Integrand size = 13, antiderivative size = 109 \[ \int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx=-\frac {b \left (a^2-2 b^2\right ) \text {arctanh}(\cosh (x))}{2 a^4}-\frac {2 b^4 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}+\frac {\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a} \]
-1/2*b*(a^2-2*b^2)*arctanh(cosh(x))/a^4+1/3*(2*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-2*b^4*arctanh((b-a*tanh(1/2* x))/(a^2+b^2)^(1/2))/a^4/(a^2+b^2)^(1/2)
Time = 1.00 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.94 \[ \int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx=\frac {\frac {48 b^4 \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+4 a \left (2 a^2-3 b^2\right ) \coth \left (\frac {x}{2}\right )+3 a^2 b \text {csch}^2\left (\frac {x}{2}\right )-12 a^2 b \log \left (\cosh \left (\frac {x}{2}\right )\right )+24 b^3 \log \left (\cosh \left (\frac {x}{2}\right )\right )+12 a^2 b \log \left (\sinh \left (\frac {x}{2}\right )\right )-24 b^3 \log \left (\sinh \left (\frac {x}{2}\right )\right )+3 a^2 b \text {sech}^2\left (\frac {x}{2}\right )+8 a^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {1}{2} a^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)+8 a^3 \tanh \left (\frac {x}{2}\right )-12 a b^2 \tanh \left (\frac {x}{2}\right )}{24 a^4} \]
((48*b^4*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + 4* a*(2*a^2 - 3*b^2)*Coth[x/2] + 3*a^2*b*Csch[x/2]^2 - 12*a^2*b*Log[Cosh[x/2] ] + 24*b^3*Log[Cosh[x/2]] + 12*a^2*b*Log[Sinh[x/2]] - 24*b^3*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*S inh[x])/2 + 8*a^3*Tanh[x/2] - 12*a*b^2*Tanh[x/2])/(24*a^4)
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.30, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.538, Rules used = {3042, 3281, 25, 3042, 26, 3534, 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.
| \(\displaystyle \int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (i x)^4 (a-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {\int -\frac {\text {csch}^3(x) \left (2 b \sinh ^2(x)+2 a \sinh (x)+3 b\right )}{a+b \sinh (x)}dx}{3 a}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\text {csch}^3(x) \left (2 b \sinh ^2(x)+2 a \sinh (x)+3 b\right )}{a+b \sinh (x)}dx}{3 a}-\frac {\coth (x) \text {csch}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}-\frac {\int -\frac {i \left (-2 b \sin (i x)^2-2 i a \sin (i x)+3 b\right )}{\sin (i x)^3 (a-i b \sin (i x))}dx}{3 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \int \frac {-2 b \sin (i x)^2-2 i a \sin (i x)+3 b}{\sin (i x)^3 (a-i b \sin (i x))}dx}{3 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (\frac {\int \frac {\text {csch}^2(x) \left (-3 i b^2 \sinh ^2(x)+i a b \sinh (x)+2 \left (2 i a^2-3 i b^2\right )\right )}{a+b \sinh (x)}dx}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (\frac {\int -\frac {3 i b^2 \sin (i x)^2+a b \sin (i x)+2 \left (2 i a^2-3 i b^2\right )}{\sin (i x)^2 (a-i b \sin (i x))}dx}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\int \frac {3 i b^2 \sin (i x)^2+a b \sin (i x)+2 i \left (2 a^2-3 b^2\right )}{\sin (i x)^2 (a-i b \sin (i x))}dx}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {\int \frac {3 i \text {csch}(x) \left (a \sinh (x) b^2+\left (a^2-2 b^2\right ) b\right )}{a+b \sinh (x)}dx}{a}+\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {3 i \int \frac {\text {csch}(x) \left (a \sinh (x) b^2+\left (a^2-2 b^2\right ) b\right )}{a+b \sinh (x)}dx}{a}+\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {3 i \int \frac {i \left (b \left (a^2-2 b^2\right )-i a b^2 \sin (i x)\right )}{\sin (i x) (a-i b \sin (i x))}dx}{a}+\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \int \frac {b \left (a^2-2 b^2\right )-i a b^2 \sin (i x)}{\sin (i