \coth (c+d x))}-\frac {4 a b \left (a^2+b^2\right ) \log (b \cosh (c+d x)+a \sinh (c+d x))}{\left (a^2-b^2\right )^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.24 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=-\frac {\log (1-\tanh (c+d x))}{2 (a+b)^4 d}+\frac {\log (1+\tanh (c+d x))}{2 (a-b)^4 d}-\frac {4 a b \left (a^2+b^2\right ) \log (b+a \tanh (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {b^4}{3 a^3 \left (a^2-b^2\right ) d (b+a \tanh (c+d x))^3}+\frac {b^3 \left (2 a^2-b^2\right )}{a^3 \left (a^2-b^2\right )^2 d (b+a \tanh (c+d x))^2}-\frac {b^2 \left (6 a^4-3 a^2 b^2+b^4\right )}{a^3 \left (a^2-b^2\right )^3 d (b+a \tanh (c+d x))} \]

[In]

Integrate[(a + b*Coth[c + d*x])^(-4),x]

[Out]

-1/2*Log[1 - Tanh[c + d*x]]/((a + b)^4*d) + Log[1 + Tanh[c + d*x]]/(2*(a - b)^4*d) - (4*a*b*(a^2 + b^2)*Log[b
+ a*Tanh[c + d*x]])/((a^2 - b^2)^4*d) - b^4/(3*a^3*(a^2 - b^2)*d*(b + a*Tanh[c + d*x])^3) + (b^3*(2*a^2 - b^2)
)/(a^3*(a^2 - b^2)^2*d*(b + a*Tanh[c + d*x])^2) - (b^2*(6*a^4 - 3*a^2*b^2 + b^4))/(a^3*(a^2 - b^2)^3*d*(b + a*
Tanh[c + d*x]))

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{4}}+\frac {b}{3 \left (a -b \right ) \left (a +b \right ) \left (a +b \coth \left (d x +c \right )\right )^{3}}+\frac {a b}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \coth \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}+b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \coth \left (d x +c \right )\right )}-\frac {4 b a \left (a^{2}+b^{2}\right ) \ln \left (a +b \coth \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{4}}}{d}\)	\(163\)
default	\(\frac {\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{4}}+\frac {b}{3 \left (a -b \right ) \left (a +b \right ) \left (a +b \coth \left (d x +c \right )\right )^{3}}+\frac {a b}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \coth \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}+b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \coth \left (d x +c \right )\right )}-\frac {4 b a \left (a^{2}+b^{2}\right ) \ln \left (a +b \coth \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{4}}}{d}\)	\(163\)
parallelrisch	\(\frac {-4 b \,a^{2} \left (a^{2}+b^{2}\right ) \left (b +a \tanh \left (d x +c \right )\right )^{3} \ln \left (b +a \tanh \left (d x +c \right )\right )+4 b \,a^{2} \left (a^{2}+b^{2}\right ) \left (b +a \tanh \left (d x +c \right )\right )^{3} \ln \left (1-\tanh \left (d x +c \right )\right )+\left (\left (a^{6} d x +b \left (3 d x +2\right ) a^{5}+b^{2} \left (3 d x -2\right ) a^{4}+b^{3} \left (d x -1\right ) a^{3}+a^{2} b^{4}+\frac {a \,b^{5}}{3}-\frac {b^{6}}{3}\right ) a \tanh \left (d x +c \right )^{3}+3 a^{3} b d x \left (a +b \right )^{3} \tanh \left (d x +c \right )^{2}+3 \left (a^{3} d x +b \left (3 d x -\frac {4}{3}\right ) a^{2}+3 \left (d x +\frac {4}{9}\right ) b^{2} a +b^{3} d x \right ) b^{2} a^{2} \tanh \left (d x +c \right )+\left (a^{4} d x +b \left (3 d x -\frac {7}{3}\right ) a^{3}+3 \left (d x +\frac {7}{9}\right ) b^{2} a^{2}+b^{3} \left (d x -\frac {1}{3}\right ) a +\frac {b^{4}}{3}\right ) b^{3}\right ) \left (a +b \right )}{a \left (a -b \right )^{4} \left (a +b \right )^{4} d \left (b +a \tanh \left (d x +c \right )\right )^{3}}\)	\(301\)
risch	\(\frac {x}{a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}}+\frac {8 b \,a^{3} x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}+\frac {8 b^{3} a x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}+\frac {8 b \,a^{3} c}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}+\frac {8 b^{3} a c}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}-\frac {4 b^{2} \left (9 a^{4} {\mathrm e}^{4 d x +4 c}+12 a^{3} b \,{\mathrm e}^{4 d x +4 c}+3 b^{4} {\mathrm e}^{4 d x +4 c}-18 a^{4} {\mathrm e}^{2 d x +2 c}+6 a^{3} b \,{\mathrm e}^{2 d x +2 c}+15 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}-6 a \,b^{3} {\mathrm e}^{2 d x +2 c}+3 b^{4} {\mathrm e}^{2 d x +2 c}+9 a^{4}-18 a^{3} b +11 a^{2} b^{2}-4 a \,b^{3}+2 b^{4}\right )}{3 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \left ({\mathrm e}^{2 d x +2 c} a +b \,{\mathrm e}^{2 d x +2 c}-a +b \right )^{3} d \left (a -b \right )^{3}}-\frac {4 b \,a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a -b}{a +b}\right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}-\frac {4 b^{3} a \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a -b}{a +b}\right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}\)	\(551\)

