Integrand size = 13, antiderivative size = 100 \[ \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {a x}{a^2+b^2}+\frac {b^2 x}{a \left (a^2+b^2\right )}+\frac {2 b^3 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2}}+\frac {b \text {sech}(x)}{a^2+b^2}-\frac {a \tanh (x)}{a^2+b^2} \] Output:
a*x/(a^2+b^2)+b^2*x/a/(a^2+b^2)+2*b^3*arctanh((a-b*tanh(1/2*x))/(a^2+b^2)^ (1/2))/a/(a^2+b^2)^(3/2)+b*sech(x)/(a^2+b^2)-a*tanh(x)/(a^2+b^2)
Time = 0.36 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82 \[ \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x+\frac {2 b^3 \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}}{a}+\frac {b \text {sech}(x)}{a^2+b^2}-\frac {a \tanh (x)}{a^2+b^2} \] Input:
Integrate[Tanh[x]^2/(a + b*Csch[x]),x]
Output:
(x + (2*b^3*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a^2 - b^2]])/(-a^2 - b^2)^(3/2) )/a + (b*Sech[x])/(a^2 + b^2) - (a*Tanh[x])/(a^2 + b^2)
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.692, Rules used = {3042, 25, 4386, 26, 26, 3042, 26, 3381, 25, 26, 3042, 25, 26, 3086, 24, 3214, 3042, 3139, 1083, 219, 3954, 24}
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.
| \(\displaystyle \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cot (i x)^2 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cot (i x)^2 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 4386 |
| \(\displaystyle -\int -\frac {i \sinh (x) \tanh ^2(x)}{i b+i a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int -\frac {i \sinh (x) \tanh ^2(x)}{b+a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\sinh (x) \tanh ^2(x)}{a \sinh (x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i x)^3}{\cos (i x)^2 (b-i a \sin (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i x)^3}{\cos (i x)^2 (b-i a \sin (i x))}dx\) |
| \(\Big \downarrow \) 3381 |
| \(\displaystyle i \left (-\frac {b^2 \int \frac {i \sinh (x)}{b+a \sinh (x)}dx}{a^2+b^2}+\frac {i a \int -\tanh ^2(x)dx}{a^2+b^2}+\frac {b \int i \text {sech}(x) \tanh (x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {b^2 \int \frac {i \sinh (x)}{b+a \sinh (x)}dx}{a^2+b^2}-\frac {i a \int \tanh ^2(x)dx}{a^2+b^2}+\frac {b \int i \text {sech}(x) \tanh (x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i b^2 \int \frac {\sinh (x)}{b+a \sinh (x)}dx}{a^2+b^2}-\frac {i a \int \tanh ^2(x)dx}{a^2+b^2}+\frac {i b \int \text {sech}(x) \tanh (x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i b^2 \int -\frac {i \sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}-\frac {i a \int -\tan (i x)^2dx}{a^2+b^2}+\frac {i b \int -i \sec (i x) \tan (i x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i b^2 \int -\frac {i \sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}+\frac {i b \int -i \sec (i x) \tan (i x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {b^2 \int \frac {\sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}+\frac {b \int \sec (i x) \tan (i x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle i \left (-\frac {b^2 \int \frac {\sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {i b \int 1d\text {sech}(x)}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (-\frac {b^2 \int \frac {\sin (i x)}{b-i a \sin (i x)}dx}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle i \left (\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {b^2 \left (\frac {i x}{a}-\frac {i b \int \frac {1}{b+a \sinh (x)}dx}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {b^2 \left (\frac {i x}{a}-\frac {i b \int \frac {1}{b-i a \sin (i x)}dx}{a}\right )}{a^2+b^2}+\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle i \left (\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {b^2 \left (\frac {i x}{a}-\frac {2 i b \int \frac {1}{-b \tanh ^2\left (\frac {x}{2}\right )+2 a \tanh \left (\frac {x}{2}\right )+b}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle i \left (\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {b^2 \left (\frac {4 i b \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 a-2 b \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 a-2 b \tanh \left (\frac {x}{2}\right )\right )}{a}+\frac {i x}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle i \left (\frac {i a \int \tan (i x)^2dx}{a^2+b^2}-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle i \left (\frac {i a (\tanh (x)-\int 1dx)}{a^2+b^2}-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (-\frac {b^2 \left (\frac {2 i b \text {arctanh}\left (\frac {2 a-2 b \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i x}{a}\right )}{a^2+b^2}+\frac {i a (\tanh (x)-x)}{a^2+b^2}-\frac {i b \text {sech}(x)}{a^2+b^2}\right )\) |
Input:
Int[Tanh[x]^2/(a + b*Csch[x]),x]
Output:
I*(-((b^2*((I*x)/a + ((2*I)*b*ArcTanh[(2*a - 2*b*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2])))/(a^2 + b^2)) - (I*b*Sech[x])/(a^2 + b^2) + ( I*a*(-x + Tanh[x]))/(a^2 + b^2))
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 = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {-2 a \tanh \left (\frac {x}{2}\right )+2 b}{\left (a^{2}+b^{2}\right ) \left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {2 b^{3} \operatorname {arctanh}\left (\frac {-2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(95\) |
| risch | \(\frac {x}{a}+\frac {2 b \,{\mathrm e}^{x}+2 a}{\left ({\mathrm e}^{2 x}+1\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b +a^{4}+2 a^{2} b^{2}+b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a}-\frac {b^{3} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b -a^{4}-2 a^{2} b^{2}-b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a}\) | \(155\) |
Input:
int(tanh(x)^2/(a+b*csch(x)),x,method=_RETURNVERBOSE)
Output:
2/(a^2+b^2)*(-a*tanh(1/2*x)+b)/(tanh(1/2*x)^2+1)+1/a*ln(tanh(1/2*x)+1)-1/a *ln(tanh(1/2*x)-1)+2/a*b^3/(a^2+b^2)^(3/2)*arctanh(1/2*(-2*tanh(1/2*x)*b+2 *a)/(a^2+b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (96) = 192\).
Time = 0.10 (sec) , antiderivative size = 349, normalized size of antiderivative = 3.49 \[ \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \, a^{4} + 2 \, a^{2} b^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} x \cosh \left (x\right )^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} x \sinh \left (x\right )^{2} + {\left (b^{3} \cosh \left (x\right )^{2} + 2 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right ) + b^{3} \sinh \left (x\right )^{2} + b^{3}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} x + 2 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} b + a b^{3} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sinh \left (x\right )^{2}} \] Input:
integrate(tanh(x)^2/(a+b*csch(x)),x, algorithm="fricas")
Output:
(2*a^4 + 2*a^2*b^2 + (a^4 + 2*a^2*b^2 + b^4)*x*cosh(x)^2 + (a^4 + 2*a^2*b^ 2 + b^4)*x*sinh(x)^2 + (b^3*cosh(x)^2 + 2*b^3*cosh(x)*sinh(x) + b^3*sinh(x )^2 + b^3)*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*c osh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*c osh(x) + b)*sinh(x) - a)) + (a^4 + 2*a^2*b^2 + b^4)*x + 2*(a^3*b + a*b^3)* cosh(x) + 2*(a^3*b + a*b^3 + (a^4 + 2*a^2*b^2 + b^4)*x*cosh(x))*sinh(x))/( a^5 + 2*a^3*b^2 + a*b^4 + (a^5 + 2*a^3*b^2 + a*b^4)*cosh(x)^2 + 2*(a^5 + 2 *a^3*b^2 + a*b^4)*cosh(x)*sinh(x) + (a^5 + 2*a^3*b^2 + a*b^4)*sinh(x)^2)
\[ \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \] Input:
