\(\int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx\) [123]

Optimal result

Integrand size = 13, antiderivative size = 183 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a}-\frac {3 \text {arctanh}(\cosh (x))}{8 b}+\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cosh (x))}{2 b^3}-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cosh (x))}{b^5}+\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b^5}-\frac {a \coth (x)}{b^2}+\frac {a \left (a^2+3 b^2\right ) \coth (x)}{b^4}+\frac {a \coth ^3(x)}{3 b^2}+\frac {3 \coth (x) \text {csch}(x)}{8 b}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 b^3}-\frac {\coth (x) \text {csch}^3(x)}{4 b} \]

[Out]

x/a-3/8*arctanh(cosh(x))/b+1/2*(a^2+3*b^2)*arctanh(cosh(x))/b^3-(a^4+3*a^2*b^2+3*b^4)*arctanh(cosh(x))/b^5+2*(
a^2+b^2)^(5/2)*arctanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1/2))/a/b^5-a*coth(x)/b^2+a*(a^2+3*b^2)*coth(x)/b^4+1/3*a*
coth(x)^3/b^2+3/8*coth(x)*csch(x)/b-1/2*(a^2+3*b^2)*coth(x)*csch(x)/b^3-1/4*coth(x)*csch(x)^3/b

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3983, 2976, 3855, 3852, 8, 3853, 2739, 632, 210} \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b^5}+\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cosh (x))}{2 b^3}+\frac {a \left (a^2+3 b^2\right ) \coth (x)}{b^4}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 b^3}-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cosh (x))}{b^5}+\frac {a \coth ^3(x)}{3 b^2}-\frac {a \coth (x)}{b^2}+\frac {x}{a}-\frac {3 \text {arctanh}(\cosh (x))}{8 b}-\frac {\coth (x) \text {csch}^3(x)}{4 b}+\frac {3 \coth (x) \text {csch}(x)}{8 b} \]

[In]

Int[Coth[x]^6/(a + b*Csch[x]),x]

[Out]

x/a - (3*ArcTanh[Cosh[x]])/(8*b) + ((a^2 + 3*b^2)*ArcTanh[Cosh[x]])/(2*b^3) - ((a^4 + 3*a^2*b^2 + 3*b^4)*ArcTa
nh[Cosh[x]])/b^5 + (2*(a^2 + b^2)^(5/2)*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*b^5) - (a*Coth[x])/b^2
+ (a*(a^2 + 3*b^2)*Coth[x])/b^4 + (a*Coth[x]^3)/(3*b^2) + (3*Coth[x]*Csch[x])/(8*b) - ((a^2 + 3*b^2)*Coth[x]*C
sch[x])/(2*b^3) - (Coth[x]*Csch[x]^3)/(4*b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*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]

Rule 2976

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x]
/; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || (EqQ[m, -1] && G
tQ[p, 0]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3983

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[Cos[c + d*x]^m
*((b + a*Sin[c + d*x])^n/Sin[c + d*x]^(m + n)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[
n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])

