Integrand size = 13, antiderivative size = 113 \[ \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx=\frac {b \left (3 a^2-2 b^2\right ) \arctan (\sinh (x))}{2 a^4}+\frac {2 (a-b)^{3/2} (a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^4}-\frac {\left (4 a^2-3 b^2\right ) \tanh (x)}{3 a^3}-\frac {b \text {sech}(x) \tanh (x)}{2 a^2}+\frac {\text {sech}^2(x) \tanh (x)}{3 a} \] Output:
1/2*b*(3*a^2-2*b^2)*arctan(sinh(x))/a^4+2*(a-b)^(3/2)*(a+b)^(3/2)*arctanh( (a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/a^4-1/3*(4*a^2-3*b^2)*tanh(x)/a^3-1/2 *b*sech(x)*tanh(x)/a^2+1/3*sech(x)^2*tanh(x)/a
Time = 0.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx=\frac {6 b \left (3 a^2-2 b^2\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )-12 \left (-a^2+b^2\right )^{3/2} \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )+a \left (-8 a^2+6 b^2-3 a b \text {sech}(x)+2 a^2 \text {sech}^2(x)\right ) \tanh (x)}{6 a^4} \] Input:
Integrate[Tanh[x]^4/(a + b*Cosh[x]),x]
Output:
(6*b*(3*a^2 - 2*b^2)*ArcTan[Tanh[x/2]] - 12*(-a^2 + b^2)^(3/2)*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]] + a*(-8*a^2 + 6*b^2 - 3*a*b*Sech[x] + 2* a^2*Sech[x]^2)*Tanh[x])/(6*a^4)
Time = 1.00 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 3204, 3042, 3534, 27, 3042, 3480, 3042, 3138, 221, 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.
| \(\displaystyle \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan \left (-\frac {\pi }{2}+i x\right )^4 \left (a-b \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 3204 |
| \(\displaystyle -\frac {\int \frac {\left (-3 \left (2 a^2-b^2\right ) \cosh ^2(x)-a b \cosh (x)+2 \left (4 a^2-3 b^2\right )\right ) \text {sech}^2(x)}{a+b \cosh (x)}dx}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-3 \left (2 a^2-b^2\right ) \sin \left (i x+\frac {\pi }{2}\right )^2-a b \sin \left (i x+\frac {\pi }{2}\right )+2 \left (4 a^2-3 b^2\right )}{\sin \left (i x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (i x+\frac {\pi }{2}\right )\right )}dx}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\int -\frac {3 \left (b \left (3 a^2-2 b^2\right )+a \left (2 a^2-b^2\right ) \cosh (x)\right ) \text {sech}(x)}{a+b \cosh (x)}dx}{a}+\frac {2 \left (4 a^2-3 b^2\right ) \tanh (x)}{a}}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2-3 b^2\right ) \tanh (x)}{a}-\frac {3 \int \frac {\left (b \left (3 a^2-2 b^2\right )+a \left (2 a^2-b^2\right ) \cosh (x)\right ) \text {sech}(x)}{a+b \cosh (x)}dx}{a}}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2-3 b^2\right ) \tanh (x)}{a}-\frac {3 \int \frac {b \left (3 a^2-2 b^2\right )+a \left (2 a^2-b^2\right ) \sin \left (i x+\frac {\pi }{2}\right )}{\sin \left (i x+\frac {\pi }{2}\right ) \left (a+b \sin \left (i x+\frac {\pi }{2}\right )\right )}dx}{a}}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2-3 b^2\right ) \tanh (x)}{a}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \int \frac {1}{a+b \cosh (x)}dx}{a}+\frac {b \left (3 a^2-2 b^2\right ) \int \text {sech}(x)dx}{a}\right )}{a}}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2-3 b^2\right ) \tanh (x)}{a}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^2 \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a}+\frac {b \left (3 a^2-2 b^2\right ) \int \csc \left (i x+\frac {\pi }{2}\right )dx}{a}\right )}{a}}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2-3 b^2\right ) \tanh (x)}{a}-\frac {3 \left (\frac {4 \left (a^2-b^2\right )^2 \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a}+\frac {b \left (3 a^2-2 b^2\right ) \int \csc \left (i x+\frac {\pi }{2}\right )dx}{a}\right )}{a}}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2-3 b^2\right ) \tanh (x)}{a}-\frac {3 \left (\frac {4 \left (a^2-b^2\right )^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}+\frac {b \left (3 a^2-2 b^2\right ) \int \csc \left (i x+\frac {\pi }{2}\right )dx}{a}\right )}{a}}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\frac {2 \left (4 a^2-3 b^2\right ) \tanh (x)}{a}-\frac {3 \left (\frac {b \left (3 a^2-2 b^2\right ) \arctan (\sinh (x))}{a}+\frac {4 \left (a^2-b^2\right )^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{6 a^2}-\frac {b \tanh (x) \text {sech}(x)}{2 a^2}+\frac {\tanh (x) \text {sech}^2(x)}{3 a}\) |
Input:
Int[Tanh[x]^4/(a + b*Cosh[x]),x]
Output:
-1/2*(b*Sech[x]*Tanh[x])/a^2 + (Sech[x]^2*Tanh[x])/(3*a) - ((-3*((b*(3*a^2 - 2*b^2)*ArcTan[Sinh[x]])/a + (4*(a^2 - b^2)^2*ArcTanh[(Sqrt[a - b]*Tanh[ x/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b])))/a + (2*(4*a^2 - 3*b^2)*T anh[x])/a)/(6*a^2)
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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*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_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ e + f*x]^3)), x] + (-Simp[b*(m - 2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(6*a^2*f*Sin[e + f*x]^2)), x] - Simp[1/(6*a^2) Int[((a + b*Sin[e + f* x])^m/Sin[e + f*x]^2)*Simp[8*a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f , m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1] && IntegerQ[2*m]
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 = 1.38 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(\frac {2 \left (a +b \right )^{2} \left (a -b \right )^{2} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\frac {2 \left (\left (-a^{3}+\frac {1}{2} a^{2} b +b^{2} a \right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-\frac {10}{3} a^{3}+2 b^{2} a \right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{3}+b^{2} a -\frac {1}{2} a^{2} b \right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{3}}+b \left (3 a^{2}-2 b^{2}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{4}}\) | \(154\) |
| risch | \(\frac {-3 a b \,{\mathrm e}^{5 x}+12 a^{2} {\mathrm e}^{4 x}-6 b^{2} {\mathrm e}^{4 x}+12 a^{2} {\mathrm e}^{2 x}-12 b^{2} {\mathrm e}^{2 x}+3 b \,{\mathrm e}^{x} a +8 a^{2}-6 b^{2}}{3 a^{3} \left ({\mathrm e}^{2 x}+1\right )^{3}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}{a^{2}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right ) b^{2}}{a^{4}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right )}{a^{2}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b^{2}}{a^{4}}+\frac {3 i b \ln \left ({\mathrm e}^{x}+i\right )}{2 a^{2}}-\frac {i b^{3} \ln \left ({\mathrm e}^{x}+i\right )}{a^{4}}-\frac {3 i b \ln \left ({\mathrm e}^{x}-i\right )}{2 a^{2}}+\frac {i b^{3} \ln \left ({\mathrm e}^{x}-i\right )}{a^{4}}\) | \(290\) |
Input:
int(tanh(x)^4/(a+b*cosh(x)),x,method=_RETURNVERBOSE)
Output:
2*(a+b)^2*(a-b)^2/a^4/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b) *(a-b))^(1/2))+2/a^4*(((-a^3+1/2*a^2*b+b^2*a)*tanh(1/2*x)^5+(-10/3*a^3+2*b ^2*a)*tanh(1/2*x)^3+(-a^3+b^2*a-1/2*a^2*b)*tanh(1/2*x))/(tanh(1/2*x)^2+1)^ 3+1/2*b*(3*a^2-2*b^2)*arctan(tanh(1/2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 967 vs. \(2 (95) = 190\).
