Integrand size = 13, antiderivative size = 108 \[ \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx=\frac {b \left (3 a^2+2 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^4}-\frac {2 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4}-\frac {\left (4 a^2+3 b^2\right ) \coth (x)}{3 a^3}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a} \] Output:
1/2*b*(3*a^2+2*b^2)*arctanh(cosh(x))/a^4-2*(a^2+b^2)^(3/2)*arctanh((b-a*ta nh(1/2*x))/(a^2+b^2)^(1/2))/a^4-1/3*(4*a^2+3*b^2)*coth(x)/a^3+1/2*b*coth(x )*csch(x)/a^2-1/3*coth(x)*csch(x)^2/a
Time = 0.35 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.82 \[ \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx=\frac {48 \left (-a^2-b^2\right )^{3/2} \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )-4 a \left (4 a^2+3 b^2\right ) \coth \left (\frac {x}{2}\right )+3 a^2 b \text {csch}^2\left (\frac {x}{2}\right )+12 b \left (3 a^2+2 b^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )-12 b \left (3 a^2+2 b^2\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )+3 a^2 b \text {sech}^2\left (\frac {x}{2}\right )+8 a^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {1}{2} a^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)-4 a \left (4 a^2+3 b^2\right ) \tanh \left (\frac {x}{2}\right )}{24 a^4} \] Input:
Integrate[Coth[x]^4/(a + b*Sinh[x]),x]
Output:
(48*(-a^2 - b^2)^(3/2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]] - 4*a*(4 *a^2 + 3*b^2)*Coth[x/2] + 3*a^2*b*Csch[x/2]^2 + 12*b*(3*a^2 + 2*b^2)*Log[C osh[x/2]] - 12*b*(3*a^2 + 2*b^2)*Log[Sinh[x/2]] + 3*a^2*b*Sech[x/2]^2 + 8* a^3*Csch[x]^3*Sinh[x/2]^4 - (a^3*Csch[x/2]^4*Sinh[x])/2 - 4*a*(4*a^2 + 3*b ^2)*Tanh[x/2])/(24*a^4)
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.19, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.308, Rules used = {3042, 3204, 25, 3042, 25, 3534, 27, 3042, 26, 3480, 26, 3042, 26, 3139, 1083, 219, 4257}
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.
| \(\displaystyle \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (i x)^4 (a-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 3204 |
| \(\displaystyle -\frac {\int -\frac {\text {csch}^2(x) \left (3 \left (2 a^2+b^2\right ) \sinh ^2(x)-a b \sinh (x)+2 \left (4 a^2+3 b^2\right )\right )}{a+b \sinh (x)}dx}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\text {csch}^2(x) \left (3 \left (2 a^2+b^2\right ) \sinh ^2(x)-a b \sinh (x)+2 \left (4 a^2+3 b^2\right )\right )}{a+b \sinh (x)}dx}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\frac {-3 \left (2 a^2+b^2\right ) \sin (i x)^2+i a b \sin (i x)+2 \left (4 a^2+3 b^2\right )}{\sin (i x)^2 (a-i b \sin (i x))}dx}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {-3 \left (2 a^2+b^2\right ) \sin (i x)^2+i a b \sin (i x)+2 \left (4 a^2+3 b^2\right )}{\sin (i x)^2 (a-i b \sin (i x))}dx}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\int \frac {3 \text {csch}(x) \left (b \left (3 a^2+2 b^2\right )-a \left (2 a^2+b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{a}+\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \int \frac {\text {csch}(x) \left (b \left (3 a^2+2 b^2\right )-a \left (2 a^2+b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{a}+\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 \int \frac {i \left (b \left (3 a^2+2 b^2\right )+i a \left (2 a^2+b^2\right ) \sin (i x)\right )}{\sin (i x) (a-i b \sin (i x))}dx}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \int \frac {b \left (3 a^2+2 b^2\right )+i a \left (2 a^2+b^2\right ) \sin (i x)}{\sin (i x) (a-i b \sin (i x))}dx}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {2 i \left (a^2+b^2\right )^2 \int \frac {1}{a+b \sinh (x)}dx}{a}+\frac {b \left (3 a^2+2 b^2\right ) \int -i \text {csch}(x)dx}{a}\right )}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {2 i \left (a^2+b^2\right )^2 \int \frac {1}{a+b \sinh (x)}dx}{a}-\frac {i b \left (3 a^2+2 b^2\right ) \int \text {csch}(x)dx}{a}\right )}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {2 i \left (a^2+b^2\right )^2 \int \frac {1}{a-i b \sin (i x)}dx}{a}-\frac {i b \left (3 a^2+2 b^2\right ) \int i \csc (i x)dx}{a}\right )}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {2 i \left (a^2+b^2\right )^2 \int \frac {1}{a-i b \sin (i x)}dx}{a}+\frac {b \left (3 a^2+2 b^2\right ) \int \csc (i x)dx}{a}\right )}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b \left (3 a^2+2 b^2\right ) \int \csc (i x)dx}{a}+\frac {4 i \left (a^2+b^2\right )^2 \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b \left (3 a^2+2 b^2\right ) \int \csc (i x)dx}{a}-\frac {8 i \left (a^2+b^2\right )^2 \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}\right )}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {b \left (3 a^2+2 b^2\right ) \int \csc (i x)dx}{a}-\frac {4 i \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}\right )}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2+3 b^2\right ) \coth (x)}{a}+\frac {3 i \left (\frac {i b \left (3 a^2+2 b^2\right ) \text {arctanh}(\cosh (x))}{a}-\frac {4 i \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}\right )}{a}}{6 a^2}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
Input:
Int[Coth[x]^4/(a + b*Sinh[x]),x]
Output:
-1/6*(((3*I)*((I*b*(3*a^2 + 2*b^2)*ArcTanh[Cosh[x]])/a - ((4*I)*(a^2 + b^2 )^(3/2)*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/a))/a + (2*(4* a^2 + 3*b^2)*Coth[x])/a)/a^2 + (b*Coth[x]*Csch[x])/(2*a^2) - (Coth[x]*Csch [x]^2)/(3*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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ e + f*x]^3)), x] + (-Simp[b*(m - 2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(6*a^2*f*Sin[e + f*x]^2)), x] - Simp[1/(6*a^2) Int[((a + b*Sin[e + f* x])^m/Sin[e + f*x]^2)*Simp[8*a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f , m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1] && IntegerQ[2*m]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[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*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[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] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 2.20 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3} a^{2}}{3}+\tanh \left (\frac {x}{2}\right )^{2} a b +5 \tanh \left (\frac {x}{2}\right ) a^{2}+4 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{3}}-\frac {1}{24 a \tanh \left (\frac {x}{2}\right )^{3}}-\frac {5 a^{2}+4 b^{2}}{8 a^{3} \tanh \left (\frac {x}{2}\right )}+\frac {b}{8 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {b \left (3 a^{2}+2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a^{4}}-\frac {\left (-16 a^{4}-32 a^{2} b^{2}-16 b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{8 a^{4} \sqrt {a^{2}+b^{2}}}\) | \(169\) |
| risch | \(-\frac {-3 a b \,{\mathrm e}^{5 x}+12 \,{\mathrm e}^{4 x} a^{2}+6 b^{2} {\mathrm e}^{4 x}-12 \,{\mathrm e}^{2 x} a^{2}-12 \,{\mathrm e}^{2 x} b^{2}+3 b \,{\mathrm e}^{x} a +8 a^{2}+6 b^{2}}{3 \left ({\mathrm e}^{2 x}-1\right )^{3} a^{3}}+\frac {3 b \ln \left ({\mathrm e}^{x}+1\right )}{2 a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{x}+1\right )}{a^{4}}-\frac {3 b \ln \left ({\mathrm e}^{x}-1\right )}{2 a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{x}-1\right )}{a^{4}}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \ln \left ({\mathrm e}^{x}-\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{3}-a \,b^{2}}{b \left (a^{2}+b^{2}\right )}\right )}{a^{4}}-\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{3}+a \,b^{2}}{b \left (a^{2}+b^{2}\right )}\right )}{a^{4}}\) | \(224\) |
Input:
int(coth(x)^4/(a+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
-1/8/a^3*(1/3*tanh(1/2*x)^3*a^2+tanh(1/2*x)^2*a*b+5*tanh(1/2*x)*a^2+4*b^2* tanh(1/2*x))-1/24/a/tanh(1/2*x)^3-1/8*(5*a^2+4*b^2)/a^3/tanh(1/2*x)+1/8/a^ 2*b/tanh(1/2*x)^2-1/2/a^4*b*(3*a^2+2*b^2)*ln(tanh(1/2*x))-1/8/a^4*(-16*a^4 -32*a^2*b^2-16*b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2 +b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1303 vs. \(2 (96) = 192\).