Rubi steps \begin{align*} \text {integral}& = i \int \frac {\cosh (x) \coth ^5(x)}{i b+i a \sinh (x)} \, dx \\ & = -\int \left (-\frac {1}{a}-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {csch}(x)}{b^5}-\frac {a \left (-a^2-3 b^2\right ) \text {csch}^2(x)}{b^4}-\frac {\left (a^2+3 b^2\right ) \text {csch}^3(x)}{b^3}+\frac {a \text {csch}^4(x)}{b^2}-\frac {\text {csch}^5(x)}{b}+\frac {i \left (a^2+b^2\right )^3}{a b^5 (i b+i a \sinh (x))}\right ) \, dx \\ & = \frac {x}{a}-\frac {a \int \text {csch}^4(x) \, dx}{b^2}+\frac {\int \text {csch}^5(x) \, dx}{b}-\frac {\left (i \left (a^2+b^2\right )^3\right ) \int \frac {1}{i b+i a \sinh (x)} \, dx}{a b^5}-\frac {\left (a \left (a^2+3 b^2\right )\right ) \int \text {csch}^2(x) \, dx}{b^4}+\frac {\left (a^2+3 b^2\right ) \int \text {csch}^3(x) \, dx}{b^3}+\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \int \text {csch}(x) \, dx}{b^5} \\ & = \frac {x}{a}-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cosh (x))}{b^5}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 b^3}-\frac {\coth (x) \text {csch}^3(x)}{4 b}-\frac {(i a) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right )}{b^2}-\frac {3 \int \text {csch}^3(x) \, dx}{4 b}-\frac {\left (2 i \left (a^2+b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{i b+2 i a x-i b x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a b^5}+\frac {\left (i a \left (a^2+3 b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,-i \coth (x))}{b^4}-\frac {\left (a^2+3 b^2\right ) \int \text {csch}(x) \, dx}{2 b^3} \\ & = \frac {x}{a}+\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cosh (x))}{2 b^3}-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cosh (x))}{b^5}-\frac {a \coth (x)}{b^2}+\frac {a \left (a^2+3 b^2\right ) \coth (x)}{b^4}+\frac {a \coth ^3(x)}{3 b^2}+\frac {3 \coth (x) \text {csch}(x)}{8 b}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 b^3}-\frac {\coth (x) \text {csch}^3(x)}{4 b}+\frac {3 \int \text {csch}(x) \, dx}{8 b}+\frac {\left (4 i \left (a^2+b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,2 i a-2 i b \tanh \left (\frac {x}{2}\right )\right )}{a b^5} \\ & = \frac {x}{a}-\frac {3 \text {arctanh}(\cosh (x))}{8 b}+\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cosh (x))}{2 b^3}-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cosh (x))}{b^5}+\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b^5}-\frac {a \coth (x)}{b^2}+\frac {a \left (a^2+3 b^2\right ) \coth (x)}{b^4}+\frac {a \coth ^3(x)}{3 b^2}+\frac {3 \coth (x) \text {csch}(x)}{8 b}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 b^3}-\frac {\coth (x) \text {csch}^3(x)}{4 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.63 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {\text {csch}(x) (b+a \sinh (x)) \left (192 b^5 x+384 \left (-a^2-b^2\right )^{5/2} \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )+32 a^2 b \left (3 a^2+7 b^2\right ) \coth \left (\frac {x}{2}\right )-6 a b^2 \left (4 a^2+9 b^2\right ) \text {csch}^2\left (\frac {x}{2}\right )-3 a b^4 \text {csch}^4\left (\frac {x}{2}\right )-24 a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )+24 a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )-6 a b^2 \left (4 a^2+9 b^2\right ) \text {sech}^2\left (\frac {x}{2}\right )+3 a b^4 \text {sech}^4\left (\frac {x}{2}\right )-64 a^2 b^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )+4 a^2 b^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)+32 a^2 b \left (3 a^2+7 b^2\right ) \tanh \left (\frac {x}{2}\right )\right )}{192 a b^5 (a+b \text {csch}(x))} \]

[In]

Integrate[Coth[x]^6/(a + b*Csch[x]),x]

[Out]

(Csch[x]*(b + a*Sinh[x])*(192*b^5*x + 384*(-a^2 - b^2)^(5/2)*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a^2 - b^2]] + 32*a
^2*b*(3*a^2 + 7*b^2)*Coth[x/2] - 6*a*b^2*(4*a^2 + 9*b^2)*Csch[x/2]^2 - 3*a*b^4*Csch[x/2]^4 - 24*a*(8*a^4 + 20*
a^2*b^2 + 15*b^4)*Log[Cosh[x/2]] + 24*a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*Log[Sinh[x/2]] - 6*a*b^2*(4*a^2 + 9*b^2)
*Sech[x/2]^2 + 3*a*b^4*Sech[x/2]^4 - 64*a^2*b^3*Csch[x]^3*Sinh[x/2]^4 + 4*a^2*b^3*Csch[x/2]^4*Sinh[x] + 32*a^2
*b*(3*a^2 + 7*b^2)*Tanh[x/2]))/(192*a*b^5*(a + b*Csch[x]))

