Integrand size = 13, antiderivative size = 124 \[ \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx=-\frac {2 a^4 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {a^2 b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {b \text {sech}(x)}{a^2+b^2}+\frac {b \text {sech}^3(x)}{3 \left (a^2+b^2\right )}-\frac {a^3 \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )} \] Output:
-2*a^4*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(5/2)-a^2*b*se ch(x)/(a^2+b^2)^2-b*sech(x)/(a^2+b^2)+b*sech(x)^3/(3*a^2+3*b^2)-a^3*tanh(x )/(a^2+b^2)^2-a*tanh(x)^3/(3*a^2+3*b^2)
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.87 \[ \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx=\frac {\frac {6 a^4 \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-3 b \left (2 a^2+b^2\right ) \text {sech}(x)+\left (a^2+b^2\right ) \text {sech}^3(x) (b+a \sinh (x))-a \left (4 a^2+b^2\right ) \tanh (x)}{3 \left (a^2+b^2\right )^2} \] Input:
Integrate[Tanh[x]^4/(a + b*Sinh[x]),x]
Output:
((6*a^4*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 3*b *(2*a^2 + b^2)*Sech[x] + (a^2 + b^2)*Sech[x]^3*(b + a*Sinh[x]) - a*(4*a^2 + b^2)*Tanh[x])/(3*(a^2 + b^2)^2)
Time = 0.83 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.692, Rules used = {3042, 3206, 25, 26, 3042, 25, 26, 3086, 2009, 3087, 15, 3206, 26, 3042, 26, 3086, 24, 3139, 1083, 219, 4254, 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.
| \(\displaystyle \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (i x)^4}{a-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 3206 |
| \(\displaystyle -\frac {a^2 \int -\frac {\tanh ^2(x)}{a+b \sinh (x)}dx}{a^2+b^2}+\frac {a \int -\text {sech}^2(x) \tanh ^2(x)dx}{a^2+b^2}+\frac {i b \int -i \text {sech}(x) \tanh ^3(x)dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a^2 \int \frac {\tanh ^2(x)}{a+b \sinh (x)}dx}{a^2+b^2}-\frac {a \int \text {sech}^2(x) \tanh ^2(x)dx}{a^2+b^2}+\frac {i b \int -i \text {sech}(x) \tanh ^3(x)dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {a^2 \int \frac {\tanh ^2(x)}{a+b \sinh (x)}dx}{a^2+b^2}-\frac {a \int \text {sech}^2(x) \tanh ^2(x)dx}{a^2+b^2}+\frac {b \int \text {sech}(x) \tanh ^3(x)dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \int -\frac {\tan (i x)^2}{a-i b \sin (i x)}dx}{a^2+b^2}-\frac {a \int -\sec (i x)^2 \tan (i x)^2dx}{a^2+b^2}+\frac {b \int i \sec (i x) \tan (i x)^3dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \int \frac {\tan (i x)^2}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {a \int \sec (i x)^2 \tan (i x)^2dx}{a^2+b^2}+\frac {b \int i \sec (i x) \tan (i x)^3dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a^2 \int \frac {\tan (i x)^2}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {a \int \sec (i x)^2 \tan (i x)^2dx}{a^2+b^2}+\frac {i b \int \sec (i x) \tan (i x)^3dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle \frac {b \int \left (\text {sech}^2(x)-1\right )d\text {sech}(x)}{a^2+b^2}-\frac {a^2 \int \frac {\tan (i x)^2}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {a \int \sec (i x)^2 \tan (i x)^2dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \int \frac {\tan (i x)^2}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {a \int \sec (i x)^2 \tan (i x)^2dx}{a^2+b^2}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle -\frac {i a \int -\tanh ^2(x)d(i \tanh (x))}{a^2+b^2}-\frac {a^2 \int \frac {\tan (i x)^2}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a^2 \int \frac {\tan (i x)^2}{a-i b \sin (i x)}dx}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3206 |
