Integrand size = 13, antiderivative size = 125 \[ \int \frac {\cosh ^4(x)}{a+b \text {csch}(x)} \, dx=\frac {\left (3 a^4+12 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {2 b \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5}-\frac {\cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}-\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{8 a^4} \]
1/8*(3*a^4+12*a^2*b^2+8*b^4)*x/a^5+2*b*(a^2+b^2)^(3/2)*arctanh((a-b*tanh(1 /2*x))/(a^2+b^2)^(1/2))/a^5-1/12*cosh(x)^3*(4*b-3*a*sinh(x))/a^2-1/8*cosh( x)*(8*b*(a^2+b^2)-a*(3*a^2+4*b^2)*sinh(x))/a^4
Time = 0.83 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.44 \[ \int \frac {\cosh ^4(x)}{a+b \text {csch}(x)} \, dx=\frac {-24 a b \left (5 a^2+4 b^2\right ) \cosh (x)-8 a^3 b \cosh (3 x)+3 \left (12 a^4 x+48 a^2 b^2 x+32 b^4 x+64 a^2 b \sqrt {-a^2-b^2} \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )+64 b^3 \sqrt {-a^2-b^2} \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )+8 a^2 \left (a^2+b^2\right ) \sinh (2 x)+a^4 \sinh (4 x)\right )}{96 a^5} \]
(-24*a*b*(5*a^2 + 4*b^2)*Cosh[x] - 8*a^3*b*Cosh[3*x] + 3*(12*a^4*x + 48*a^ 2*b^2*x + 32*b^4*x + 64*a^2*b*Sqrt[-a^2 - b^2]*ArcTan[(a - b*Tanh[x/2])/Sq rt[-a^2 - b^2]] + 64*b^3*Sqrt[-a^2 - b^2]*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a ^2 - b^2]] + 8*a^2*(a^2 + b^2)*Sinh[2*x] + a^4*Sinh[4*x]))/(96*a^5)
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.22, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.308, Rules used = {3042, 4360, 26, 26, 3042, 26, 3344, 26, 3042, 3344, 25, 3042, 3214, 3042, 3139, 1083, 219}
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.
| \(\displaystyle \int \frac {\cosh ^4(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i x)^4}{a+i b \csc (i x)}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {i \sinh (x) \cosh ^4(x)}{i a \sinh (x)+i b}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int -\frac {i \cosh ^4(x) \sinh (x)}{b+a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\sinh (x) \cosh ^4(x)}{a \sinh (x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i x) \cos (i x)^4}{b-i a \sin (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\cos (i x)^4 \sin (i x)}{b-i a \sin (i x)}dx\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -i \left (-\frac {\int \frac {i \cosh ^2(x) \left (a b-\left (3 a^2+4 b^2\right ) \sinh (x)\right )}{b+a \sinh (x)}dx}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (-\frac {i \int \frac {\cosh ^2(x) \left (a b-\left (3 a^2+4 b^2\right ) \sinh (x)\right )}{b+a \sinh (x)}dx}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-\frac {i \int \frac {\cos (i x)^2 \left (a b+i \left (3 a^2+4 b^2\right ) \sin (i x)\right )}{b-i a \sin (i x)}dx}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -i \left (-\frac {i \left (\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{2 a^2}-\frac {\int -\frac {a b \left (5 a^2+4 b^2\right )-\left (3 a^4+12 b^2 a^2+8 b^4\right ) \sinh (x)}{b+a \sinh (x)}dx}{2 a^2}\right )}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (-\frac {i \left (\frac {\int \frac {a b \left (5 a^2+4 b^2\right )-\left (3 a^4+12 b^2 a^2+8 b^4\right ) \sinh (x)}{b+a \sinh (x)}dx}{2 a^2}+\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{2 a^2}\right )}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-\frac {i \left (\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{2 a^2}+\frac {\int \frac {a b \left (5 a^2+4 b^2\right )+i \left (3 a^4+12 b^2 a^2+8 b^4\right ) \sin (i x)}{b-i a \sin (i x)}dx}{2 a^2}\right )}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -i \left (-\frac {i \left (\frac {\frac {8 b \left (a^2+b^2\right )^2 \int \frac {1}{b+a \sinh (x)}dx}{a}-\frac {x \left (3 a^4+12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}+\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{2 a^2}\right )}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-\frac {i \left (\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{2 a^2}+\frac {-\frac {x \left (3 a^4+12 a^2 b^2+8 b^4\right )}{a}+\frac {8 b \left (a^2+b^2\right )^2 \int \frac {1}{b-i a \sin (i x)}dx}{a}}{2 a^2}\right )}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -i \left (-\frac {i \left (\frac {\frac {16 b \left (a^2+b^2\right )^2 \int \frac {1}{-b \tanh ^2\left (\frac {x}{2}\right )+2 a \tanh \left (\frac {x}{2}\right )+b}d\tanh \left (\frac {x}{2}\right )}{a}-\frac {x \left (3 a^4+12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}+\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{2 a^2}\right )}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -i \left (-\frac {i \left (\frac {-\frac {32 b \left (a^2+b^2\right )^2 \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 a-2 b \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 a-2 b \tanh \left (\frac {x}{2}\right )\right )}{a}-\frac {x \left (3 a^4+12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}+\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{2 a^2}\right )}{4 a^2}-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -i \left (-\frac {i \cosh ^3(x) (4 b-3 a \sinh (x))}{12 a^2}-\frac {i \left (\frac {\cosh (x) \left (8 b \left (a^2+b^2\right )-a \left (3 a^2+4 b^2\right ) \sinh (x)\right )}{2 a^2}+\frac {-\frac {16 b \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}-\frac {x \left (3 a^4+12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}\right )}{4 a^2}\right )\) |
(-I)*(((-1/12*I)*Cosh[x]^3*(4*b - 3*a*Sinh[x]))/a^2 - ((I/4)*((-(((3*a^4 + 12*a^2*b^2 + 8*b^4)*x)/a) - (16*b*(a^2 + b^2)^(3/2)*ArcTanh[(2*a - 2*b*Ta nh[x/2])/(2*Sqrt[a^2 + b^2])])/a)/(2*a^2) + (Cosh[x]*(8*b*(a^2 + b^2) - a* (3*a^2 + 4*b^2)*Sinh[x]))/(2*a^2)))/a^2)
3.1.93.3.1 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_) + (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_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && IntegerQ[2*m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 60.04 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.72
| method | result | size |
| risch | \(\frac {3 x}{8 a}+\frac {3 x \,b^{2}}{2 a^{3}}+\frac {x \,b^{4}}{a^{5}}+\frac {{\mathrm e}^{4 x}}{64 a}-\frac {b \,{\mathrm e}^{3 x}}{24 a^{2}}+\frac {{\mathrm e}^{2 x}}{8 a}+\frac {{\mathrm e}^{2 x} b^{2}}{8 a^{3}}-\frac {5 b \,{\mathrm e}^{x}}{8 a^{2}}-\frac {b^{3} {\mathrm e}^{x}}{2 a^{4}}-\frac {5 b \,{\mathrm e}^{-x}}{8 a^{2}}-\frac {b^{3} {\mathrm e}^{-x}}{2 a^{4}}-\frac {{\mathrm e}^{-2 x}}{8 a}-\frac {{\mathrm e}^{-2 x} b^{2}}{8 a^{3}}-\frac {b \,{\mathrm e}^{-3 x}}{24 a^{2}}-\frac {{\mathrm e}^{-4 x}}{64 a}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b \ln \left ({\mathrm e}^{x}+\frac {b +\sqrt {a^{2}+b^{2}}}{a}\right )}{a^{5}}-\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b \ln \left ({\mathrm e}^{x}-\frac {-b +\sqrt {a^{2}+b^{2}}}{a}\right )}{a^{5}}\) | \(215\) |
| default | \(\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {-3 a -2 b}{6 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {-7 a^{2}-4 a b -4 b^{2}}{8 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\left (-3 a^{4}-12 a^{2} b^{2}-8 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 a^{5}}-\frac {-5 a^{3}-12 a^{2} b -4 a \,b^{2}-8 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {-3 a +2 b}{6 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {7 a^{2}-4 a b +4 b^{2}}{8 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {\left (3 a^{4}+12 a^{2} b^{2}+8 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 a^{5}}-\frac {-5 a^{3}+12 a^{2} b -4 a \,b^{2}+8 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {2 b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{5} \sqrt {a^{2}+b^{2}}}\) | \(311\) |
3/8*x/a+3/2*x/a^3*b^2+x/a^5*b^4+1/64/a*exp(x)^4-1/24/a^2*b*exp(x)^3+1/8/a* exp(x)^2+1/8/a^3*exp(x)^2*b^2-5/8*b/a^2*exp(x)-1/2*b^3/a^4*exp(x)-5/8*b/a^ 2/exp(x)-1/2*b^3/a^4/exp(x)-1/8/a/exp(x)^2-1/8/a^3/exp(x)^2*b^2-1/24/a^2*b /exp(x)^3-1/64/a/exp(x)^4+(a^2+b^2)^(3/2)*b/a^5*ln(exp(x)+(b+(a^2+b^2)^(1/ 2))/a)-(a^2+b^2)^(3/2)*b/a^5*ln(exp(x)-(-b+(a^2+b^2)^(1/2))/a)
Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (114) = 228\).