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.42

method	result	size
default	\(\frac {\frac {b^{3} \tanh \left (\frac {x}{2}\right )^{4}}{4}+\frac {2 a \tanh \left (\frac {x}{2}\right )^{3} b^{2}}{3}+2 a^{2} b \tanh \left (\frac {x}{2}\right )^{2}+4 b^{3} \tanh \left (\frac {x}{2}\right )^{2}+8 a^{3} \tanh \left (\frac {x}{2}\right )+18 a \,b^{2} \tanh \left (\frac {x}{2}\right )}{16 b^{4}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\left (32 a^{6}+96 a^{4} b^{2}+96 a^{2} b^{4}+32 b^{6}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{16 a \,b^{5} \sqrt {a^{2}+b^{2}}}-\frac {1}{64 b \tanh \left (\frac {x}{2}\right )^{4}}-\frac {4 a^{2}+8 b^{2}}{32 b^{3} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (16 a^{4}+40 a^{2} b^{2}+30 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 b^{5}}+\frac {a}{24 b^{2} \tanh \left (\frac {x}{2}\right )^{3}}+\frac {a \left (4 a^{2}+9 b^{2}\right )}{8 b^{4} \tanh \left (\frac {x}{2}\right )}\)	\(260\)
risch	\(\frac {x}{a}+\frac {-12 a^{2} b \,{\mathrm e}^{7 x}-27 b^{3} {\mathrm e}^{7 x}+24 a^{3} {\mathrm e}^{6 x}+72 a \,b^{2} {\mathrm e}^{6 x}+12 a^{2} b \,{\mathrm e}^{5 x}+3 b^{3} {\mathrm e}^{5 x}-72 a^{3} {\mathrm e}^{4 x}-168 a \,b^{2} {\mathrm e}^{4 x}+12 a^{2} b \,{\mathrm e}^{3 x}+3 b^{3} {\mathrm e}^{3 x}+72 a^{3} {\mathrm e}^{2 x}+152 a \,b^{2} {\mathrm e}^{2 x}-12 a^{2} b \,{\mathrm e}^{x}-27 b^{3} {\mathrm e}^{x}-24 a^{3}-56 a \,b^{2}}{12 b^{4} \left ({\mathrm e}^{2 x}-1\right )^{4}}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}}+a^{4} b +2 a^{2} b^{3}+b^{5}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a}\right )}{b^{5} a}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{x}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}}-a^{4} b -2 a^{2} b^{3}-b^{5}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a}\right )}{b^{5} a}-\frac {\ln \left ({\mathrm e}^{x}+1\right ) a^{4}}{b^{5}}-\frac {5 \ln \left ({\mathrm e}^{x}+1\right ) a^{2}}{2 b^{3}}-\frac {15 \ln \left ({\mathrm e}^{x}+1\right )}{8 b}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) a^{4}}{b^{5}}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right ) a^{2}}{2 b^{3}}+\frac {15 \ln \left ({\mathrm e}^{x}-1\right )}{8 b}\)	\(371\)

[In]

int(coth(x)^6/(a+b*csch(x)),x,method=_RETURNVERBOSE)

[Out]