[In]

int(1/(a+b*coth(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/2/(a-b)^4*ln(coth(d*x+c)+1)+1/3*b/(a-b)/(a+b)/(a+b*coth(d*x+c))^3+a*b/(a+b)^2/(a-b)^2/(a+b*coth(d*x+c))
^2+b*(3*a^2+b^2)/(a+b)^3/(a-b)^3/(a+b*coth(d*x+c))-4*b*a*(a^2+b^2)/(a+b)^4/(a-b)^4*ln(a+b*coth(d*x+c))-1/2/(a+
b)^4*ln(coth(d*x+c)-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3698 vs. \(2 (167) = 334\).

Time = 0.32 (sec) , antiderivative size = 3698, normalized size of antiderivative = 21.88 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*coth(d*x+c))^4,x, algorithm="fricas")

[Out]

1/3*(3*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^6
+ 18*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)*si
nh(d*x + c)^5 + 3*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*sinh
(d*x + c)^6 - 36*a^5*b^2 + 108*a^4*b^3 - 116*a^3*b^4 + 60*a^2*b^5 - 24*a*b^6 + 8*b^7 - 3*(12*a^5*b^2 + 4*a^4*b
^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6
- b^7)*d*x)*cosh(d*x + c)^4 - 3*(12*a^5*b^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 - 15*(a^7 + 7*a^6*b +
21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^2 + 3*(a^7 + 5*a^6*b + 9*
a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7)*d*x)*sinh(d*x + c)^4 + 12*(5*(a^7 + 7*a^6*b + 21*
a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^3 - (12*a^5*b^2 + 4*a^4*b^3
- 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 -
b^7)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*
b^6 - b^7)*d*x + 3*(24*a^5*b^2 - 32*a^4*b^3 - 12*a^3*b^4 + 28*a^2*b^5 - 12*a*b^6 + 4*b^7 + 3*(a^7 + 3*a^6*b +
a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*d*x)*cosh(d*x + c)^2 + 3*(24*a^5*b^2 - 32*a^4*b^3 -
12*a^3*b^4 + 28*a^2*b^5 - 12*a*b^6 + 4*b^7 + 15*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^
2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^4 + 3*(a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*
a*b^6 + b^7)*d*x - 6*(12*a^5*b^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5
*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 12*(a^6*b - 3*a^5*b^
2 + 4*a^4*b^3 - 4*a^3*b^4 + 3*a^2*b^5 - a*b^6 - (a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6
)*cosh(d*x + c)^6 - 6*(a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*x + c)*sinh(d*x +
c)^5 - (a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*sinh(d*x + c)^6 + 3*(a^6*b + a^5*b^2 -
a^2*b^5 - a*b^6)*cosh(d*x + c)^4 + 3*(a^6*b + a^5*b^2 - a^2*b^5 - a*b^6 - 5*(a^6*b + 3*a^5*b^2 + 4*a^4*b^3 +
4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 4*(5*(a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*
b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*x + c)^3 - 3*(a^6*b + a^5*b^2 - a^2*b^5 - a*b^6)*cosh(d*x + c))*sinh(d*x + c)^
3 - 3*(a^6*b - a^5*b^2 - a^2*b^5 + a*b^6)*cosh(d*x + c)^2 - 3*(a^6*b - a^5*b^2 - a^2*b^5 + a*b^6 + 5*(a^6*b +
3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*x + c)^4 - 6*(a^6*b + a^5*b^2 - a^2*b^5 - a*b^6)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 6*((a^6*b + 3*a^5*b^2 + 4*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5 + a*b^6)*cosh(d*