x) (a-i b \sin (i x))}dx}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int -i \text {csch}(x)dx}{a}-\frac {2 i b^4 \int \frac {1}{a+b \sinh (x)}dx}{a}\right )}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \left (-\frac {i b \left (a^2-2 b^2\right ) \int \text {csch}(x)dx}{a}-\frac {2 i b^4 \int \frac {1}{a+b \sinh (x)}dx}{a}\right )}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \left (-\frac {i b \left (a^2-2 b^2\right ) \int i \csc (i x)dx}{a}-\frac {2 i b^4 \int \frac {1}{a-i b \sin (i x)}dx}{a}\right )}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int \csc (i x)dx}{a}-\frac {2 i b^4 \int \frac {1}{a-i b \sin (i x)}dx}{a}\right )}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int \csc (i x)dx}{a}-\frac {4 i b^4 \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int \csc (i x)dx}{a}+\frac {8 i b^4 \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}\right )}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int \csc (i x)dx}{a}+\frac {4 i b^4 \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\coth (x) \text {csch}^2(x)}{3 a}+\frac {i \left (-\frac {\frac {2 i \left (2 a^2-3 b^2\right ) \coth (x)}{a}-\frac {3 \left (\frac {i b \left (a^2-2 b^2\right ) \text {arctanh}(\cosh (x))}{a}+\frac {4 i b^4 \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )}{a}}{2 a}-\frac {3 i b \coth (x) \text {csch}(x)}{2 a}\right )}{3 a}\) |
-1/3*(Coth[x]*Csch[x]^2)/a + ((I/3)*(-1/2*((-3*((I*b*(a^2 - 2*b^2)*ArcTanh [Cosh[x]])/a + ((4*I)*b^4*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2] )])/(a*Sqrt[a^2 + b^2])))/a + ((2*I)*(2*a^2 - 3*b^2)*Coth[x])/a)/a - (((3* I)/2)*b*Coth[x]*Csch[x])/a))/a
3.1.79.3.1 Defintions of rubi rules used
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_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*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[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si n[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] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2* n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
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 = 0.88 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.39
| method | result | size |
| default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3} a^{2}}{3}+a b \tanh \left (\frac {x}{2}\right )^{2}-3 a^{2} \tanh \left (\frac {x}{2}\right )+4 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{3}}+\frac {2 b^{4} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4} \sqrt {a^{2}+b^{2}}}-\frac {1}{24 a \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-3 a^{2}+4 b^{2}}{8 a^{3} \tanh \left (\frac {x}{2}\right )}+\frac {b}{8 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {b \left (a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a^{4}}\) | \(151\) |
| risch | \(-\frac {-3 a b \,{\mathrm e}^{5 x}+6 b^{2} {\mathrm e}^{4 x}+12 a^{2} {\mathrm e}^{2 x}-12 b^{2} {\mathrm e}^{2 x}+3 b \,{\mathrm e}^{x} a -4 a^{2}+6 b^{2}}{3 a^{3} \left ({\mathrm e}^{2 x}-1\right )^{3}}+\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a^{4}}-\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a^{4}}+\frac {b \ln \left ({\mathrm e}^{x}-1\right )}{2 a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{x}-1\right )}{a^{4}}-\frac {b \ln \left ({\mathrm e}^{x}+1\right )}{2 a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{x}+1\right )}{a^{4}}\) | \(221\) |
-1/8/a^3*(1/3*tanh(1/2*x)^3*a^2+a*b*tanh(1/2*x)^2-3*a^2*tanh(1/2*x)+4*b^2* tanh(1/2*x))+2/a^4*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))-1/24/a/tanh(1/2*x)^3-1/8/a^3*(-3*a^2+4*b^2)/tanh(1/2*x)+1/ 8/a^2*b/tanh(1/2*x)^2+1/2/a^4*b*(a^2-2*b^2)*ln(tanh(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 1676 vs. \(2 (97) = 194\).