1/16/b^4*(1/4*b^3*tanh(1/2*x)^4+2/3*a*tanh(1/2*x)^3*b^2+2*a^2*b*tanh(1/2*x)^2+4*b^3*tanh(1/2*x)^2+8*a^3*tanh(1
/2*x)+18*a*b^2*tanh(1/2*x))+1/a*ln(tanh(1/2*x)+1)-1/a*ln(tanh(1/2*x)-1)+1/16*(32*a^6+96*a^4*b^2+96*a^2*b^4+32*
b^6)/a/b^5/(a^2+b^2)^(1/2)*arctanh(1/2*(-2*b*tanh(1/2*x)+2*a)/(a^2+b^2)^(1/2))-1/64/b/tanh(1/2*x)^4-1/32*(4*a^
2+8*b^2)/b^3/tanh(1/2*x)^2+1/16/b^5*(16*a^4+40*a^2*b^2+30*b^4)*ln(tanh(1/2*x))+1/24*a/b^2/tanh(1/2*x)^3+1/8*a*
(4*a^2+9*b^2)/b^4/tanh(1/2*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3160 vs. \(2 (167) = 334\).

Time = 0.41 (sec) , antiderivative size = 3160, normalized size of antiderivative = 17.27 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]

[In]

integrate(coth(x)^6/(a+b*csch(x)),x, algorithm="fricas")

[Out]