integrate(tanh(x)**2/(a+b*csch(x)),x)
Output:
Integral(tanh(x)**2/(a + b*csch(x)), x)
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08 \[ \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^{3} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (b e^{\left (-x\right )} - a\right )}}{a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{a} \] Input:
integrate(tanh(x)^2/(a+b*csch(x)),x, algorithm="maxima")
Output:
-b^3*log((a*e^(-x) - b - sqrt(a^2 + b^2))/(a*e^(-x) - b + sqrt(a^2 + b^2)) )/((a^3 + a*b^2)*sqrt(a^2 + b^2)) + 2*(b*e^(-x) - a)/(a^2 + b^2 + (a^2 + b ^2)*e^(-2*x)) + x/a
Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^{3} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {x}{a} + \frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}} \] Input:
integrate(tanh(x)^2/(a+b*csch(x)),x, algorithm="giac")
Output:
-b^3*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^3 + a*b^2)*sqrt(a^2 + b^2)) + x/a + 2*(b*e^x + a)/((a^2 + b^2)*(e^(2*x) + 1))
Time = 3.70 (sec) , antiderivative size = 376, normalized size of antiderivative = 3.76 \[ \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a}+\frac {\frac {2\,a}{a^2+b^2}+\frac {2\,b\,{\mathrm {e}}^x}{a^2+b^2}}{{\mathrm {e}}^{2\,x}+1}+\frac {2\,\mathrm {atan}\left (\left (\frac {a^4\,\sqrt {-a^8-3\,a^6\,b^2-3\,a^4\,b^4-a^2\,b^6}}{2}+\frac {a^2\,b^2\,\sqrt {-a^8-3\,a^6\,b^2-3\,a^4\,b^4-a^2\,b^6}}{2}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,b^3}{a^3\,\left (a^3+a\,b^2\right )\,\sqrt {b^6}\,\left (a^2+b^2\right )}+\frac {2\,\left (a\,b^3\,\sqrt {b^6}+a^3\,b\,\sqrt {b^6}\right )}{a^2\,b^2\,\sqrt {-a^2\,{\left (a^2+b^2\right )}^3}\,\left (a^3+a\,b^2\right )\,\sqrt {-a^8-3\,a^6\,b^2-3\,a^4\,b^4-a^2\,b^6}}\right )-\frac {2\,\left (a^4\,\sqrt {b^6}+a^2\,b^2\,\sqrt {b^6}\right )}{a^2\,b^2\,\sqrt {-a^2\,{\left (a^2+b^2\right )}^3}\,\left (a^3+a\,b^2\right )\,\sqrt {-a^8-3\,a^6\,b^2-3\,a^4\,b^4-a^2\,b^6}}\right )\right )\,\sqrt {b^6}}{\sqrt {-a^8-3\,a^6\,b^2-3\,a^4\,b^4-a^2\,b^6}} \] Input:
int(tanh(x)^2/(a + b/sinh(x)),x)
Output:
x/a + ((2*a)/(a^2 + b^2) + (2*b*exp(x))/(a^2 + b^2))/(exp(2*x) + 1) + (2*a tan(((a^4*(- a^8 - a^2*b^6 - 3*a^4*b^4 - 3*a^6*b^2)^(1/2))/2 + (a^2*b^2*(- a^8 - a^2*b^6 - 3*a^4*b^4 - 3*a^6*b^2)^(1/2))/2)*(exp(x)*((2*b^3)/(a^3*(a *b^2 + a^3)*(b^6)^(1/2)*(a^2 + b^2)) + (2*(a*b^3*(b^6)^(1/2) + a^3*b*(b^6) ^(1/2)))/(a^2*b^2*(-a^2*(a^2 + b^2)^3)^(1/2)*(a*b^2 + a^3)*(- a^8 - a^2*b^ 6 - 3*a^4*b^4 - 3*a^6*b^2)^(1/2))) - (2*(a^4*(b^6)^(1/2) + a^2*b^2*(b^6)^( 1/2)))/(a^2*b^2*(-a^2*(a^2 + b^2)^3)^(1/2)*(a*b^2 + a^3)*(- a^8 - a^2*b^6 - 3*a^4*b^4 - 3*a^6*b^2)^(1/2))))*(b^6)^(1/2))/(- a^8 - a^2*b^6 - 3*a^4*b^ 4 - 3*a^6*b^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.24 \[ \int \frac {\tanh ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {-2 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) b^{3} i -2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) b^{3} i +e^{2 x} a^{4} x -2 e^{2 x} a^{4}+2 e^{2 x} a^{2} b^{2} x -2 e^{2 x} a^{2} b^{2}+e^{2 x} b^{4} x +2 e^{x} a^{3} b +2 e^{x} a \,b^{3}+a^{4} x +2 a^{2} b^{2} x +b^{4} x}{a \left (e^{2 x} a^{4}+2 e^{2 x} a^{2} b^{2}+e^{2 x} b^{4}+a^{4}+2 a^{2} b^{2}+b^{4}\right )} \] Input:
int(tanh(x)^2/(a+b*csch(x)),x)
Output:
( - 2*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))* b**3*i - 2*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))*b**3 *i + e**(2*x)*a**4*x - 2*e**(2*x)*a**4 + 2*e**(2*x)*a**2*b**2*x - 2*e**(2* x)*a**2*b**2 + e**(2*x)*b**4*x + 2*e**x*a**3*b + 2*e**x*a*b**3 + a**4*x + 2*a**2*b**2*x + b**4*x)/(a*(e**(2*x)*a**4 + 2*e**(2*x)*a**2*b**2 + e**(2*x )*b**4 + a**4 + 2*a**2*b**2 + b**4))