Time = 0.33 (sec) , antiderivative size = 1676, normalized size of antiderivative = 15.38 \[ \int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \]
1/6*(6*(a^4*b + a^2*b^3)*cosh(x)^5 + 6*(a^4*b + a^2*b^3)*sinh(x)^5 + 8*a^5 - 4*a^3*b^2 - 12*a*b^4 - 12*(a^3*b^2 + a*b^4)*cosh(x)^4 - 6*(2*a^3*b^2 + 2*a*b^4 - 5*(a^4*b + a^2*b^3)*cosh(x))*sinh(x)^4 + 12*(5*(a^4*b + a^2*b^3) *cosh(x)^2 - 4*(a^3*b^2 + a*b^4)*cosh(x))*sinh(x)^3 - 24*(a^5 - a*b^4)*cos h(x)^2 - 12*(2*a^5 - 2*a*b^4 - 5*(a^4*b + a^2*b^3)*cosh(x)^3 + 6*(a^3*b^2 + a*b^4)*cosh(x)^2)*sinh(x)^2 + 6*(b^4*cosh(x)^6 + 6*b^4*cosh(x)*sinh(x)^5 + b^4*sinh(x)^6 - 3*b^4*cosh(x)^4 + 3*b^4*cosh(x)^2 + 3*(5*b^4*cosh(x)^2 - b^4)*sinh(x)^4 - b^4 + 4*(5*b^4*cosh(x)^3 - 3*b^4*cosh(x))*sinh(x)^3 + 3 *(5*b^4*cosh(x)^4 - 6*b^4*cosh(x)^2 + b^4)*sinh(x)^2 + 6*(b^4*cosh(x)^5 - 2*b^4*cosh(x)^3 + b^4*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)*sin h(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sin h(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - 6*(a^4*b + a^2*b^ 3)*cosh(x) - 3*((a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^6 + 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)*sinh(x)^5 + (a^4*b - a^2*b^3 - 2*b^5)*sinh(x)^6 - a^4*b + a^2*b^3 + 2*b^5 - 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^4 - 3*(a^4*b - a^2*b ^3 - 2*b^5 - 5*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^4* b - a^2*b^3 - 2*b^5)*cosh(x)^3 - 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x))*sinh (x)^3 + 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2 + 3*(a^4*b - a^2*b^3 - 2*b^5 + 5*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^4 - 6*(a^4*b - a^2*b^3 - 2*b^5)*...
\[ \int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.78 \[ \int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx=\frac {b^{4} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} - \frac {3 \, a b e^{\left (-x\right )} - 6 \, b^{2} e^{\left (-4 \, x\right )} - 3 \, a b e^{\left (-5 \, x\right )} + 4 \, a^{2} - 6 \, b^{2} - 12 \, {\left (a^{2} - b^{2}\right )} e^{\left (-2 \, x\right )}}{3 \, {\left (3 \, a^{3} e^{\left (-2 \, x\right )} - 3 \, a^{3} e^{\left (-4 \, x\right )} + a^{3} e^{\left (-6 \, x\right )} - a^{3}\right )}} - \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, a^{4}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, a^{4}} \]
b^4*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2))) /(sqrt(a^2 + b^2)*a^4) - 1/3*(3*a*b*e^(-x) - 6*b^2*e^(-4*x) - 3*a*b*e^(-5* x) + 4*a^2 - 6*b^2 - 12*(a^2 - b^2)*e^(-2*x))/(3*a^3*e^(-2*x) - 3*a^3*e^(- 4*x) + a^3*e^(-6*x) - a^3) - 1/2*(a^2*b - 2*b^3)*log(e^(-x) + 1)/a^4 + 1/2 *(a^2*b - 2*b^3)*log(e^(-x) - 1)/a^4
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.57 \[ \int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx=\frac {b^{4} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} - \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{4}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{4}} + \frac {3 \, a b e^{\left (5 \, x\right )} - 6 \, b^{2} e^{\left (4 \, x\right )} - 12 \, a^{2} e^{\left (2 \, x\right )} + 12 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} + 4 \, a^{2} - 6 \, b^{2}}{3 \, a^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]
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) - 1/2*(a^2*b - 2*b^3)*log(e^x + 1)/a^4 + 1/2*(a^2*b - 2*b^3)*log(abs(e^x - 1))/a^4 + 1/3*(3*a*b*e^(5*x) - 6*b^2*e ^(4*x) - 12*a^2*e^(2*x) + 12*b^2*e^(2*x) - 3*a*b*e^x + 4*a^2 - 6*b^2)/(a^3 *(e^(2*x) - 1)^3)
