Integrand size = 13, antiderivative size = 146 \[ \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 b^5 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b}}-\frac {b \left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^4}+\frac {\left (3 a^2+4 b^2\right ) \cosh (x) \sinh (x)}{8 a^3}-\frac {b \cosh ^2(x) \sinh (x)}{3 a^2}+\frac {\cosh ^3(x) \sinh (x)}{4 a} \] Output:
1/8*(3*a^4+4*a^2*b^2+8*b^4)*x/a^5-2*b^5*arctan((a-b)^(1/2)*tanh(1/2*x)/(a+ b)^(1/2))/a^5/(a-b)^(1/2)/(a+b)^(1/2)-1/3*b*(2*a^2+3*b^2)*sinh(x)/a^4+1/8* (3*a^2+4*b^2)*cosh(x)*sinh(x)/a^3-1/3*b*cosh(x)^2*sinh(x)/a^2+1/4*cosh(x)^ 3*sinh(x)/a
Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {12 \left (3 a^4+4 a^2 b^2+8 b^4\right ) x+\frac {192 b^5 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-24 a b \left (3 a^2+4 b^2\right ) \sinh (x)+24 a^2 \left (a^2+b^2\right ) \sinh (2 x)-8 a^3 b \sinh (3 x)+3 a^4 \sinh (4 x)}{96 a^5} \] Input:
Integrate[Cosh[x]^4/(a + b*Sech[x]),x]
Output:
(12*(3*a^4 + 4*a^2*b^2 + 8*b^4)*x + (192*b^5*ArcTan[((-a + b)*Tanh[x/2])/S qrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 24*a*b*(3*a^2 + 4*b^2)*Sinh[x] + 24*a^2 *(a^2 + b^2)*Sinh[2*x] - 8*a^3*b*Sinh[3*x] + 3*a^4*Sinh[4*x])/(96*a^5)
Time = 1.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.17, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.308, Rules used = {3042, 4340, 25, 3042, 4592, 3042, 4592, 3042, 4592, 27, 3042, 4407, 3042, 4318, 3042, 3138, 218}
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.
| \(\displaystyle \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc \left (\frac {\pi }{2}+i x\right )^4 \left (a+b \csc \left (\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 4340 |
| \(\displaystyle \frac {\int -\frac {\cosh ^3(x) \left (-3 b \text {sech}^2(x)-3 a \text {sech}(x)+4 b\right )}{a+b \text {sech}(x)}dx}{4 a}+\frac {\sinh (x) \cosh ^3(x)}{4 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\int \frac {\cosh ^3(x) \left (-3 b \text {sech}^2(x)-3 a \text {sech}(x)+4 b\right )}{a+b \text {sech}(x)}dx}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\int \frac {-3 b \csc \left (i x+\frac {\pi }{2}\right )^2-3 a \csc \left (i x+\frac {\pi }{2}\right )+4 b}{\csc \left (i x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (i x+\frac {\pi }{2}\right )\right )}dx}{4 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\int \frac {\cosh ^2(x) \left (-8 b^2 \text {sech}^2(x)+a b \text {sech}(x)+3 \left (3 a^2+4 b^2\right )\right )}{a+b \text {sech}(x)}dx}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\int \frac {-8 b^2 \csc \left (i x+\frac {\pi }{2}\right )^2+a b \csc \left (i x+\frac {\pi }{2}\right )+3 \left (3 a^2+4 b^2\right )}{\csc \left (i x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (i x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\int \frac {\cosh (x) \left (-3 b \left (3 a^2+4 b^2\right ) \text {sech}^2(x)-a \left (9 a^2-4 b^2\right ) \text {sech}(x)+8 b \left (2 a^2+3 b^2\right )\right )}{a+b \text {sech}(x)}dx}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\int \frac {-3 b \left (3 a^2+4 b^2\right ) \csc \left (i x+\frac {\pi }{2}\right )^2-a \left (9 a^2-4 b^2\right ) \csc \left (i x+\frac {\pi }{2}\right )+8 b \left (2 a^2+3 b^2\right )}{\csc \left (i