x + c)^5 - 2*(a^6*b + a^5*b^2 - a^2*b^5 - a*b^6)*cosh(d*x + c)^3 + (a^6*b - a^5*b^2 - a^2*b^5 + a*b^6)*cosh(d*
x + c))*sinh(d*x + c))*log(2*(b*cosh(d*x + c) + a*sinh(d*x + c))/(cosh(d*x + c) - sinh(d*x + c))) + 6*(3*(a^7
+ 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^5 - 2*(12*a^5
*b^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2
*b^5 - 5*a*b^6 - b^7)*d*x)*cosh(d*x + c)^3 + (24*a^5*b^2 - 32*a^4*b^3 - 12*a^3*b^4 + 28*a^2*b^5 - 12*a*b^6 + 4
*b^7 + 3*(a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*d*x)*cosh(d*x + c))*sinh(
d*x + c))/((a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*
b^8 - a^2*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)^6 + 6*(a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 1
4*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)*sinh(d*x + c)^5 +
(a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2*
b^9 + 3*a*b^10 + b^11)*d*sinh(d*x + c)^6 - 3*(a^11 + a^10*b - 5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5
- 10*a^5*b^6 - 10*a^4*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 - b^11)*d*cosh(d*x + c)^4 + 3*(5*(a^11 + 3*a^10*b -
a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2*b^9 + 3*a*b^10 + b^
11)*d*cosh(d*x + c)^2 - (a^11 + a^10*b - 5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5 - 10*a^5*b^6 - 10*a^4
*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 - b^11)*d)*sinh(d*x + c)^4 + 3*(a^11 - a^10*b - 5*a^9*b^2 + 5*a^8*b^3 +
10*a^7*b^4 - 10*a^6*b^5 - 10*a^5*b^6 + 10*a^4*b^7 + 5*a^3*b^8 - 5*a^2*b^9 - a*b^10 + b^11)*d*cosh(d*x + c)^2 +
4*(5*(a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 -
a^2*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)^3 - 3*(a^11 + a^10*b - 5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6
*b^5 - 10*a^5*b^6 - 10*a^4*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 - b^11)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(
5*(a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2
*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)^4 - 6*(a^11 + a^10*b - 5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5
- 10*a^5*b^6 - 10*a^4*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 - b^11)*d*cosh(d*x + c)^2 + (a^11 - a^10*b - 5*a^9
*b^2 + 5*a^8*b^3 + 10*a^7*b^4 - 10*a^6*b^5 - 10*a^5*b^6 + 10*a^4*b^7 + 5*a^3*b^8 - 5*a^2*b^9 - a*b^10 + b^11)*
d)*sinh(d*x + c)^2 - (a^11 - 3*a^10*b - a^9*b^2 + 11*a^8*b^3 - 6*a^7*b^4 - 14*a^6*b^5 + 14*a^5*b^6 + 6*a^4*b^7
- 11*a^3*b^8 + a^2*b^9 + 3*a*b^10 - b^11)*d + 6*((a^11 + 3*a^10*b - a^9*b^2 - 11*a^8*b^3 - 6*a^7*b^4 + 14*a^6
*b^5 + 14*a^5*b^6 - 6*a^4*b^7 - 11*a^3*b^8 - a^2*b^9 + 3*a*b^10 + b^11)*d*cosh(d*x + c)^5 - 2*(a^11 + a^10*b -
5*a^9*b^2 - 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5 - 10*a^5*b^6 - 10*a^4*b^7 + 5*a^3*b^8 + 5*a^2*b^9 - a*b^10 -
b^11)*d*cosh(d*x + c)^3 + (a^11 - a^10*b - 5*a^9*b^2 + 5*a^8*b^3 + 10*a^7*b^4 - 10*a^6*b^5 - 10*a^5*b^6 + 10*a
^4*b^7 + 5*a^3*b^8 - 5*a^2*b^9 - a*b^10 + b^11)*d*cosh(d*x + c))*sinh(d*x + c))