Time = 0.26 (sec) , antiderivative size = 924, normalized size of antiderivative = 7.39 \[ \int \frac {\cosh ^4(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]
1/192*(3*a^4*cosh(x)^8 + 3*a^4*sinh(x)^8 - 8*a^3*b*cosh(x)^7 + 8*(3*a^4*co sh(x) - a^3*b)*sinh(x)^7 + 24*(a^4 + a^2*b^2)*cosh(x)^6 + 4*(21*a^4*cosh(x )^2 - 14*a^3*b*cosh(x) + 6*a^4 + 6*a^2*b^2)*sinh(x)^6 + 24*(3*a^4 + 12*a^2 *b^2 + 8*b^4)*x*cosh(x)^4 - 24*(5*a^3*b + 4*a*b^3)*cosh(x)^5 + 24*(7*a^4*c osh(x)^3 - 7*a^3*b*cosh(x)^2 - 5*a^3*b - 4*a*b^3 + 6*(a^4 + a^2*b^2)*cosh( x))*sinh(x)^5 - 8*a^3*b*cosh(x) + 2*(105*a^4*cosh(x)^4 - 140*a^3*b*cosh(x) ^3 + 180*(a^4 + a^2*b^2)*cosh(x)^2 + 12*(3*a^4 + 12*a^2*b^2 + 8*b^4)*x - 6 0*(5*a^3*b + 4*a*b^3)*cosh(x))*sinh(x)^4 - 3*a^4 - 24*(5*a^3*b + 4*a*b^3)* cosh(x)^3 + 8*(21*a^4*cosh(x)^5 - 35*a^3*b*cosh(x)^4 - 15*a^3*b - 12*a*b^3 + 60*(a^4 + a^2*b^2)*cosh(x)^3 + 12*(3*a^4 + 12*a^2*b^2 + 8*b^4)*x*cosh(x ) - 30*(5*a^3*b + 4*a*b^3)*cosh(x)^2)*sinh(x)^3 - 24*(a^4 + a^2*b^2)*cosh( x)^2 + 12*(7*a^4*cosh(x)^6 - 14*a^3*b*cosh(x)^5 + 30*(a^4 + a^2*b^2)*cosh( x)^4 - 2*a^4 - 2*a^2*b^2 + 12*(3*a^4 + 12*a^2*b^2 + 8*b^4)*x*cosh(x)^2 - 2 0*(5*a^3*b + 4*a*b^3)*cosh(x)^3 - 6*(5*a^3*b + 4*a*b^3)*cosh(x))*sinh(x)^2 + 192*((a^2*b + b^3)*cosh(x)^4 + 4*(a^2*b + b^3)*cosh(x)^3*sinh(x) + 6*(a ^2*b + b^3)*cosh(x)^2*sinh(x)^2 + 4*(a^2*b + b^3)*cosh(x)*sinh(x)^3 + (a^2 *b + b^3)*sinh(x)^4)*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)) + 8*(3*a^4*cosh(x)^7 - 7*a^3*b*cosh...