x+\frac {\pi }{2}\right ) \left (a+b \csc \left (i x+\frac {\pi }{2}\right )\right )}dx}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {\int \frac {3 \left (3 a^4+4 b^2 a^2+b \left (3 a^2+4 b^2\right ) \text {sech}(x) a+8 b^4\right )}{a+b \text {sech}(x)}dx}{a}}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \int \frac {3 a^4+4 b^2 a^2+b \left (3 a^2+4 b^2\right ) \text {sech}(x) a+8 b^4}{a+b \text {sech}(x)}dx}{a}}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \int \frac {3 a^4+4 b^2 a^2+b \left (3 a^2+4 b^2\right ) \csc \left (i x+\frac {\pi }{2}\right ) a+8 b^4}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx}{a}}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{a}-\frac {8 b^5 \int \frac {\text {sech}(x)}{a+b \text {sech}(x)}dx}{a}\right )}{a}}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{a}-\frac {8 b^5 \int \frac {\csc \left (i x+\frac {\pi }{2}\right )}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{a}-\frac {8 b^4 \int \frac {1}{\frac {a \cosh (x)}{b}+1}dx}{a}\right )}{a}}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{a}-\frac {8 b^4 \int \frac {1}{\frac {a \sin \left (i x+\frac {\pi }{2}\right )}{b}+1}dx}{a}\right )}{a}}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{a}-\frac {16 b^4 \int \frac {1}{\frac {a+b}{b}-\left (1-\frac {a}{b}\right ) \tanh ^2\left (\frac {x}{2}\right )}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a}}{2 a}}{3 a}}{4 a}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\frac {4 b \sinh (x) \cosh ^2(x)}{3 a}-\frac {\frac {3 \left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{2 a}-\frac {\frac {8 b \left (2 a^2+3 b^2\right ) \sinh (x)}{a}-\frac {3 \left (\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{a}-\frac {16 b^5 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{2 a}}{3 a}}{4 a}\) |
Input:
Int[Cosh[x]^4/(a + b*Sech[x]),x]
Output:
(Cosh[x]^3*Sinh[x])/(4*a) - ((4*b*Cosh[x]^2*Sinh[x])/(3*a) - ((3*(3*a^2 + 4*b^2)*Cosh[x]*Sinh[x])/(2*a) - ((-3*(((3*a^4 + 4*a^2*b^2 + 8*b^4)*x)/a - (16*b^5*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b])))/a + (8*b*(2*a^2 + 3*b^2)*Sinh[x])/a)/(2*a))/(3*a))/(4*a)
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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[Cot[e + f*x]*((d*Csc[e + f*x])^n/(a*f*n)), x] - Sim p[1/(a*d*n) Int[((d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x]))*Simp[b*n - a*(n + 1)*Csc[e + f*x] - b*(n + 1)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(126)=252\).
Time = 0.68 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.81
| method | result | size |
| risch | \(\frac {3 x}{8 a}+\frac {x \,b^{2}}{2 a^{3}}+\frac {x \,b^{4}}{a^{5}}+\frac {{\mathrm e}^{4 x}}{64 a}-\frac {b \,{\mathrm e}^{3 x}}{24 a^{2}}+\frac {{\mathrm e}^{2 x}}{8 a}+\frac {{\mathrm e}^{2 x} b^{2}}{8 a^{3}}-\frac {3 b \,{\mathrm e}^{x}}{8 a^{2}}-\frac {b^{3} {\mathrm e}^{x}}{2 a^{4}}+\frac {3 b \,{\mathrm e}^{-x}}{8 a^{2}}+\frac {b^{3} {\mathrm e}^{-x}}{2 a^{4}}-\frac {{\mathrm e}^{-2 x}}{8 a}-\frac {{\mathrm e}^{-2 x} b^{2}}{8 a^{3}}+\frac {b \,{\mathrm e}^{-3 x}}{24 a^{2}}-\frac {{\mathrm e}^{-4 x}}{64 a}-\frac {b^{5} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{5}}+\frac {b^{5} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{5}}\) | \(264\) |