Time = 0.16 (sec) , antiderivative size = 1303, normalized size of antiderivative = 12.06 \[ \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^4/(a+b*sinh(x)),x, algorithm="fricas")
Output:
1/6*(6*a^2*b*cosh(x)^5 + 6*a^2*b*sinh(x)^5 - 12*(2*a^3 + a*b^2)*cosh(x)^4 + 6*(5*a^2*b*cosh(x) - 4*a^3 - 2*a*b^2)*sinh(x)^4 - 6*a^2*b*cosh(x) + 12*( 5*a^2*b*cosh(x)^2 - 4*(2*a^3 + a*b^2)*cosh(x))*sinh(x)^3 - 16*a^3 - 12*a*b ^2 + 24*(a^3 + a*b^2)*cosh(x)^2 + 12*(5*a^2*b*cosh(x)^3 + 2*a^3 + 2*a*b^2 - 6*(2*a^3 + a*b^2)*cosh(x)^2)*sinh(x)^2 + 6*((a^2 + b^2)*cosh(x)^6 + 6*(a ^2 + b^2)*cosh(x)*sinh(x)^5 + (a^2 + b^2)*sinh(x)^6 - 3*(a^2 + b^2)*cosh(x )^4 + 3*(5*(a^2 + b^2)*cosh(x)^2 - a^2 - b^2)*sinh(x)^4 + 4*(5*(a^2 + b^2) *cosh(x)^3 - 3*(a^2 + b^2)*cosh(x))*sinh(x)^3 + 3*(a^2 + b^2)*cosh(x)^2 + 3*(5*(a^2 + b^2)*cosh(x)^4 - 6*(a^2 + b^2)*cosh(x)^2 + a^2 + b^2)*sinh(x)^ 2 - a^2 - b^2 + 6*((a^2 + b^2)*cosh(x)^5 - 2*(a^2 + b^2)*cosh(x)^3 + (a^2 + b^2)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^ 2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a ^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*co sh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) + 3*((3*a^2*b + 2*b^3)*cosh(x)^6 + 6*(3*a^2*b + 2*b^3)*cosh(x)*sinh(x)^5 + (3*a^2*b + 2*b^3)*sinh(x)^6 - 3*( 3*a^2*b + 2*b^3)*cosh(x)^4 - 3*(3*a^2*b + 2*b^3 - 5*(3*a^2*b + 2*b^3)*cosh (x)^2)*sinh(x)^4 + 4*(5*(3*a^2*b + 2*b^3)*cosh(x)^3 - 3*(3*a^2*b + 2*b^3)* cosh(x))*sinh(x)^3 - 3*a^2*b - 2*b^3 + 3*(3*a^2*b + 2*b^3)*cosh(x)^2 + 3*( 5*(3*a^2*b + 2*b^3)*cosh(x)^4 + 3*a^2*b + 2*b^3 - 6*(3*a^2*b + 2*b^3)*cosh (x)^2)*sinh(x)^2 + 6*((3*a^2*b + 2*b^3)*cosh(x)^5 - 2*(3*a^2*b + 2*b^3)...
\[ \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx=\int \frac {\coth ^{4}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \] Input:
integrate(coth(x)**4/(a+b*sinh(x)),x)
Output:
Integral(coth(x)**4/(a + b*sinh(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (96) = 192\).
Time = 0.13 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.96 \[ \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx=-\frac {3 \, a b e^{\left (-x\right )} - 3 \, a b e^{\left (-5 \, x\right )} - 8 \, a^{2} - 6 \, b^{2} + 12 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )} - 6 \, {\left (2 \, a^{2} + b^{2}\right )} e^{\left (-4 \, x\right )}}{3 \, {\left (3 \, a^{3} e^{\left (-2 \, x\right )} - 3 \, a^{3} e^{\left (-4 \, x\right )} + a^{3} e^{\left (-6 \, x\right )} - a^{3}\right )}} + \frac {{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, a^{4}} - \frac {{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, a^{4}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} \] Input:
integrate(coth(x)^4/(a+b*sinh(x)),x, algorithm="maxima")
Output:
-1/3*(3*a*b*e^(-x) - 3*a*b*e^(-5*x) - 8*a^2 - 6*b^2 + 12*(a^2 + b^2)*e^(-2 *x) - 6*(2*a^2 + b^2)*e^(-4*x))/(3*a^3*e^(-2*x) - 3*a^3*e^(-4*x) + a^3*e^( -6*x) - a^3) + 1/2*(3*a^2*b + 2*b^3)*log(e^(-x) + 1)/a^4 - 1/2*(3*a^2*b + 2*b^3)*log(e^(-x) - 1)/a^4 + (a^4 + 2*a^2*b^2 + b^4)*log((b*e^(-x) - a - s qrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^4)
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (96) = 192\).