| \(\displaystyle -\frac {a^2 \left (-\frac {a^2 \int \frac {1}{a+b \sinh (x)}dx}{a^2+b^2}+\frac {a \int \text {sech}^2(x)dx}{a^2+b^2}+\frac {i b \int i \text {sech}(x) \tanh (x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a^2 \left (-\frac {a^2 \int \frac {1}{a+b \sinh (x)}dx}{a^2+b^2}+\frac {a \int \text {sech}^2(x)dx}{a^2+b^2}-\frac {b \int \text {sech}(x) \tanh (x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a^2 \left (-\frac {a^2 \int \frac {1}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {a \int \csc \left (i x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}-\frac {b \int -i \sec (i x) \tan (i x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a^2 \left (-\frac {a^2 \int \frac {1}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {a \int \csc \left (i x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {i b \int \sec (i x) \tan (i x)dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -\frac {a^2 \left (-\frac {a^2 \int \frac {1}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {a \int \csc \left (i x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {b \int 1d\text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a^2 \left (-\frac {a^2 \int \frac {1}{a-i b \sin (i x)}dx}{a^2+b^2}+\frac {a \int \csc \left (i x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {a^2 \left (\frac {a \int \csc \left (i x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}-\frac {2 a^2 \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a^2+b^2}+\frac {b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {a^2 \left (\frac {a \int \csc \left (i x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {4 a^2 \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2}+\frac {b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^2 \left (\frac {a \int \csc \left (i x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {2 a^2 \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {a^2 \left (\frac {i a \int 1d(-i \tanh (x))}{a^2+b^2}+\frac {2 a^2 \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a^2 \left (\frac {2 a^2 \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \tanh (x)}{a^2+b^2}+\frac {b \text {sech}(x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \tanh ^3(x)}{3 \left (a^2+b^2\right )}+\frac {b \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a^2+b^2}\) |
Input:
Int[Tanh[x]^4/(a + b*Sinh[x]),x]
Output:
(b*(-Sech[x] + Sech[x]^3/3))/(a^2 + b^2) - (a*Tanh[x]^3)/(3*(a^2 + b^2)) - (a^2*((2*a^2*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/(a^2 + b ^2)^(3/2) + (b*Sech[x])/(a^2 + b^2) + (a*Tanh[x])/(a^2 + b^2)))/(a^2 + b^2 )
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 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[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[a/(a^2 - b^2) Int[(g*Tan[e + f*x])^p/Sin[e + f*x] ^2, x], x] + (-Simp[b*(g/(a^2 - b^2)) Int[(g*Tan[e + f*x])^(p - 1)/Cos[e + f*x], x], x] - Simp[a^2*(g^2/(a^2 - b^2)) Int[(g*Tan[e + f*x])^(p - 2)/ (a + b*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2 , 0] && IntegersQ[2*p] && GtQ[p, 1]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 1.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(\frac {32 a^{4} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (16 a^{4}+32 a^{2} b^{2}+16 b^{4}\right ) \sqrt {a^{2}+b^{2}}}+\frac {-2 a^{3} \tanh \left (\frac {x}{2}\right )^{5}-2 a^{2} b \tanh \left (\frac {x}{2}\right )^{4}+2 \left (-\frac {10}{3} a^{3}-\frac {4}{3} a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}+2 \left (-4 a^{2} b -2 b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{2}-2 a^{3} \tanh \left (\frac {x}{2}\right )-\frac {10 a^{2} b}{3}-\frac {4 b^{3}}{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{3}}\) | \(171\) |
| risch | \(\frac {-4 a^{2} b \,{\mathrm e}^{5 x}-2 b^{3} {\mathrm e}^{5 x}+4 \,{\mathrm e}^{4 x} a^{3}+2 \,{\mathrm e}^{4 x} a \,b^{2}-\frac {16 a^{2} b \,{\mathrm e}^{3 x}}{3}-\frac {4 \,{\mathrm e}^{3 x} b^{3}}{3}+4 a^{3} {\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} a^{2} b -2 b^{3} {\mathrm e}^{x}+\frac {8 a^{3}}{3}+\frac {2 a \,b^{2}}{3}}{\left (a^{2}+b^{2}\right )^{2} \left ({\mathrm e}^{2 x}+1\right )^{3}}+\frac {a^{4} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {a^{4} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\) | \(246\) |
Input:
int(tanh(x)^4/(a+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
32*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))+2/(a^4+2*a^2*b^2+b^4)*(-a^3*tanh(1/2*x)^5-a^2*b *tanh(1/2*x)^4+(-10/3*a^3-4/3*a*b^2)*tanh(1/2*x)^3+(-4*a^2*b-2*b^3)*tanh(1 /2*x)^2-a^3*tanh(1/2*x)-5/3*a^2*b-2/3*b^3)/(1+tanh(1/2*x)^2)^3
Leaf count of result is larger than twice the leaf count of optimal. 1199 vs. \(2 (116) = 232\).