| default | \(-\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {-3 a -2 b}{6 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {7 a^{2}+4 a b +4 b^{2}}{8 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {\left (3 a^{4}+4 a^{2} b^{2}+8 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 a^{5}}-\frac {-5 a^{3}-8 a^{2} b -4 a \,b^{2}-8 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 b^{5} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {-3 a -2 b}{6 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {-7 a^{2}-4 a b -4 b^{2}}{8 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\left (-3 a^{4}-4 a^{2} b^{2}-8 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 a^{5}}-\frac {-5 a^{3}-8 a^{2} b -4 a \,b^{2}-8 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(299\) |
Input:
int(cosh(x)^4/(a+b*sech(x)),x,method=_RETURNVERBOSE)
Output:
3/8*x/a+1/2*x/a^3*b^2+x/a^5*b^4+1/64/a*exp(x)^4-1/24*b/a^2*exp(x)^3+1/8/a* exp(x)^2+1/8/a^3*exp(x)^2*b^2-3/8*b/a^2*exp(x)-1/2*b^3/a^4*exp(x)+3/8*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*b/a^2 /exp(x)^3-1/64/a/exp(x)^4-1/(-a^2+b^2)^(1/2)*b^5/a^5*ln(exp(x)+(b*(-a^2+b^ 2)^(1/2)+a^2-b^2)/(-a^2+b^2)^(1/2)/a)+1/(-a^2+b^2)^(1/2)*b^5/a^5*ln(exp(x) +(b*(-a^2+b^2)^(1/2)-a^2+b^2)/(-a^2+b^2)^(1/2)/a)
Leaf count of result is larger than twice the leaf count of optimal. 1161 vs. \(2 (126) = 252\).
Time = 0.11 (sec) , antiderivative size = 2402, normalized size of antiderivative = 16.45 \[ \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \] Input:
integrate(cosh(x)^4/(a+b*sech(x)),x, algorithm="fricas")
Output:
[1/192*(3*(a^6 - a^4*b^2)*cosh(x)^8 + 3*(a^6 - a^4*b^2)*sinh(x)^8 - 8*(a^5 *b - a^3*b^3)*cosh(x)^7 - 8*(a^5*b - a^3*b^3 - 3*(a^6 - a^4*b^2)*cosh(x))* sinh(x)^7 + 24*(a^6 - a^2*b^4)*cosh(x)^6 + 4*(6*a^6 - 6*a^2*b^4 + 21*(a^6 - a^4*b^2)*cosh(x)^2 - 14*(a^5*b - a^3*b^3)*cosh(x))*sinh(x)^6 - 3*a^6 + 3 *a^4*b^2 + 24*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x*cosh(x)^4 - 24*(3*a^ 5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^5 - 24*(3*a^5*b + a^3*b^3 - 4*a*b^5 - 7*( a^6 - a^4*b^2)*cosh(x)^3 + 7*(a^5*b - a^3*b^3)*cosh(x)^2 - 6*(a^6 - a^2*b^ 4)*cosh(x))*sinh(x)^5 + 2*(105*(a^6 - a^4*b^2)*cosh(x)^4 - 140*(a^5*b - a^ 3*b^3)*cosh(x)^3 + 180*(a^6 - a^2*b^4)*cosh(x)^2 + 12*(3*a^6 + a^4*b^2 + 4 *a^2*b^4 - 8*b^6)*x - 60*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x))*sinh(x)^4 + 24*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^3 + 8*(9*a^5*b + 3*a^3*b^3 - 12 *a*b^5 + 21*(a^6 - a^4*b^2)*cosh(x)^5 - 35*(a^5*b - a^3*b^3)*cosh(x)^4 + 6 0*(a^6 - a^2*b^4)*cosh(x)^3 + 12*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x*c osh(x) - 30*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^2)*sinh(x)^3 - 24*(a^6 - a^2*b^4)*cosh(x)^2 + 12*(7*(a^6 - a^4*b^2)*cosh(x)^6 - 2*a^6 + 2*a^2*b^4 - 14*(a^5*b - a^3*b^3)*cosh(x)^5 + 30*(a^6 - a^2*b^4)*cosh(x)^4 + 12*(3*a^ 6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x*cosh(x)^2 - 20*(3*a^5*b + a^3*b^3 - 4*a *b^5)*cosh(x)^3 + 6*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x))*sinh(x)^2 - 192 *(b^5*cosh(x)^4 + 4*b^5*cosh(x)^3*sinh(x) + 6*b^5*cosh(x)^2*sinh(x)^2 + 4* b^5*cosh(x)*sinh(x)^3 + b^5*sinh(x)^4)*sqrt(-a^2 + b^2)*log((a^2*cosh(x...