Time = 0.14 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.80 \[ \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx=\frac {{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{4}} - \frac {{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{4}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} + \frac {3 \, a b e^{\left (5 \, x\right )} - 12 \, a^{2} e^{\left (4 \, x\right )} - 6 \, b^{2} e^{\left (4 \, x\right )} + 12 \, a^{2} e^{\left (2 \, x\right )} + 12 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} - 8 \, a^{2} - 6 \, b^{2}}{3 \, a^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \] Input:
integrate(coth(x)^4/(a+b*sinh(x)),x, algorithm="giac")
Output:
1/2*(3*a^2*b + 2*b^3)*log(e^x + 1)/a^4 - 1/2*(3*a^2*b + 2*b^3)*log(abs(e^x - 1))/a^4 + (a^4 + 2*a^2*b^2 + b^4)*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^4) + 1/3* (3*a*b*e^(5*x) - 12*a^2*e^(4*x) - 6*b^2*e^(4*x) + 12*a^2*e^(2*x) + 12*b^2* e^(2*x) - 3*a*b*e^x - 8*a^2 - 6*b^2)/(a^3*(e^(2*x) - 1)^3)
Time = 2.31 (sec) , antiderivative size = 778, normalized size of antiderivative = 7.20 \[ \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx =\text {Too large to display} \] Input:
int(coth(x)^4/(a + b*sinh(x)),x)
Output:
(log(- (8*(18*a^8*b + 8*b^9 + 40*a^2*b^7 + 74*a^4*b^5 + 60*a^6*b^3 - 30*a^ 9*exp(x) - 14*a*b^8*exp(x) - 69*a^3*b^6*exp(x) - 126*a^5*b^4*exp(x) - 101* a^7*b^2*exp(x)))/(a^9*b^3) - (((a^2 + b^2)^3)^(1/2)*((8*(4*a^8 + 8*b^8 + 3 6*a^2*b^6 + 57*a^4*b^4 + 34*a^6*b^2 - 12*a*b^7*exp(x) - 36*a^7*b*exp(x) - 52*a^3*b^5*exp(x) - 75*a^5*b^3*exp(x)))/(a^6*b^4) - (((16*(4*a^4*b + 4*b^5 + 8*a^2*b^3 - 8*a^5*exp(x) - 7*a*b^4*exp(x) - 15*a^3*b^2*exp(x)))/(a*b^5) + (32*((a^2 + b^2)^3)^(1/2)*(3*a^4*b + 2*a^2*b^3 - 4*a^5*exp(x) - 3*a^3*b ^2*exp(x)))/(a^4*b^5))*((a^2 + b^2)^3)^(1/2))/a^4))/a^4)*((a^2 + b^2)^3)^( 1/2))/a^4 - ((2*(2*a^2 + b^2))/a^3 - (b*exp(x))/a^2)/(exp(2*x) - 1) - (4/a - (2*b*exp(x))/a^2)/(exp(4*x) - 2*exp(2*x) + 1) - (log((((a^2 + b^2)^3)^( 1/2)*((8*(4*a^8 + 8*b^8 + 36*a^2*b^6 + 57*a^4*b^4 + 34*a^6*b^2 - 12*a*b^7* exp(x) - 36*a^7*b*exp(x) - 52*a^3*b^5*exp(x) - 75*a^5*b^3*exp(x)))/(a^6*b^ 4) + (((16*(4*a^4*b + 4*b^5 + 8*a^2*b^3 - 8*a^5*exp(x) - 7*a*b^4*exp(x) - 15*a^3*b^2*exp(x)))/(a*b^5) - (32*((a^2 + b^2)^3)^(1/2)*(3*a^4*b + 2*a^2*b ^3 - 4*a^5*exp(x) - 3*a^3*b^2*exp(x)))/(a^4*b^5))*((a^2 + b^2)^3)^(1/2))/a ^4))/a^4 - (8*(18*a^8*b + 8*b^9 + 40*a^2*b^7 + 74*a^4*b^5 + 60*a^6*b^3 - 3 0*a^9*exp(x) - 14*a*b^8*exp(x) - 69*a^3*b^6*exp(x) - 126*a^5*b^4*exp(x) - 101*a^7*b^2*exp(x)))/(a^9*b^3))*((a^2 + b^2)^3)^(1/2))/a^4 - 8/(3*a*(3*exp (2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (log(exp(x) - 1)*(3*a^2*b + 2*b^3))/ (2*a^4) + (log(exp(x) + 1)*(3*a^2*b + 2*b^3))/(2*a^4)