1/24*(24*b^5*x*cosh(x)^8 + 24*b^5*x*sinh(x)^8 - 6*(4*a^3*b^2 + 9*a*b^4)*cosh(x)^7 + 6*(32*b^5*x*cosh(x) - 4*a^
3*b^2 - 9*a*b^4)*sinh(x)^7 - 48*(2*b^5*x - a^4*b - 3*a^2*b^3)*cosh(x)^6 + 6*(112*b^5*x*cosh(x)^2 - 16*b^5*x +
8*a^4*b + 24*a^2*b^3 - 7*(4*a^3*b^2 + 9*a*b^4)*cosh(x))*sinh(x)^6 + 24*b^5*x + 6*(4*a^3*b^2 + a*b^4)*cosh(x)^5
+ 6*(224*b^5*x*cosh(x)^3 + 4*a^3*b^2 + a*b^4 - 21*(4*a^3*b^2 + 9*a*b^4)*cosh(x)^2 - 48*(2*b^5*x - a^4*b - 3*a
^2*b^3)*cosh(x))*sinh(x)^5 - 48*a^4*b - 112*a^2*b^3 + 48*(3*b^5*x - 3*a^4*b - 7*a^2*b^3)*cosh(x)^4 + 6*(280*b^
5*x*cosh(x)^4 + 24*b^5*x - 24*a^4*b - 56*a^2*b^3 - 35*(4*a^3*b^2 + 9*a*b^4)*cosh(x)^3 - 120*(2*b^5*x - a^4*b -
3*a^2*b^3)*cosh(x)^2 + 5*(4*a^3*b^2 + a*b^4)*cosh(x))*sinh(x)^4 + 6*(4*a^3*b^2 + a*b^4)*cosh(x)^3 + 6*(224*b^
5*x*cosh(x)^5 + 4*a^3*b^2 + a*b^4 - 35*(4*a^3*b^2 + 9*a*b^4)*cosh(x)^4 - 160*(2*b^5*x - a^4*b - 3*a^2*b^3)*cos
h(x)^3 + 10*(4*a^3*b^2 + a*b^4)*cosh(x)^2 + 32*(3*b^5*x - 3*a^4*b - 7*a^2*b^3)*cosh(x))*sinh(x)^3 - 16*(6*b^5*
x - 9*a^4*b - 19*a^2*b^3)*cosh(x)^2 + 2*(336*b^5*x*cosh(x)^6 - 48*b^5*x - 63*(4*a^3*b^2 + 9*a*b^4)*cosh(x)^5 +
72*a^4*b + 152*a^2*b^3 - 360*(2*b^5*x - a^4*b - 3*a^2*b^3)*cosh(x)^4 + 30*(4*a^3*b^2 + a*b^4)*cosh(x)^3 + 144
*(3*b^5*x - 3*a^4*b - 7*a^2*b^3)*cosh(x)^2 + 9*(4*a^3*b^2 + a*b^4)*cosh(x))*sinh(x)^2 + 24*((a^4 + 2*a^2*b^2 +
b^4)*cosh(x)^8 + 8*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^7 + (a^4 + 2*a^2*b^2 + b^4)*sinh(x)^8 - 4*(a^4 + 2
*a^2*b^2 + b^4)*cosh(x)^6 - 4*(a^4 + 2*a^2*b^2 + b^4 - 7*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(
a^4 + 2*a^2*b^2 + b^4)*cosh(x)^3 - 3*(a^4 + 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^5 + 6*(a^4 + 2*a^2*b^2 + b^4)*co
sh(x)^4 + 2*(35*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^4 + 3*a^4 + 6*a^2*b^2 + 3*b^4 - 30*(a^4 + 2*a^2*b^2 + b^4)*cos
h(x)^2)*sinh(x)^4 + a^4 + 2*a^2*b^2 + b^4 + 8*(7*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^5 - 10*(a^4 + 2*a^2*b^2 + b^4
)*cosh(x)^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^3 - 4*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2 + 4*(7*(a^4 +
2*a^2*b^2 + b^4)*cosh(x)^6 - 15*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^4 - a^4 - 2*a^2*b^2 - b^4 + 9*(a^4 + 2*a^2*b^
2 + b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 2*a^2*b^2 + b^4)*cosh(x)^7 - 3*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^5 + 3
*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^3 - (a^4 + 2*a^2*b^2 + b^4)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((a^2*cosh(x
)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(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)) - 6*(4*a^3*b^2
+ 9*a*b^4)*cosh(x) - 3*((8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 8*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)
*sinh(x)^7 + (8*a^5 + 20*a^3*b^2 + 15*a*b^4)*sinh(x)^8 - 4*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 - 4*(8*a^
5 + 20*a^3*b^2 + 15*a*b^4 - 7*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(8*a^5 + 20*a^3*b^2
+ 15*a*b^4)*cosh(x)^3 - 3*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^5 + 8*a^5 + 20*a^3*b^2 + 15*a*b^4 +
6*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 2*(24*a^5 + 60*a^3*b^2 + 45*a*b^4 + 35*(8*a^5 + 20*a^3*b^2 + 15
*a*b^4)*cosh(x)^4 - 30*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^4 + 8*(7*(8*a^5 + 20*a^3*b^2 + 15*a*
b^4)*cosh(x)^5 - 10*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 3*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sin
h(x)^3 - 4*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2 + 4*(7*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 - 8*a^5
- 20*a^3*b^2 - 15*a*b^4 - 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 9*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cos
h(x)^2)*sinh(x)^2 + 8*((8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^7 - 3*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5
+ 3*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^3 - (8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x))*log(cosh(x)
+ sinh(x) + 1) + 3*((8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 8*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)*sin
h(x)^7 + (8*a^5 + 20*a^3*b^2 + 15*a*b^4)*sinh(x)^8 - 4*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 - 4*(8*a^5 +
20*a^3*b^2 + 15*a*b^4 - 7*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(8*a^5 + 20*a^3*b^2 + 15
*a*b^4)*cosh(x)^3 - 3*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^5 + 8*a^5 + 20*a^3*b^2 + 15*a*b^4 + 6*(
8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 2*(24*a^5 + 60*a^3*b^2 + 45*a*b^4 + 35*(8*a^5 + 20*a^3*b^2 + 15*a*b
^4)*cosh(x)^4 - 30*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^4 + 8*(7*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)
*cosh(x)^5 - 10*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 3*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)
^3 - 4*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2 + 4*(7*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 - 8*a^5 - 20
*a^3*b^2 - 15*a*b^4 - 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 9*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)
^2)*sinh(x)^2 + 8*((8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^7 - 3*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 3
*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x)^3 - (8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x))*log(cosh(x) + s
inh(x) - 1) + 2*(96*b^5*x*cosh(x)^7 - 21*(4*a^3*b^2 + 9*a*b^4)*cosh(x)^6 - 144*(2*b^5*x - a^4*b - 3*a^2*b^3)*c
osh(x)^5 - 12*a^3*b^2 - 27*a*b^4 + 15*(4*a^3*b^2 + a*b^4)*cosh(x)^4 + 96*(3*b^5*x - 3*a^4*b - 7*a^2*b^3)*cosh(
x)^3 + 9*(4*a^3*b^2 + a*b^4)*cosh(x)^2 - 16*(6*b^5*x - 9*a^4*b - 19*a^2*b^3)*cosh(x))*sinh(x))/(a*b^5*cosh(x)^
8 + 8*a*b^5*cosh(x)*sinh(x)^7 + a*b^5*sinh(x)^8 - 4*a*b^5*cosh(x)^6 + 6*a*b^5*cosh(x)^4 - 4*a*b^5*cosh(x)^2 +
4*(7*a*b^5*cosh(x)^2 - a*b^5)*sinh(x)^6 + a*b^5 + 8*(7*a*b^5*cosh(x)^3 - 3*a*b^5*cosh(x))*sinh(x)^5 + 2*(35*a*
b^5*cosh(x)^4 - 30*a*b^5*cosh(x)^2 + 3*a*b^5)*sinh(x)^4 + 8*(7*a*b^5*cosh(x)^5 - 10*a*b^5*cosh(x)^3 + 3*a*b^5*
cosh(x))*sinh(x)^3 + 4*(7*a*b^5*cosh(x)^6 - 15*a*b^5*cosh(x)^4 + 9*a*b^5*cosh(x)^2 - a*b^5)*sinh(x)^2 + 8*(a*b
^5*cosh(x)^7 - 3*a*b^5*cosh(x)^5 + 3*a*b^5*cosh(x)^3 - a*b^5*cosh(x))*sinh(x))