Time = 0.15 (sec) , antiderivative size = 2003, normalized size of antiderivative = 17.73 \[ \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)^4/(a+b*cosh(x)),x, algorithm="fricas")
Output:
[-1/3*(3*a^2*b*cosh(x)^5 + 3*a^2*b*sinh(x)^5 - 6*(2*a^3 - a*b^2)*cosh(x)^4 + 3*(5*a^2*b*cosh(x) - 4*a^3 + 2*a*b^2)*sinh(x)^4 - 3*a^2*b*cosh(x) + 6*( 5*a^2*b*cosh(x)^2 - 4*(2*a^3 - a*b^2)*cosh(x))*sinh(x)^3 - 8*a^3 + 6*a*b^2 - 12*(a^3 - a*b^2)*cosh(x)^2 + 6*(5*a^2*b*cosh(x)^3 - 2*a^3 + 2*a*b^2 - 6 *(2*a^3 - a*b^2)*cosh(x)^2)*sinh(x)^2 + 3*((a^2 - b^2)*cosh(x)^6 + 6*(a^2 - b^2)*cosh(x)*sinh(x)^5 + (a^2 - b^2)*sinh(x)^6 + 3*(a^2 - b^2)*cosh(x)^4 + 3*(5*(a^2 - b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^4 + 4*(5*(a^2 - b^2)*co sh(x)^3 + 3*(a^2 - b^2)*cosh(x))*sinh(x)^3 + 3*(a^2 - b^2)*cosh(x)^2 + 3*( 5*(a^2 - b^2)*cosh(x)^4 + 6*(a^2 - b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + a^2 - b^2 + 6*((a^2 - b^2)*cosh(x)^5 + 2*(a^2 - b^2)*cosh(x)^3 + (a^2 - b ^2)*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)*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)) - 3*((3*a^2*b - 2*b^3)*cosh(x)^6 + 6* (3*a^2*b - 2*b^3)*cosh(x)*sinh(x)^5 + (3*a^2*b - 2*b^3)*sinh(x)^6 + 3*(3*a ^2*b - 2*b^3)*cosh(x)^4 + 3*(3*a^2*b - 2*b^3 + 5*(3*a^2*b - 2*b^3)*cosh(x) ^2)*sinh(x)^4 + 4*(5*(3*a^2*b - 2*b^3)*cosh(x)^3 + 3*(3*a^2*b - 2*b^3)*cos h(x))*sinh(x)^3 + 3*a^2*b - 2*b^3 + 3*(3*a^2*b - 2*b^3)*cosh(x)^2 + 3*(5*( 3*a^2*b - 2*b^3)*cosh(x)^4 + 3*a^2*b - 2*b^3 + 6*(3*a^2*b - 2*b^3)*cosh(x) ^2)*sinh(x)^2 + 6*((3*a^2*b - 2*b^3)*cosh(x)^5 + 2*(3*a^2*b - 2*b^3)*co...
\[ \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx=\int \frac {\tanh ^{4}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \] Input:
integrate(tanh(x)**4/(a+b*cosh(x)),x)
Output:
Integral(tanh(x)**4/(a + b*cosh(x)), x)
Exception generated. \[ \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(tanh(x)^4/(a+b*cosh(x)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.27 \[ \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx=\frac {{\left (3 \, a^{2} b - 2 \, b^{3}\right )} \arctan \left (e^{x}\right )}{a^{4}} + \frac {2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} a^{4}} - \frac {3 \, a b e^{\left (5 \, x\right )} - 12 \, a^{2} e^{\left (4 \, x\right )} + 6 \, b^{2} e^{\left (4 \, x\right )} - 12 \, a^{2} e^{\left (2 \, x\right )} + 12 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} - 8 \, a^{2} + 6 \, b^{2}}{3 \, a^{3} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \] Input:
integrate(tanh(x)^4/(a+b*cosh(x)),x, algorithm="giac")
Output:
(3*a^2*b - 2*b^3)*arctan(e^x)/a^4 + 2*(a^4 - 2*a^2*b^2 + b^4)*arctan((b*e^ x + a)/sqrt(-a^2 + b^2))/(sqrt(-a^2 + b^2)*a^4) - 1/3*(3*a*b*e^(5*x) - 12* a^2*e^(4*x) + 6*b^2*e^(4*x) - 12*a^2*e^(2*x) + 12*b^2*e^(2*x) - 3*a*b*e^x - 8*a^2 + 6*b^2)/(a^3*(e^(2*x) + 1)^3)
Time = 7.27 (sec) , antiderivative size = 722, normalized size of antiderivative = 6.39 \[ \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx =\text {Too large to display} \] Input:
int(tanh(x)^4/(a + b*cosh(x)),x)
Output:
