Integrand size = 13, antiderivative size = 183 \[ \int \frac {\tanh ^4(x)}{a+b \text {csch}(x)} \, dx=\frac {a b^2 x}{\left (a^2+b^2\right )^2}+\frac {b^4 x}{a \left (a^2+b^2\right )^2}+\frac {a x}{a^2+b^2}+\frac {2 b^5 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{5/2}}+\frac {b^3 \text {sech}(x)}{\left (a^2+b^2\right )^2}+\frac {b \text {sech}(x)}{a^2+b^2}-\frac {b \text {sech}^3(x)}{3 \left (a^2+b^2\right )}-\frac {a b^2 \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {a \tanh (x)}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )} \]
a*b^2*x/(a^2+b^2)^2+b^4*x/a/(a^2+b^2)^2+a*x/(a^2+b^2)+2*b^5*arctanh((a-b*t anh(1/2*x))/(a^2+b^2)^(1/2))/a/(a^2+b^2)^(5/2)+b^3*sech(x)/(a^2+b^2)^2+b*s ech(x)/(a^2+b^2)-1/3*b*sech(x)^3/(a^2+b^2)-a*b^2*tanh(x)/(a^2+b^2)^2-a*tan h(x)/(a^2+b^2)-1/3*a*tanh(x)^3/(a^2+b^2)
Time = 0.62 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.77 \[ \int \frac {\tanh ^4(x)}{a+b \text {csch}(x)} \, dx=\frac {1}{3} \left (\frac {3 \left (x-\frac {2 b^5 \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{5/2}}\right )}{a}+\frac {3 b \left (a^2+2 b^2\right ) \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {b \text {sech}^3(x)}{a^2+b^2}-\frac {a \left (4 a^2+7 b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}+\frac {a \text {sech}^2(x) \tanh (x)}{a^2+b^2}\right ) \]
((3*(x - (2*b^5*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a^2 - b^2]])/(-a^2 - b^2)^( 5/2)))/a + (3*b*(a^2 + 2*b^2)*Sech[x])/(a^2 + b^2)^2 - (b*Sech[x]^3)/(a^2 + b^2) - (a*(4*a^2 + 7*b^2)*Tanh[x])/(a^2 + b^2)^2 + (a*Sech[x]^2*Tanh[x]) /(a^2 + b^2))/3
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.
| \(\displaystyle \int \frac {\tanh ^4(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot (i x)^4 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 4386 |
| \(\displaystyle \int \frac {i \sinh (x) \tanh ^4(x)}{i a \sinh (x)+i b}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int -\frac {i \sinh (x) \tanh ^4(x)}{b+a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\sinh (x) \tanh ^4(x)}{a \sinh (x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i x)^5}{\cos (i x)^4 (b-i a \sin (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i x)^5}{\cos (i x)^4 (b-i a \sin (i x))}dx\) |
| \(\Big \downarrow \) 3381 |
| \(\displaystyle -i \left (\frac {i a \int \tanh ^4(x)dx}{a^2+b^2}-\frac {b^2 \int -\frac {i \sinh (x) \tanh ^2(x)}{b+a \sinh (x)}dx}{a^2+b^2}+\frac {b \int -i \text {sech}(x) \tanh ^3(x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i a \int \tanh ^4(x)dx}{a^2+b^2}+\frac {i b^2 \int \frac {\sinh (x) \tanh ^2(x)}{b+a \sinh (x)}dx}{a^2+b^2}-\frac {i b \int \text {sech}(x) \tanh ^3(x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}+\frac {i b^2 \int \frac {i \sin (i x)^3}{\cos (i x)^2 (b-i a \sin (i x))}dx}{a^2+b^2}-\frac {i b \int i \sec (i x) \tan (i x)^3dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \int \frac {\sin (i x)^3}{\cos (i x)^2 (b-i a \sin (i x))}dx}{a^2+b^2}+\frac {b \int \sec (i x) \tan (i x)^3dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {i b \int \left (\text {sech}^2(x)-1\right )d\text {sech}(x)}{a^2+b^2}-\frac {b^2 \int \frac {\sin (i x)^3}{\cos (i x)^2 (b-i a \sin (i x))}dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \int \frac {\sin (i x)^3}{\cos (i x)^2 (b-i a \sin (i x))}dx}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3381 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {i \sinh (x)}{b+a \sinh (x)}dx}{a^2+b^2}+\frac {i a \int -\tanh ^2(x)dx}{a^2+b^2}+\frac {b \int i \text {sech}(x) \tanh (x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {i \sinh (x)}{b+a \sinh (x)}dx}{a^2+b^2}-\frac {i a \int \tanh ^2(x)dx}{a^2+b^2}+\frac {b \int i \text {sech}(x) \tanh (x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {i b^2 \int \frac {\sinh (x)}{b+a \sinh (x)}dx}{a^2+b^2}-\frac {i a \int \tanh ^2(x)dx}{a^2+b^2}+\frac {i b \int \text {sech}(x) \tanh (x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {i b^2 \int -\frac {i \sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}-\frac {i a \int -\tan (i x)^2dx}{a^2+b^2}+\frac {i b \int -i \sec (i x) \tan (i x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {i b^2 \int -\frac {i \sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}+\frac {i b \int -i \sec (i x) \tan (i x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {\sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}+\frac {b \int \sec (i x) \tan (i x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {\sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {i b \int 1d\text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {\sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {b^2 \left (\frac {i x}{a}-\frac {i b \int \frac {1}{b+a \sinh (x)}dx}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {i x}{a}-\frac {i b \int \frac {1}{b-i a \sin (i x)}dx}{a}\right )}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {b^2 \left (\frac {i x}{a}-\frac {2 i b \int \frac {1}{-b \tanh ^2\left (\frac {x}{2}\right )+2 a \tanh \left (\frac {x}{2}\right )+b}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -i \left (\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {b^2 \left (\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {b^2 \left (\frac {4 i b \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 a-2 b \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 a-2 b \tanh \left (\frac {x}{2}\right )\right )}{a}+\frac {i x}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -i \left (-\frac {b^2 \left (\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {i a \int \tan (i x)^4dx}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle -i \left (-\frac {b^2 \left (\frac {i a (\tanh (x)-\int 1dx)}{a^2+b^2}-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {i a \left (-\int -\tanh ^2(x)dx-\frac {1}{3} \tanh ^3(x)\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -i \left (\frac {i a \left (-\int -\tanh ^2(x)dx-\frac {1}{3} \tanh ^3(x)\right )}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}+\frac {i a (\tanh (x)-x)}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i a \left (\int \tanh ^2(x)dx-\frac {\tanh ^3(x)}{3}\right )}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}+\frac {i a (\tanh (x)-x)}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i a \left (-\frac {\tanh ^3(x)}{3}+\int -\tan (i x)^2dx\right )}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}+\frac {i a (\tanh (x)-x)}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i a \left (-\frac {1}{3} \tanh ^3(x)-\int \tan (i x)^2dx\right )}{a^2+b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}+\frac {i a (\tanh (x)-x)}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {i b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\right )\) |
3.2.14.3.1 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_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
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_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a*(d^2/(a^2 - b^2)) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x], x] + (-Simp[ b*(d/(a^2 - b^2)) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a^2*(d^2/(g^2*(a^2 - b^2))) Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[ e + f*x])^(n - 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d, e, f, g }, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1 ]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