Time = 0.17 (sec) , antiderivative size = 654, normalized size of antiderivative = 6.06 \[ \int \frac {\coth ^4(x)}{a+b \sinh (x)} \, dx=\frac {-6 e^{6 x} \mathrm {log}\left (e^{x}-1\right ) b^{3}+6 e^{6 x} \mathrm {log}\left (e^{x}+1\right ) b^{3}-4 e^{6 x} a \,b^{2}+6 e^{5 x} a^{2} b +18 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) b^{3}-18 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) b^{3}+36 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} i -6 e^{x} a^{2} b -12 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} i -8 a \,b^{2}-8 e^{6 x} a^{3}-8 a^{3}+12 e^{6 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} i +12 e^{6 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{2} i -36 e^{4 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} i -36 e^{4 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{2} i +6 \,\mathrm {log}\left (e^{x}-1\right ) b^{3}-6 \,\mathrm {log}\left (e^{x}+1\right ) b^{3}+36 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{2} i -9 e^{6 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b +9 e^{6 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b +27 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b -27 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b +12 e^{2 x} a \,b^{2}-27 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b +18 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) b^{3}-18 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) b^{3}+27 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b +9 \,\mathrm {log}\left (e^{x}-1\right ) a^{2} b -9 \,\mathrm {log}\left (e^{x}+1\right ) a^{2} b -12 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{2} i}{6 a^{4} \left (e^{6 x}-3 e^{4 x}+3 e^{2 x}-1\right )} \] Input:
int(coth(x)^4/(a+b*sinh(x)),x)
Output:
(12*e**(6*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a* *2*i + 12*e**(6*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b** 2))*b**2*i - 36*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*i - 36*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqr t(a**2 + b**2))*b**2*i + 36*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a* i)/sqrt(a**2 + b**2))*a**2*i + 36*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b* i + a*i)/sqrt(a**2 + b**2))*b**2*i - 12*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*i - 12*sqrt(a**2 + b**2)*atan((e**x*b*i + a* i)/sqrt(a**2 + b**2))*b**2*i - 9*e**(6*x)*log(e**x - 1)*a**2*b - 6*e**(6*x )*log(e**x - 1)*b**3 + 9*e**(6*x)*log(e**x + 1)*a**2*b + 6*e**(6*x)*log(e* *x + 1)*b**3 - 8*e**(6*x)*a**3 - 4*e**(6*x)*a*b**2 + 6*e**(5*x)*a**2*b + 2 7*e**(4*x)*log(e**x - 1)*a**2*b + 18*e**(4*x)*log(e**x - 1)*b**3 - 27*e**( 4*x)*log(e**x + 1)*a**2*b - 18*e**(4*x)*log(e**x + 1)*b**3 - 27*e**(2*x)*l og(e**x - 1)*a**2*b - 18*e**(2*x)*log(e**x - 1)*b**3 + 27*e**(2*x)*log(e** x + 1)*a**2*b + 18*e**(2*x)*log(e**x + 1)*b**3 + 12*e**(2*x)*a*b**2 - 6*e* *x*a**2*b + 9*log(e**x - 1)*a**2*b + 6*log(e**x - 1)*b**3 - 9*log(e**x + 1 )*a**2*b - 6*log(e**x + 1)*b**3 - 8*a**3 - 8*a*b**2)/(6*a**4*(e**(6*x) - 3 *e**(4*x) + 3*e**(2*x) - 1))