8/(3*a*(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1)) - (4/a - (2*b*exp(x))/a^2 )/(2*exp(2*x) + exp(4*x) + 1) + ((2*(2*a^2 - b^2))/a^3 - (b*exp(x))/a^2)/( exp(2*x) + 1) + (log((((32*a^8 + 64*b^8 - 288*a^2*b^6 + 456*a^4*b^4 - 272* a^6*b^2 + 96*a*b^7*exp(x) - 288*a^7*b*exp(x) - 416*a^3*b^5*exp(x) + 600*a^ 5*b^3*exp(x))/(a^6*b^4) - (((32*((a + b)^3*(a - b)^3)^(1/2)*(3*a^2*b - 2*b ^3 + 4*a^3*exp(x) - 3*a*b^2*exp(x)))/(a^2*b^5) + (16*(a^2 - b^2)*(4*a^2*b - 4*b^3 + 8*a^3*exp(x) - 7*a*b^2*exp(x)))/(a*b^5))*((a + b)^3*(a - b)^3)^( 1/2))/a^4)*((a + b)^3*(a - b)^3)^(1/2))/a^4 - (8*(a^2 - b^2)^2*(3*a^2 - 2* b^2)*(6*a^2*b - 4*b^3 + 10*a^3*exp(x) - 7*a*b^2*exp(x)))/(a^9*b^3))*((a + b)^3*(a - b)^3)^(1/2))/a^4 - (log(- (((32*a^8 + 64*b^8 - 288*a^2*b^6 + 456 *a^4*b^4 - 272*a^6*b^2 + 96*a*b^7*exp(x) - 288*a^7*b*exp(x) - 416*a^3*b^5* exp(x) + 600*a^5*b^3*exp(x))/(a^6*b^4) - (((32*((a + b)^3*(a - b)^3)^(1/2) *(3*a^2*b - 2*b^3 + 4*a^3*exp(x) - 3*a*b^2*exp(x)))/(a^2*b^5) - (16*(a^2 - b^2)*(4*a^2*b - 4*b^3 + 8*a^3*exp(x) - 7*a*b^2*exp(x)))/(a*b^5))*((a + b) ^3*(a - b)^3)^(1/2))/a^4)*((a + b)^3*(a - b)^3)^(1/2))/a^4 - (8*(a^2 - b^2 )^2*(3*a^2 - 2*b^2)*(6*a^2*b - 4*b^3 + 10*a^3*exp(x) - 7*a*b^2*exp(x)))/(a ^9*b^3))*((a + b)^3*(a - b)^3)^(1/2))/a^4 - (b*log(exp(x) - 1i)*(3*a^2 - 2 *b^2)*1i)/(2*a^4) + (b*log(exp(x) + 1i)*(3*a^2 - 2*b^2)*1i)/(2*a^4)
Time = 0.24 (sec) , antiderivative size = 516, normalized size of antiderivative = 4.57 \[ \int \frac {\tanh ^4(x)}{a+b \cosh (x)} \, dx=\frac {9 e^{6 x} \mathit {atan} \left (e^{x}\right ) a^{2} b -6 e^{6 x} \mathit {atan} \left (e^{x}\right ) b^{3}+27 e^{4 x} \mathit {atan} \left (e^{x}\right ) a^{2} b -18 e^{4 x} \mathit {atan} \left (e^{x}\right ) b^{3}+27 e^{2 x} \mathit {atan} \left (e^{x}\right ) a^{2} b -18 e^{2 x} \mathit {atan} \left (e^{x}\right ) b^{3}+9 \mathit {atan} \left (e^{x}\right ) a^{2} b -6 \mathit {atan} \left (e^{x}\right ) b^{3}-6 e^{6 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}+6 e^{6 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{2}-18 e^{4 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}+18 e^{4 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{2}-18 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}+18 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{2}-6 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}+6 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{2}-4 e^{6 x} a^{3}+2 e^{6 x} a \,b^{2}-3 e^{5 x} a^{2} b -6 e^{2 x} a \,b^{2}+3 e^{x} a^{2} b +4 a^{3}-4 a \,b^{2}}{3 a^{4} \left (e^{6 x}+3 e^{4 x}+3 e^{2 x}+1\right )} \] Input:
int(tanh(x)^4/(a+b*cosh(x)),x)
Output:
(9*e**(6*x)*atan(e**x)*a**2*b - 6*e**(6*x)*atan(e**x)*b**3 + 27*e**(4*x)*a tan(e**x)*a**2*b - 18*e**(4*x)*atan(e**x)*b**3 + 27*e**(2*x)*atan(e**x)*a* *2*b - 18*e**(2*x)*atan(e**x)*b**3 + 9*atan(e**x)*a**2*b - 6*atan(e**x)*b* *3 - 6*e**(6*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2 ))*a**2 + 6*e**(6*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*b**2 - 18*e**(4*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**2 + 18*e**(4*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sq rt( - a**2 + b**2))*b**2 - 18*e**(2*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**2 + 18*e**(2*x)*sqrt( - a**2 + b**2)*atan((e* *x*b + a)/sqrt( - a**2 + b**2))*b**2 - 6*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**2 + 6*sqrt( - a**2 + b**2)*atan((e**x*b + a )/sqrt( - a**2 + b**2))*b**2 - 4*e**(6*x)*a**3 + 2*e**(6*x)*a*b**2 - 3*e** (5*x)*a**2*b - 6*e**(2*x)*a*b**2 + 3*e**x*a**2*b + 4*a**3 - 4*a*b**2)/(3*a **4*(e**(6*x) + 3*e**(4*x) + 3*e**(2*x) + 1))