\[ \int \frac {\cosh ^4(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\cosh ^{4}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.74 \[ \int \frac {\cosh ^4(x)}{a+b \text {csch}(x)} \, dx=-\frac {{\left (8 \, a^{2} b e^{\left (-x\right )} - 3 \, a^{3} - 24 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-2 \, x\right )} + 24 \, {\left (5 \, a^{2} b + 4 \, b^{3}\right )} e^{\left (-3 \, x\right )}\right )} e^{\left (4 \, x\right )}}{192 \, a^{4}} - \frac {8 \, a^{2} b e^{\left (-3 \, x\right )} + 3 \, a^{3} e^{\left (-4 \, x\right )} + 24 \, {\left (5 \, a^{2} b + 4 \, b^{3}\right )} e^{\left (-x\right )} + 24 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-2 \, x\right )}}{192 \, a^{4}} + \frac {{\left (3 \, a^{4} + 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} - \frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{5}} \]
-1/192*(8*a^2*b*e^(-x) - 3*a^3 - 24*(a^3 + a*b^2)*e^(-2*x) + 24*(5*a^2*b + 4*b^3)*e^(-3*x))*e^(4*x)/a^4 - 1/192*(8*a^2*b*e^(-3*x) + 3*a^3*e^(-4*x) + 24*(5*a^2*b + 4*b^3)*e^(-x) + 24*(a^3 + a*b^2)*e^(-2*x))/a^4 + 1/8*(3*a^4 + 12*a^2*b^2 + 8*b^4)*x/a^5 - (a^4*b + 2*a^2*b^3 + b^5)*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^5 )
Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.77 \[ \int \frac {\cosh ^4(x)}{a+b \text {csch}(x)} \, dx=\frac {3 \, a^{3} e^{\left (4 \, x\right )} - 8 \, a^{2} b e^{\left (3 \, x\right )} + 24 \, a^{3} e^{\left (2 \, x\right )} + 24 \, a b^{2} e^{\left (2 \, x\right )} - 120 \, a^{2} b e^{x} - 96 \, b^{3} e^{x}}{192 \, a^{4}} + \frac {{\left (3 \, a^{4} + 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} - \frac {{\left (8 \, a^{3} b e^{x} + 3 \, a^{4} + 24 \, {\left (5 \, a^{3} b + 4 \, a b^{3}\right )} e^{\left (3 \, x\right )} + 24 \, {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-4 \, x\right )}}{192 \, a^{5}} - \frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{5}} \]
1/192*(3*a^3*e^(4*x) - 8*a^2*b*e^(3*x) + 24*a^3*e^(2*x) + 24*a*b^2*e^(2*x) - 120*a^2*b*e^x - 96*b^3*e^x)/a^4 + 1/8*(3*a^4 + 12*a^2*b^2 + 8*b^4)*x/a^ 5 - 1/192*(8*a^3*b*e^x + 3*a^4 + 24*(5*a^3*b + 4*a*b^3)*e^(3*x) + 24*(a^4 + a^2*b^2)*e^(2*x))*e^(-4*x)/a^5 - (a^4*b + 2*a^2*b^3 + b^5)*log(abs(2*a*e ^x + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^x + 2*b + 2*sqrt(a^2 + b^2)))/(sqr t(a^2 + b^2)*a^5)
Time = 2.61 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.98 \[ \int \frac {\cosh ^4(x)}{a+b \text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}+\frac {x\,\left (3\,a^4+12\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {{\mathrm {e}}^{-2\,x}\,\left (a^2+b^2\right )}{8\,a^3}+\frac {{\mathrm {e}}^{2\,x}\,\left (a^2+b^2\right )}{8\,a^3}-\frac {{\mathrm {e}}^{-x}\,\left (5\,a^2\,b+4\,b^3\right )}{8\,a^4}-\frac {b\,{\mathrm {e}}^{-3\,x}}{24\,a^2}-\frac {b\,{\mathrm {e}}^{3\,x}}{24\,a^2}-\frac {{\mathrm {e}}^x\,\left (5\,a^2\,b+4\,b^3\right )}{8\,a^4}-\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^2}{a^6}-\frac {2\,b\,\left (a-b\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{3/2}}{a^6}\right )\,{\left (a^2+b^2\right )}^{3/2}}{a^5}+\frac {b\,\ln \left (\frac {2\,b\,\left (a-b\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{3/2}}{a^6}+\frac {2\,b\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^2}{a^6}\right )\,{\left (a^2+b^2\right )}^{3/2}}{a^5} \]
exp(4*x)/(64*a) - exp(-4*x)/(64*a) + (x*(3*a^4 + 8*b^4 + 12*a^2*b^2))/(8*a ^5) - (exp(-2*x)*(a^2 + b^2))/(8*a^3) + (exp(2*x)*(a^2 + b^2))/(8*a^3) - ( exp(-x)*(5*a^2*b + 4*b^3))/(8*a^4) - (b*exp(-3*x))/(24*a^2) - (b*exp(3*x)) /(24*a^2) - (exp(x)*(5*a^2*b + 4*b^3))/(8*a^4) - (b*log((2*b*exp(x)*(a^2 + b^2)^2)/a^6 - (2*b*(a - b*exp(x))*(a^2 + b^2)^(3/2))/a^6)*(a^2 + b^2)^(3/ 2))/a^5 + (b*log((2*b*(a - b*exp(x))*(a^2 + b^2)^(3/2))/a^6 + (2*b*exp(x)* (a^2 + b^2)^2)/a^6)*(a^2 + b^2)^(3/2))/a^5