Time = 1.64 (sec) , antiderivative size = 694, normalized size of antiderivative = 6.37 \[ \int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx=\frac {8}{3\,\left (a-3\,a\,{\mathrm {e}}^{2\,x}+3\,a\,{\mathrm {e}}^{4\,x}-a\,{\mathrm {e}}^{6\,x}\right )}-\frac {4}{a-2\,a\,{\mathrm {e}}^{2\,x}+a\,{\mathrm {e}}^{4\,x}}-\frac {2\,b^2}{a^3\,{\mathrm {e}}^{2\,x}-a^3}+\frac {b\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x-24\,b^4\,{\mathrm {e}}^x+20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{2\,a^2}-\frac {b\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^x+24\,b^4\,{\mathrm {e}}^x-20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{2\,a^2}-\frac {b^3\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x-24\,b^4\,{\mathrm {e}}^x+20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^4}+\frac {b^3\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^x+24\,b^4\,{\mathrm {e}}^x-20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^4}+\frac {2\,b\,{\mathrm {e}}^x}{a^2\,{\mathrm {e}}^{4\,x}-2\,a^2\,{\mathrm {e}}^{2\,x}+a^2}+\frac {b\,{\mathrm {e}}^x}{a^2\,{\mathrm {e}}^{2\,x}-a^2}+\frac {b^4\,\ln \left (16\,a^5\,b^2-48\,a\,b^6-32\,a^3\,b^4-24\,b^6\,\sqrt {a^2+b^2}+24\,b^7\,{\mathrm {e}}^x-40\,a^2\,b^4\,\sqrt {a^2+b^2}+16\,a^4\,b^2\,\sqrt {a^2+b^2}-32\,a^6\,b\,{\mathrm {e}}^x+112\,a^2\,b^5\,{\mathrm {e}}^x+56\,a^4\,b^3\,{\mathrm {e}}^x+72\,a\,b^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-32\,a^5\,b\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+72\,a^3\,b^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^6+a^4\,b^2}-\frac {b^4\,\ln \left (24\,b^6\,\sqrt {a^2+b^2}-48\,a\,b^6-32\,a^3\,b^4+16\,a^5\,b^2+24\,b^7\,{\mathrm {e}}^x+40\,a^2\,b^4\,\sqrt {a^2+b^2}-16\,a^4\,b^2\,\sqrt {a^2+b^2}-32\,a^6\,b\,{\mathrm {e}}^x+112\,a^2\,b^5\,{\mathrm {e}}^x+56\,a^4\,b^3\,{\mathrm {e}}^x-72\,a\,b^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^5\,b\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-72\,a^3\,b^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^6+a^4\,b^2} \]
8/(3*(a - 3*a*exp(2*x) + 3*a*exp(4*x) - a*exp(6*x))) - 4/(a - 2*a*exp(2*x) + a*exp(4*x)) - (2*b^2)/(a^3*exp(2*x) - a^3) + (b*log(4*a^4 + 24*b^4 - 20 *a^2*b^2 - 4*a^4*exp(x) - 24*b^4*exp(x) + 20*a^2*b^2*exp(x)))/(2*a^2) - (b *log(4*a^4 + 24*b^4 - 20*a^2*b^2 + 4*a^4*exp(x) + 24*b^4*exp(x) - 20*a^2*b ^2*exp(x)))/(2*a^2) - (b^3*log(4*a^4 + 24*b^4 - 20*a^2*b^2 - 4*a^4*exp(x) - 24*b^4*exp(x) + 20*a^2*b^2*exp(x)))/a^4 + (b^3*log(4*a^4 + 24*b^4 - 20*a ^2*b^2 + 4*a^4*exp(x) + 24*b^4*exp(x) - 20*a^2*b^2*exp(x)))/a^4 + (2*b*exp (x))/(a^2*exp(4*x) - 2*a^2*exp(2*x) + a^2) + (b*exp(x))/(a^2*exp(2*x) - a^ 2) + (b^4*log(16*a^5*b^2 - 48*a*b^6 - 32*a^3*b^4 - 24*b^6*(a^2 + b^2)^(1/2 ) + 24*b^7*exp(x) - 40*a^2*b^4*(a^2 + b^2)^(1/2) + 16*a^4*b^2*(a^2 + b^2)^ (1/2) - 32*a^6*b*exp(x) + 112*a^2*b^5*exp(x) + 56*a^4*b^3*exp(x) + 72*a*b^ 5*exp(x)*(a^2 + b^2)^(1/2) - 32*a^5*b*exp(x)*(a^2 + b^2)^(1/2) + 72*a^3*b^ 3*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^6 + a^4*b^2) - (b^4*log( 24*b^6*(a^2 + b^2)^(1/2) - 48*a*b^6 - 32*a^3*b^4 + 16*a^5*b^2 + 24*b^7*exp (x) + 40*a^2*b^4*(a^2 + b^2)^(1/2) - 16*a^4*b^2*(a^2 + b^2)^(1/2) - 32*a^6 *b*exp(x) + 112*a^2*b^5*exp(x) + 56*a^4*b^3*exp(x) - 72*a*b^5*exp(x)*(a^2 + b^2)^(1/2) + 32*a^5*b*exp(x)*(a^2 + b^2)^(1/2) - 72*a^3*b^3*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^6 + a^4*b^2)