Sympy [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a+b*coth(d*x+c))**4,x)

[Out]

Exception raised: TypeError >> Invalid NaN comparison

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (167) = 334\).

Time = 0.24 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.09 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=-\frac {4 \, {\left (a^{3} b + a b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} d} - \frac {4 \, {\left (9 \, a^{4} b^{2} + 18 \, a^{3} b^{3} + 11 \, a^{2} b^{4} + 4 \, a b^{5} + 2 \, b^{6} - 3 \, {\left (6 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 5 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (3 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{3 \, {\left (a^{10} + 2 \, a^{9} b - 3 \, a^{8} b^{2} - 8 \, a^{7} b^{3} + 2 \, a^{6} b^{4} + 12 \, a^{5} b^{5} + 2 \, a^{4} b^{6} - 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} + 2 \, a b^{9} + b^{10} - 3 \, {\left (a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (a^{10} - 2 \, a^{9} b - 3 \, a^{8} b^{2} + 8 \, a^{7} b^{3} + 2 \, a^{6} b^{4} - 12 \, a^{5} b^{5} + 2 \, a^{4} b^{6} + 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} - 2 \, a b^{9} + b^{10}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{10} - 4 \, a^{9} b + 3 \, a^{8} b^{2} + 8 \, a^{7} b^{3} - 14 \, a^{6} b^{4} + 14 \, a^{4} b^{6} - 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} + 4 \, a b^{9} - b^{10}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} \]

[In]

integrate(1/(a+b*coth(d*x+c))^4,x, algorithm="maxima")

[Out]

-4*(a^3*b + a*b^3)*log(-(a - b)*e^(-2*d*x - 2*c) + a + b)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d)
- 4/3*(9*a^4*b^2 + 18*a^3*b^3 + 11*a^2*b^4 + 4*a*b^5 + 2*b^6 - 3*(6*a^4*b^2 + 2*a^3*b^3 - 5*a^2*b^4 - 2*a*b^5
- b^6)*e^(-2*d*x - 2*c) + 3*(3*a^4*b^2 - 4*a^3*b^3 + b^6)*e^(-4*d*x - 4*c))/((a^10 + 2*a^9*b - 3*a^8*b^2 - 8*a
^7*b^3 + 2*a^6*b^4 + 12*a^5*b^5 + 2*a^4*b^6 - 8*a^3*b^7 - 3*a^2*b^8 + 2*a*b^9 + b^10 - 3*(a^10 - 5*a^8*b^2 + 1
0*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*e^(-2*d*x - 2*c) + 3*(a^10 - 2*a^9*b - 3*a^8*b^2 + 8*a^7*b^3 + 2*a^
6*b^4 - 12*a^5*b^5 + 2*a^4*b^6 + 8*a^3*b^7 - 3*a^2*b^8 - 2*a*b^9 + b^10)*e^(-4*d*x - 4*c) - (a^10 - 4*a^9*b +
3*a^8*b^2 + 8*a^7*b^3 - 14*a^6*b^4 + 14*a^4*b^6 - 8*a^3*b^7 - 3*a^2*b^8 + 4*a*b^9 - b^10)*e^(-6*d*x - 6*c))*d)
+ (d*x + c)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=-\frac {\frac {12 \, {\left (a^{3} b + a b^{3}\right )} \log \left ({\left | a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (d x + c\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {4 \, {\left (3 \, {\left (3 \, a^{4} b^{2} - 2 \, a^{3} b^{3} - 2 \, a^{2} b^{4} + 2 \, a b^{5} - b^{6}\right )} e^{\left (4 \, d x + 4 \, c\right )} - 3 \, {\left (6 \, a^{4} b^{2} - 14 \, a^{3} b^{3} + 11 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right )} e^{\left (2 \, d x + 2 \, c\right )} + \frac {9 \, a^{5} b^{2} - 27 \, a^{4} b^{3} + 29 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 6 \, a b^{6} - 2 \, b^{7}}{a + b}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )}^{3} {\left (a + b\right )}^{3} {\left (a - b\right )}^{4}}}{3 \, d} \]