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])
Time = 0.47 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {2 \left (-a^{3}-2 a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{5}+2 b^{3} \tanh \left (\frac {x}{2}\right )^{4}+2 \left (-\frac {10}{3} a^{3}-\frac {16}{3} a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}+2 \left (2 a^{2} b +4 b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{2}+2 \left (-a^{3}-2 a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )+\frac {4 a^{2} b}{3}+\frac {10 b^{3}}{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{3}}+\frac {2 b^{5} \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}\) | \(207\) |
| risch | \(\frac {x}{a}+\frac {2 a^{2} b \,{\mathrm e}^{5 x}+4 b^{3} {\mathrm e}^{5 x}+4 a^{3} {\mathrm e}^{4 x}+6 a \,b^{2} {\mathrm e}^{4 x}+\frac {4 a^{2} b \,{\mathrm e}^{3 x}}{3}+\frac {16 b^{3} {\mathrm e}^{3 x}}{3}+4 a^{3} {\mathrm e}^{2 x}+8 a \,b^{2} {\mathrm e}^{2 x}+2 a^{2} b \,{\mathrm e}^{x}+4 b^{3} {\mathrm e}^{x}+\frac {8 a^{3}}{3}+\frac {14 a \,b^{2}}{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+{\mathrm e}^{2 x}\right )^{3}}+\frac {b^{5} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} b +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{a \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a}-\frac {b^{5} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} b -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{a \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a}\) | \(276\) |
-1/a*ln(tanh(1/2*x)-1)+1/a*ln(tanh(1/2*x)+1)+2/(a^4+2*a^2*b^2+b^4)*((-a^3- 2*a*b^2)*tanh(1/2*x)^5+b^3*tanh(1/2*x)^4+(-10/3*a^3-16/3*a*b^2)*tanh(1/2*x )^3+(2*a^2*b+4*b^3)*tanh(1/2*x)^2+(-a^3-2*a*b^2)*tanh(1/2*x)+2/3*a^2*b+5/3 *b^3)/(1+tanh(1/2*x)^2)^3+2/a*b^5/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)^(1/2)*arct anh(1/2*(-2*b*tanh(1/2*x)+2*a)/(a^2+b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1746 vs. \(2 (175) = 350\).
Time = 0.30 (sec) , antiderivative size = 1746, normalized size of antiderivative = 9.54 \[ \int \frac {\tanh ^4(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]
1/3*(3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^6 + 3*(a^6 + 3*a^4*b^ 2 + 3*a^2*b^4 + b^6)*x*sinh(x)^6 + 8*a^6 + 22*a^4*b^2 + 14*a^2*b^4 + 6*(a^ 5*b + 3*a^3*b^3 + 2*a*b^5)*cosh(x)^5 + 6*(a^5*b + 3*a^3*b^3 + 2*a*b^5 + 3* (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x))*sinh(x)^5 + 3*(4*a^6 + 10*a ^4*b^2 + 6*a^2*b^4 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x)^4 + 3*(4*a^6 + 10*a^4*b^2 + 6*a^2*b^4 + 15*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) *x*cosh(x)^2 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x + 10*(a^5*b + 3*a^3 *b^3 + 2*a*b^5)*cosh(x))*sinh(x)^4 + 4*(a^5*b + 5*a^3*b^3 + 4*a*b^5)*cosh( x)^3 + 4*(a^5*b + 5*a^3*b^3 + 4*a*b^5 + 15*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^3 + 15*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*cosh(x)^2 + 3*(4*a^6 + 10*a^4*b^2 + 6*a^2*b^4 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x) )*sinh(x)^3 + 3*(4*a^6 + 12*a^4*b^2 + 8*a^2*b^4 + 3*(a^6 + 3*a^4*b^2 + 3*a ^2*b^4 + b^6)*x)*cosh(x)^2 + 3*(4*a^6 + 12*a^4*b^2 + 8*a^2*b^4 + 15*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^4 + 20*(a^5*b + 3*a^3*b^3 + 2*a*b^ 5)*cosh(x)^3 + 6*(4*a^6 + 10*a^4*b^2 + 6*a^2*b^4 + 3*(a^6 + 3*a^4*b^2 + 3* a^2*b^4 + b^6)*x)*cosh(x)^2 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x + 4* (a^5*b + 5*a^3*b^3 + 4*a*b^5)*cosh(x))*sinh(x)^2 + 3*(b^5*cosh(x)^6 + 6*b^ 5*cosh(x)*sinh(x)^5 + b^5*sinh(x)^6 + 3*b^5*cosh(x)^4 + 3*b^5*cosh(x)^2 + b^5 + 3*(5*b^5*cosh(x)^2 + b^5)*sinh(x)^4 + 4*(5*b^5*cosh(x)^3 + 3*b^5*cos h(x))*sinh(x)^3 + 3*(5*b^5*cosh(x)^4 + 6*b^5*cosh(x)^2 + b^5)*sinh(x)^2...