Sympy [F]

\[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\coth ^{6}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]

[In]

integrate(coth(x)**6/(a+b*csch(x)),x)

[Out]

Integral(coth(x)**6/(a + b*csch(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.80 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=-\frac {24 \, a^{3} + 56 \, a b^{2} - 3 \, {\left (4 \, a^{2} b + 9 \, b^{3}\right )} e^{\left (-x\right )} - 8 \, {\left (9 \, a^{3} + 19 \, a b^{2}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (-3 \, x\right )} + 24 \, {\left (3 \, a^{3} + 7 \, a b^{2}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (-5 \, x\right )} - 24 \, {\left (a^{3} + 3 \, a b^{2}\right )} e^{\left (-6 \, x\right )} - 3 \, {\left (4 \, a^{2} b + 9 \, b^{3}\right )} e^{\left (-7 \, x\right )}}{12 \, {\left (4 \, b^{4} e^{\left (-2 \, x\right )} - 6 \, b^{4} e^{\left (-4 \, x\right )} + 4 \, b^{4} e^{\left (-6 \, x\right )} - b^{4} e^{\left (-8 \, x\right )} - b^{4}\right )}} + \frac {x}{a} - \frac {{\left (8 \, a^{4} + 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{8 \, b^{5}} + \frac {{\left (8 \, a^{4} + 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{8 \, b^{5}} - \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a b^{5}} \]

[In]

integrate(coth(x)^6/(a+b*csch(x)),x, algorithm="maxima")

[Out]