Time = 0.11 (sec) , antiderivative size = 1199, normalized size of antiderivative = 9.67 \[ \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)^4/(a+b*sinh(x)),x, algorithm="fricas")
Output:
-1/3*(6*(2*a^4*b + 3*a^2*b^3 + b^5)*cosh(x)^5 + 6*(2*a^4*b + 3*a^2*b^3 + b ^5)*sinh(x)^5 - 8*a^5 - 10*a^3*b^2 - 2*a*b^4 - 6*(2*a^5 + 3*a^3*b^2 + a*b^ 4)*cosh(x)^4 - 6*(2*a^5 + 3*a^3*b^2 + a*b^4 - 5*(2*a^4*b + 3*a^2*b^3 + b^5 )*cosh(x))*sinh(x)^4 + 4*(4*a^4*b + 5*a^2*b^3 + b^5)*cosh(x)^3 + 4*(4*a^4* b + 5*a^2*b^3 + b^5 + 15*(2*a^4*b + 3*a^2*b^3 + b^5)*cosh(x)^2 - 6*(2*a^5 + 3*a^3*b^2 + a*b^4)*cosh(x))*sinh(x)^3 - 12*(a^5 + a^3*b^2)*cosh(x)^2 - 1 2*(a^5 + a^3*b^2 - 5*(2*a^4*b + 3*a^2*b^3 + b^5)*cosh(x)^3 + 3*(2*a^5 + 3* a^3*b^2 + a*b^4)*cosh(x)^2 - (4*a^4*b + 5*a^2*b^3 + b^5)*cosh(x))*sinh(x)^ 2 - 3*(a^4*cosh(x)^6 + 6*a^4*cosh(x)*sinh(x)^5 + a^4*sinh(x)^6 + 3*a^4*cos h(x)^4 + 3*a^4*cosh(x)^2 + 3*(5*a^4*cosh(x)^2 + a^4)*sinh(x)^4 + a^4 + 4*( 5*a^4*cosh(x)^3 + 3*a^4*cosh(x))*sinh(x)^3 + 3*(5*a^4*cosh(x)^4 + 6*a^4*co sh(x)^2 + a^4)*sinh(x)^2 + 6*(a^4*cosh(x)^5 + 2*a^4*cosh(x)^3 + a^4*cosh(x ))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cos h(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*cosh(x) + 2*(b* cosh(x) + a)*sinh(x) - b)) + 6*(2*a^4*b + 3*a^2*b^3 + b^5)*cosh(x) + 6*(2* a^4*b + 3*a^2*b^3 + b^5 + 5*(2*a^4*b + 3*a^2*b^3 + b^5)*cosh(x)^4 - 4*(2*a ^5 + 3*a^3*b^2 + a*b^4)*cosh(x)^3 + 2*(4*a^4*b + 5*a^2*b^3 + b^5)*cosh(x)^ 2 - 4*(a^5 + a^3*b^2)*cosh(x))*sinh(x))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^ 6)*cosh(x)^6 + 6*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)*sinh(x)^5 ...
\[ \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx=\int \frac {\tanh ^{4}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \] Input:
integrate(tanh(x)**4/(a+b*sinh(x)),x)
Output:
Integral(tanh(x)**4/(a + b*sinh(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (116) = 232\).
Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.94 \[ \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx=\frac {a^{4} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (6 \, a^{3} e^{\left (-2 \, x\right )} + 4 \, a^{3} + a b^{2} + 3 \, {\left (2 \, a^{2} b + b^{3}\right )} e^{\left (-x\right )} + 2 \, {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (2 \, a^{2} b + b^{3}\right )} e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )}\right )}} \] Input:
integrate(tanh(x)^4/(a+b*sinh(x)),x, algorithm="maxima")
Output:
a^4*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2))) /((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)) - 2/3*(6*a^3*e^(-2*x) + 4*a^3 + a*b^2 + 3*(2*a^2*b + b^3)*e^(-x) + 2*(4*a^2*b + b^3)*e^(-3*x) + 3*(2*a^3 + a*b^2)*e^(-4*x) + 3*(2*a^2*b + 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))
Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.59 \[ \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx=\frac {a^{4} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (6 \, a^{2} b e^{\left (5 \, x\right )} + 3 \, b^{3} e^{\left (5 \, x\right )} - 6 \, a^{3} e^{\left (4 \, x\right )} - 3 \, a b^{2} e^{\left (4 \, x\right )} + 8 \, a^{2} b e^{\left (3 \, x\right )} + 2 \, b^{3} e^{\left (3 \, x\right )} - 6 \, a^{3} e^{\left (2 \, x\right )} + 6 \, a^{2} b e^{x} + 3 \, b^{3} e^{x} - 4 \, a^{3} - a b^{2}\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \] Input:
integrate(tanh(x)^4/(a+b*sinh(x)),x, algorithm="giac")
Output:
a^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)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)) - 2/3*(6*a^2*b*e^(5 *x) + 3*b^3*e^(5*x) - 6*a^3*e^(4*x) - 3*a*b^2*e^(4*x) + 8*a^2*b*e^(3*x) + 2*b^3*e^(3*x) - 6*a^3*e^(2*x) + 6*a^2*b*e^x + 3*b^3*e^x - 4*a^3 - a*b^2)/( (a^4 + 2*a^2*b^2 + b^4)*(e^(2*x) + 1)^3)
Time = 2.42 (sec) , antiderivative size = 654, normalized size of antiderivative = 5.27 \[ \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx=\frac {\frac {2\,a\,\left (2\,a^2+b^2\right )}{{\left (a^2+b^2\right )}^2}-\frac {2\,b\,{\mathrm {e}}^x\,\left (2\,a^2+b^2\right )}{{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {4\,\left (a^3+a\,b^2\right )}{{\left (a^2+b^2\right )}^2}-\frac {8\,{\mathrm {e}}^x\,\left (a^2\,b+b^3\right )}{3\,{\left (a^2+b^2\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\frac {8\,a}{3\,\left (a^2+b^2\right )}-\frac {8\,b\,{\mathrm {e}}^x}{3\,\left (a^2+b^2\right )}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,a^4}{b^2\,\sqrt {a^8}\,{\left (a^2+b^2\right )}^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {2\,\left (a^5\,\sqrt {a^8}+2\,a^3\,b^2\,\sqrt {a^8}+a\,b^4\,\sqrt {a^8}\right )}{a^3\,b^2\,\sqrt {-{\left (a^2+b^2\right )}^5}\,\left (a^4+2\,a^2\,b^2+b^4\right )\,\sqrt {-a^{10}-5\,a^8\,b^2-10\,a^6\,b^4-10\,a^4\,b^6-5\,a^2\,b^8-b^{10}}}\right )-\frac {2\,\left (b^5\,\sqrt {a^8}+2\,a^2\,b^3\,\sqrt {a^8}+a^4\,b\,\sqrt {a^8}\right )}{a^3\,b^2\,\sqrt {-{\left (a^2+b^2\right )}^5}\,\left (a^4+2\,a^2\,b^2+b^4\right )\,\sqrt {-a^{10}-5\,a^8\,b^2-10\,a^6\,b^4-10\,a^4\,b^6-5\,a^2\,b^8-b^{10}}}\right )\,\left (\frac {b^5\,\sqrt {-a^{10}-5\,a^8\,b^2-10\,a^6\,b^4-10\,a^4\,b^6-5\,a^2\,b^8-b^{10}}}{2}+\frac {a^4\,b\,\sqrt {-a^{10}-5\,a^8\,b^2-10\,a^6\,b^4-10\,a^4\,b^6-5\,a^2\,b^8-b^{10}}}{2}+a^2\,b^3\,\sqrt {-a^{10}-5\,a^8\,b^2-10\,a^6\,b^4-10\,a^4\,b^6-5\,a^2\,b^8-b^{10}}\right )\right )\,\sqrt {a^8}}{\sqrt {-a^{10}-5\,a^8\,b^2-10\,a^6\,b^4-10\,a^4\,b^6-5\,a^2\,b^8-b^{10}}} \] Input:
int(tanh(x)^4/(a + b*sinh(x)),x)
Output:
((2*a*(2*a^2 + b^2))/(a^2 + b^2)^2 - (2*b*exp(x)*(2*a^2 + b^2))/(a^2 + b^2 )^2)/(exp(2*x) + 1) - ((4*(a*b^2 + a^3))/(a^2 + b^2)^2 - (8*exp(x)*(a^2*b + b^3))/(3*(a^2 + b^2)^2))/(2*exp(2*x) + exp(4*x) + 1) + ((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) - (2*atan((exp(x)*((2*a^4)/(b^2*(a^8)^(1/2)*(a^2 + b^2)^2*(a^4 + b^4 + 2*a^2*b^2)) + (2*(a^5*(a^8)^(1/2) + 2*a^3*b^2*(a^8)^(1/2) + a*b^4*(a^8)^ (1/2)))/(a^3*b^2*(-(a^2 + b^2)^5)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)*(- a^10 - b^10 - 5*a^2*b^8 - 10*a^4*b^6 - 10*a^6*b^4 - 5*a^8*b^2)^(1/2))) - (2*(b^5* (a^8)^(1/2) + 2*a^2*b^3*(a^8)^(1/2) + a^4*b*(a^8)^(1/2)))/(a^3*b^2*(-(a^2 + b^2)^5)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)*(- a^10 - b^10 - 5*a^2*b^8 - 10*a^ 4*b^6 - 10*a^6*b^4 - 5*a^8*b^2)^(1/2)))*((b^5*(- a^10 - b^10 - 5*a^2*b^8 - 10*a^4*b^6 - 10*a^6*b^4 - 5*a^8*b^2)^(1/2))/2 + (a^4*b*(- a^10 - b^10 - 5 *a^2*b^8 - 10*a^4*b^6 - 10*a^6*b^4 - 5*a^8*b^2)^(1/2))/2 + a^2*b^3*(- a^10 - b^10 - 5*a^2*b^8 - 10*a^4*b^6 - 10*a^6*b^4 - 5*a^8*b^2)^(1/2)))*(a^8)^( 1/2))/(- a^10 - b^10 - 5*a^2*b^8 - 10*a^4*b^6 - 10*a^6*b^4 - 5*a^8*b^2)^(1 /2)
Time = 0.17 (sec) , antiderivative size = 496, normalized size of antiderivative = 4.00 \[ \int \frac {\tanh ^4(x)}{a+b \sinh (x)} \, dx=\frac {6 e^{6 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{4} i +18 e^{4 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{4} i +18 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{4} i +6 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{4} i -4 e^{6 x} a^{5}-6 e^{6 x} a^{3} b^{2}-2 e^{6 x} a \,b^{4}-12 e^{5 x} a^{4} b -18 e^{5 x} a^{2} b^{3}-6 e^{5 x} b^{5}-16 e^{3 x} a^{4} b -20 e^{3 x} a^{2} b^{3}-4 e^{3 x} b^{5}-6 e^{2 x} a^{3} b^{2}-6 e^{2 x} a \,b^{4}-12 e^{x} a^{4} b -18 e^{x} a^{2} b^{3}-6 e^{x} b^{5}+4 a^{5}+4 a^{3} b^{2}}{3 e^{6 x} a^{6}+9 e^{6 x} a^{4} b^{2}+9 e^{6 x} a^{2} b^{4}+3 e^{6 x} b^{6}+9 e^{4 x} a^{6}+27 e^{4 x} a^{4} b^{2}+27 e^{4 x} a^{2} b^{4}+9 e^{4 x} b^{6}+9 e^{2 x} a^{6}+27 e^{2 x} a^{4} b^{2}+27 e^{2 x} a^{2} b^{4}+9 e^{2 x} b^{6}+3 a^{6}+9 a^{4} b^{2}+9 a^{2} b^{4}+3 b^{6}} \] Input:
int(tanh(x)^4/(a+b*sinh(x)),x)
Output:
(2*(3*e**(6*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))* a**4*i + 9*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b* *2))*a**4*i + 9*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**4*i + 3*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b **2))*a**4*i - 2*e**(6*x)*a**5 - 3*e**(6*x)*a**3*b**2 - e**(6*x)*a*b**4 - 6*e**(5*x)*a**4*b - 9*e**(5*x)*a**2*b**3 - 3*e**(5*x)*b**5 - 8*e**(3*x)*a* *4*b - 10*e**(3*x)*a**2*b**3 - 2*e**(3*x)*b**5 - 3*e**(2*x)*a**3*b**2 - 3* e**(2*x)*a*b**4 - 6*e**x*a**4*b - 9*e**x*a**2*b**3 - 3*e**x*b**5 + 2*a**5 + 2*a**3*b**2))/(3*(e**(6*x)*a**6 + 3*e**(6*x)*a**4*b**2 + 3*e**(6*x)*a**2 *b**4 + e**(6*x)*b**6 + 3*e**(4*x)*a**6 + 9*e**(4*x)*a**4*b**2 + 9*e**(4*x )*a**2*b**4 + 3*e**(4*x)*b**6 + 3*e**(2*x)*a**6 + 9*e**(2*x)*a**4*b**2 + 9 *e**(2*x)*a**2*b**4 + 3*e**(2*x)*b**6 + a**6 + 3*a**4*b**2 + 3*a**2*b**4 + b**6))