\[ \int \frac {\tanh ^4(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\tanh ^{4}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.43 \[ \int \frac {\tanh ^4(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^{5} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (4 \, a^{3} + 7 \, a b^{2} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} e^{\left (-x\right )} + 6 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (a^{2} b + 4 \, b^{3}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-4 \, x\right )} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac {x}{a} \]
-b^5*log((a*e^(-x) - b - sqrt(a^2 + b^2))/(a*e^(-x) - b + sqrt(a^2 + b^2)) )/((a^5 + 2*a^3*b^2 + a*b^4)*sqrt(a^2 + b^2)) - 2/3*(4*a^3 + 7*a*b^2 - 3*( a^2*b + 2*b^3)*e^(-x) + 6*(a^3 + 2*a*b^2)*e^(-2*x) - 2*(a^2*b + 4*b^3)*e^( -3*x) + 3*(2*a^3 + 3*a*b^2)*e^(-4*x) - 3*(a^2*b + 2*b^3)*e^(-5*x))/(a^4 + 2*a^2*b^2 + b^4 + 3*(a^4 + 2*a^2*b^2 + b^4)*e^(-2*x) + 3*(a^4 + 2*a^2*b^2 + b^4)*e^(-4*x) + (a^4 + 2*a^2*b^2 + b^4)*e^(-6*x)) + x/a
Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.17 \[ \int \frac {\tanh ^4(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^{5} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {x}{a} + \frac {2 \, {\left (3 \, a^{2} b e^{\left (5 \, x\right )} + 6 \, b^{3} e^{\left (5 \, x\right )} + 6 \, a^{3} e^{\left (4 \, x\right )} + 9 \, a b^{2} e^{\left (4 \, x\right )} + 2 \, a^{2} b e^{\left (3 \, x\right )} + 8 \, b^{3} e^{\left (3 \, x\right )} + 6 \, a^{3} e^{\left (2 \, x\right )} + 12 \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} + 6 \, b^{3} e^{x} + 4 \, a^{3} + 7 \, a b^{2}\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]
-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)))/((a^5 + 2*a^3*b^2 + a*b^4)*sqrt(a^2 + b^2)) + x/a + 2/3*(3*a ^2*b*e^(5*x) + 6*b^3*e^(5*x) + 6*a^3*e^(4*x) + 9*a*b^2*e^(4*x) + 2*a^2*b*e ^(3*x) + 8*b^3*e^(3*x) + 6*a^3*e^(2*x) + 12*a*b^2*e^(2*x) + 3*a^2*b*e^x + 6*b^3*e^x + 4*a^3 + 7*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*(e^(2*x) + 1)^3)
Time = 3.70 (sec) , antiderivative size = 707, normalized size of antiderivative = 3.86 \[ \int \frac {\tanh ^4(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a}+\frac {\frac {8\,a}{3\,\left (a^2+b^2\right )}+\frac {8\,b\,{\mathrm {e}}^x}{3\,\left (a^2+b^2\right )}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^x\,\left (a^2\,b+b^3\right )}{3\,{\left (a^2+b^2\right )}^2}+\frac {4\,\left (a^4+a^2\,b^2\right )}{a\,{\left (a^2+b^2\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\frac {2\,{\mathrm {e}}^x\,\left (a^2\,b+2\,b^3\right )}{{\left (a^2+b^2\right )}^2}+\frac {2\,\left (2\,a^4+3\,a^2\,b^2\right )}{a\,{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,x}+1}+\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,b^5}{a^3\,\sqrt {b^{10}}\,{\left (a^2+b^2\right )}^2\,\left (a^5+2\,a^3\,b^2+a\,b^4\right )}+\frac {2\,\left (2\,a^3\,b^3\,\sqrt {b^{10}}+a\,b^5\,\sqrt {b^{10}}+a^5\,b\,\sqrt {b^{10}}\right )}{a^2\,b^4\,\sqrt {-a^2\,{\left (a^2+b^2\right )}^5}\,\left (a^5+2\,a^3\,b^2+a\,b^4\right )\,\sqrt {-a^{12}-5\,a^{10}\,b^2-10\,a^8\,b^4-10\,a^6\,b^6-5\,a^4\,b^8-a^2\,b^{10}}}\right )-\frac {2\,\left (a^6\,\sqrt {b^{10}}+a^2\,b^4\,\sqrt {b^{10}}+2\,a^4\,b^2\,\sqrt {b^{10}}\right )}{a^2\,b^4\,\sqrt {-a^2\,{\left (a^2+b^2\right )}^5}\,\left (a^5+2\,a^3\,b^2+a\,b^4\right )\,\sqrt {-a^{12}-5\,a^{10}\,b^2-10\,a^8\,b^4-10\,a^6\,b^6-5\,a^4\,b^8-a^2\,b^{10}}}\right )\,\left (\frac {a^6\,\sqrt {-a^{12}-5\,a^{10}\,b^2-10\,a^8\,b^4-10\,a^6\,b^6-5\,a^4\,b^8-a^2\,b^{10}}}{2}+\frac {a^2\,b^4\,\sqrt {-a^{12}-5\,a^{10}\,b^2-10\,a^8\,b^4-10\,a^6\,b^6-5\,a^4\,b^8-a^2\,b^{10}}}{2}+a^4\,b^2\,\sqrt {-a^{12}-5\,a^{10}\,b^2-10\,a^8\,b^4-10\,a^6\,b^6-5\,a^4\,b^8-a^2\,b^{10}}\right )\right )\,\sqrt {b^{10}}}{\sqrt {-a^{12}-5\,a^{10}\,b^2-10\,a^8\,b^4-10\,a^6\,b^6-5\,a^4\,b^8-a^2\,b^{10}}} \]
x/a + ((8*a)/(3*(a^2 + b^2)) + (8*b*exp(x))/(3*(a^2 + b^2)))/(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - ((8*exp(x)*(a^2*b + b^3))/(3*(a^2 + b^2)^2) + (4*(a^4 + a^2*b^2))/(a*(a^2 + b^2)^2))/(2*exp(2*x) + exp(4*x) + 1) + ((2 *exp(x)*(a^2*b + 2*b^3))/(a^2 + b^2)^2 + (2*(2*a^4 + 3*a^2*b^2))/(a*(a^2 + b^2)^2))/(exp(2*x) + 1) + (2*atan((exp(x)*((2*b^5)/(a^3*(b^10)^(1/2)*(a^2 + b^2)^2*(a*b^4 + a^5 + 2*a^3*b^2)) + (2*(2*a^3*b^3*(b^10)^(1/2) + a*b^5* (b^10)^(1/2) + a^5*b*(b^10)^(1/2)))/(a^2*b^4*(-a^2*(a^2 + b^2)^5)^(1/2)*(a *b^4 + a^5 + 2*a^3*b^2)*(- a^12 - a^2*b^10 - 5*a^4*b^8 - 10*a^6*b^6 - 10*a ^8*b^4 - 5*a^10*b^2)^(1/2))) - (2*(a^6*(b^10)^(1/2) + a^2*b^4*(b^10)^(1/2) + 2*a^4*b^2*(b^10)^(1/2)))/(a^2*b^4*(-a^2*(a^2 + b^2)^5)^(1/2)*(a*b^4 + a ^5 + 2*a^3*b^2)*(- a^12 - a^2*b^10 - 5*a^4*b^8 - 10*a^6*b^6 - 10*a^8*b^4 - 5*a^10*b^2)^(1/2)))*((a^6*(- a^12 - a^2*b^10 - 5*a^4*b^8 - 10*a^6*b^6 - 1 0*a^8*b^4 - 5*a^10*b^2)^(1/2))/2 + (a^2*b^4*(- a^12 - a^2*b^10 - 5*a^4*b^8 - 10*a^6*b^6 - 10*a^8*b^4 - 5*a^10*b^2)^(1/2))/2 + a^4*b^2*(- a^12 - a^2* b^10 - 5*a^4*b^8 - 10*a^6*b^6 - 10*a^8*b^4 - 5*a^10*b^2)^(1/2)))*(b^10)^(1 /2))/(- a^12 - a^2*b^10 - 5*a^4*b^8 - 10*a^6*b^6 - 10*a^8*b^4 - 5*a^10*b^2 )^(1/2)