-1/12*(24*a^3 + 56*a*b^2 - 3*(4*a^2*b + 9*b^3)*e^(-x) - 8*(9*a^3 + 19*a*b^2)*e^(-2*x) + 3*(4*a^2*b + b^3)*e^(-
3*x) + 24*(3*a^3 + 7*a*b^2)*e^(-4*x) + 3*(4*a^2*b + b^3)*e^(-5*x) - 24*(a^3 + 3*a*b^2)*e^(-6*x) - 3*(4*a^2*b +
9*b^3)*e^(-7*x))/(4*b^4*e^(-2*x) - 6*b^4*e^(-4*x) + 4*b^4*e^(-6*x) - b^4*e^(-8*x) - b^4) + x/a - 1/8*(8*a^4 +
20*a^2*b^2 + 15*b^4)*log(e^(-x) + 1)/b^5 + 1/8*(8*a^4 + 20*a^2*b^2 + 15*b^4)*log(e^(-x) - 1)/b^5 - (a^6 + 3*a
^4*b^2 + 3*a^2*b^4 + b^6)*log((a*e^(-x) - b - sqrt(a^2 + b^2))/(a*e^(-x) - b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b
^2)*a*b^5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.67 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a} - \frac {{\left (8 \, a^{4} + 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (e^{x} + 1\right )}{8 \, b^{5}} + \frac {{\left (8 \, a^{4} + 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{8 \, b^{5}} - \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a b^{5}} - \frac {12 \, a^{2} b e^{\left (7 \, x\right )} + 27 \, b^{3} e^{\left (7 \, x\right )} - 24 \, a^{3} e^{\left (6 \, x\right )} - 72 \, a b^{2} e^{\left (6 \, x\right )} - 12 \, a^{2} b e^{\left (5 \, x\right )} - 3 \, b^{3} e^{\left (5 \, x\right )} + 72 \, a^{3} e^{\left (4 \, x\right )} + 168 \, a b^{2} e^{\left (4 \, x\right )} - 12 \, a^{2} b e^{\left (3 \, x\right )} - 3 \, b^{3} e^{\left (3 \, x\right )} - 72 \, a^{3} e^{\left (2 \, x\right )} - 152 \, a b^{2} e^{\left (2 \, x\right )} + 12 \, a^{2} b e^{x} + 27 \, b^{3} e^{x} + 24 \, a^{3} + 56 \, a b^{2}}{12 \, b^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} \]

[In]

integrate(coth(x)^6/(a+b*csch(x)),x, algorithm="giac")

[Out]

x/a - 1/8*(8*a^4 + 20*a^2*b^2 + 15*b^4)*log(e^x + 1)/b^5 + 1/8*(8*a^4 + 20*a^2*b^2 + 15*b^4)*log(abs(e^x - 1))
/b^5 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(abs(2*a*e^x + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^x + 2*b + 2*sq
rt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a*b^5) - 1/12*(12*a^2*b*e^(7*x) + 27*b^3*e^(7*x) - 24*a^3*e^(6*x) - 72*a*b^2*
e^(6*x) - 12*a^2*b*e^(5*x) - 3*b^3*e^(5*x) + 72*a^3*e^(4*x) + 168*a*b^2*e^(4*x) - 12*a^2*b*e^(3*x) - 3*b^3*e^(
3*x) - 72*a^3*e^(2*x) - 152*a*b^2*e^(2*x) + 12*a^2*b*e^x + 27*b^3*e^x + 24*a^3 + 56*a*b^2)/(b^4*(e^(2*x) - 1)^
4)

Mupad [B] (verification not implemented)