[In]

integrate(1/(a+b*coth(d*x+c))^4,x, algorithm="giac")

[Out]

-1/3*(12*(a^3*b + a*b^3)*log(abs(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) - a + b))/(a^8 - 4*a^6*b^2 + 6*a^4*b^4
- 4*a^2*b^6 + b^8) - 3*(d*x + c)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 4*(3*(3*a^4*b^2 - 2*a^3*b^3 - 2
*a^2*b^4 + 2*a*b^5 - b^6)*e^(4*d*x + 4*c) - 3*(6*a^4*b^2 - 14*a^3*b^3 + 11*a^2*b^4 - 4*a*b^5 + b^6)*e^(2*d*x +
2*c) + (9*a^5*b^2 - 27*a^4*b^3 + 29*a^3*b^4 - 15*a^2*b^5 + 6*a*b^6 - 2*b^7)/(a + b))/((a*e^(2*d*x + 2*c) + b*
e^(2*d*x + 2*c) - a + b)^3*(a + b)^3*(a - b)^4))/d

Mupad [B] (verification not implemented)

Time = 2.06 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\frac {x}{{\left (a-b\right )}^4}-\frac {\ln \left (b-a+a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (4\,a^3\,b+4\,a\,b^3\right )}{d\,a^8-4\,d\,a^6\,b^2+6\,d\,a^4\,b^4-4\,d\,a^2\,b^6+d\,b^8}-\frac {4\,\left (3\,a^2\,b^2-2\,a\,b^3+b^4\right )}{d\,{\left (a+b\right )}^4\,{\left (a-b\right )}^3\,\left (b-a+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )\right )}-\frac {8\,b^4}{3\,d\,{\left (a+b\right )}^4\,\left (a-b\right )\,\left ({\mathrm {e}}^{6\,c+6\,d\,x}\,{\left (a+b\right )}^3-{\left (a-b\right )}^3+3\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )\,{\left (a-b\right )}^2-3\,{\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^2\,\left (a-b\right )\right )}+\frac {4\,\left (2\,a\,b^3-b^4\right )}{d\,{\left (a+b\right )}^4\,{\left (a-b\right )}^2\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^2+{\left (a-b\right )}^2-2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )\,\left (a-b\right )\right )} \]

[In]

int(1/(a + b*coth(c + d*x))^4,x)

[Out]

x/(a - b)^4 - (log(b - a + a*exp(2*c)*exp(2*d*x) + b*exp(2*c)*exp(2*d*x))*(4*a*b^3 + 4*a^3*b))/(a^8*d + b^8*d
- 4*a^2*b^6*d + 6*a^4*b^4*d - 4*a^6*b^2*d) - (4*(b^4 - 2*a*b^3 + 3*a^2*b^2))/(d*(a + b)^4*(a - b)^3*(b - a + e
xp(2*c + 2*d*x)*(a + b))) - (8*b^4)/(3*d*(a + b)^4*(a - b)*(exp(6*c + 6*d*x)*(a + b)^3 - (a - b)^3 + 3*exp(2*c
+ 2*d*x)*(a + b)*(a - b)^2 - 3*exp(4*c + 4*d*x)*(a + b)^2*(a - b))) + (4*(2*a*b^3 - b^4))/(d*(a + b)^4*(a - b
)^2*(exp(4*c + 4*d*x)*(a + b)^2 + (a - b)^2 - 2*exp(2*c + 2*d*x)*(a + b)*(a - b)))