Time = 3.37 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.97 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {8\,a}{3\,b^2}-\frac {6\,{\mathrm {e}}^x}{b}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {\frac {{\mathrm {e}}^x\,\left (4\,a^2+9\,b^2\right )}{4\,b^3}-\frac {2\,\left (a^4+3\,a^2\,b^2\right )}{a\,b^4}}{{\mathrm {e}}^{2\,x}-1}+\frac {\frac {4\,a}{b^2}-\frac {{\mathrm {e}}^x\,\left (4\,a^2+13\,b^2\right )}{2\,b^3}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {x}{a}+\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (8\,a^4+20\,a^2\,b^2+15\,b^4\right )}{8\,b^5}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (8\,a^4+20\,a^2\,b^2+15\,b^4\right )}{8\,b^5}-\frac {4\,{\mathrm {e}}^x}{b\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {\ln \left (a^3\,\sqrt {{\left (a^2+b^2\right )}^5}-2\,a^7\,b-2\,a\,b^7-6\,a^3\,b^5-6\,a^5\,b^3+a^8\,{\mathrm {e}}^x+4\,b^8\,{\mathrm {e}}^x+2\,a\,b^2\,\sqrt {{\left (a^2+b^2\right )}^5}-4\,b^3\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^5}+13\,a^2\,b^6\,{\mathrm {e}}^x+15\,a^4\,b^4\,{\mathrm {e}}^x+7\,a^6\,b^2\,{\mathrm {e}}^x-3\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^5}\right )\,\sqrt {{\left (a^2+b^2\right )}^5}}{a\,b^5}-\frac {\ln \left (a^8\,{\mathrm {e}}^x-2\,a^7\,b-a^3\,\sqrt {{\left (a^2+b^2\right )}^5}-6\,a^3\,b^5-6\,a^5\,b^3-2\,a\,b^7+4\,b^8\,{\mathrm {e}}^x-2\,a\,b^2\,\sqrt {{\left (a^2+b^2\right )}^5}+4\,b^3\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^5}+13\,a^2\,b^6\,{\mathrm {e}}^x+15\,a^4\,b^4\,{\mathrm {e}}^x+7\,a^6\,b^2\,{\mathrm {e}}^x+3\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^5}\right )\,\sqrt {{\left (a^2+b^2\right )}^5}}{a\,b^5} \]

[In]

int(coth(x)^6/(a + b/sinh(x)),x)

[Out]

((8*a)/(3*b^2) - (6*exp(x))/b)/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1) - ((exp(x)*(4*a^2 + 9*b^2))/(4*b^3) -
(2*(a^4 + 3*a^2*b^2))/(a*b^4))/(exp(2*x) - 1) + ((4*a)/b^2 - (exp(x)*(4*a^2 + 13*b^2))/(2*b^3))/(exp(4*x) - 2*
exp(2*x) + 1) + x/a + (log(exp(x) - 1)*(8*a^4 + 15*b^4 + 20*a^2*b^2))/(8*b^5) - (log(exp(x) + 1)*(8*a^4 + 15*b
^4 + 20*a^2*b^2))/(8*b^5) - (4*exp(x))/(b*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) + (log(a^3*((
a^2 + b^2)^5)^(1/2) - 2*a^7*b - 2*a*b^7 - 6*a^3*b^5 - 6*a^5*b^3 + a^8*exp(x) + 4*b^8*exp(x) + 2*a*b^2*((a^2 +
b^2)^5)^(1/2) - 4*b^3*exp(x)*((a^2 + b^2)^5)^(1/2) + 13*a^2*b^6*exp(x) + 15*a^4*b^4*exp(x) + 7*a^6*b^2*exp(x)
- 3*a^2*b*exp(x)*((a^2 + b^2)^5)^(1/2))*((a^2 + b^2)^5)^(1/2))/(a*b^5) - (log(a^8*exp(x) - 2*a^7*b - a^3*((a^2
+ b^2)^5)^(1/2) - 6*a^3*b^5 - 6*a^5*b^3 - 2*a*b^7 + 4*b^8*exp(x) - 2*a*b^2*((a^2 + b^2)^5)^(1/2) + 4*b^3*exp(
x)*((a^2 + b^2)^5)^(1/2) + 13*a^2*b^6*exp(x) + 15*a^4*b^4*exp(x) + 7*a^6*b^2*exp(x) + 3*a^2*b*exp(x)*((a^2 + b
^2)^5)^(1/2))*((a^2 + b^2)^5)^(1/2))/(a*b^5)