\[ \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\cosh ^{4}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \] Input:
integrate(cosh(x)**4/(a+b*sech(x)),x)
Output:
Integral(cosh(x)**4/(a + b*sech(x)), x)
Exception generated. \[ \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cosh(x)^4/(a+b*sech(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*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.25 \[ \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx=-\frac {2 \, b^{5} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {3 \, a^{3} e^{\left (4 \, x\right )} - 8 \, a^{2} b e^{\left (3 \, x\right )} + 24 \, a^{3} e^{\left (2 \, x\right )} + 24 \, a b^{2} e^{\left (2 \, x\right )} - 72 \, a^{2} b e^{x} - 96 \, b^{3} e^{x}}{192 \, a^{4}} + \frac {{\left (3 \, a^{4} + 4 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} + \frac {{\left (8 \, a^{3} b e^{x} - 3 \, a^{4} + 24 \, {\left (3 \, a^{3} b + 4 \, a b^{3}\right )} e^{\left (3 \, x\right )} - 24 \, {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-4 \, x\right )}}{192 \, a^{5}} \] Input:
integrate(cosh(x)^4/(a+b*sech(x)),x, algorithm="giac")
Output:
-2*b^5*arctan((a*e^x + b)/sqrt(a^2 - b^2))/(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) - 72*a ^2*b*e^x - 96*b^3*e^x)/a^4 + 1/8*(3*a^4 + 4*a^2*b^2 + 8*b^4)*x/a^5 + 1/192 *(8*a^3*b*e^x - 3*a^4 + 24*(3*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
Time = 2.79 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.72 \[ \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}+\frac {x\,\left (3\,a^4+4\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {{\mathrm {e}}^{-2\,x}\,\left (a^2+b^2\right )}{8\,a^3}+\frac {{\mathrm {e}}^{2\,x}\,\left (a^2+b^2\right )}{8\,a^3}+\frac {{\mathrm {e}}^{-x}\,\left (3\,a^2\,b+4\,b^3\right )}{8\,a^4}+\frac {b\,{\mathrm {e}}^{-3\,x}}{24\,a^2}-\frac {b\,{\mathrm {e}}^{3\,x}}{24\,a^2}-\frac {{\mathrm {e}}^x\,\left (3\,a^2\,b+4\,b^3\right )}{8\,a^4}+\frac {b^5\,\ln \left (\frac {2\,b^5\,{\mathrm {e}}^x}{a^6}-\frac {2\,b^5\,\left (a+b\,{\mathrm {e}}^x\right )}{a^6\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^5\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {b^5\,\ln \left (\frac {2\,b^5\,{\mathrm {e}}^x}{a^6}+\frac {2\,b^5\,\left (a+b\,{\mathrm {e}}^x\right )}{a^6\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^5\,\sqrt {a+b}\,\sqrt {b-a}} \] Input:
int(cosh(x)^4/(a + b/cosh(x)),x)
Output:
exp(4*x)/(64*a) - exp(-4*x)/(64*a) + (x*(3*a^4 + 8*b^4 + 4*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) + (e xp(-x)*(3*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)*(3*a^2*b + 4*b^3))/(8*a^4) + (b^5*log((2*b^5*exp(x))/a^ 6 - (2*b^5*(a + b*exp(x)))/(a^6*(a + b)^(1/2)*(b - a)^(1/2))))/(a^5*(a + b )^(1/2)*(b - a)^(1/2)) - (b^5*log((2*b^5*exp(x))/a^6 + (2*b^5*(a + b*exp(x )))/(a^6*(a + b)^(1/2)*(b - a)^(1/2))))/(a^5*(a + b)^(1/2)*(b - a)^(1/2))
Time = 0.21 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.13 \[ \int \frac {\cosh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {-384 e^{4 x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +b}{\sqrt {a^{2}-b^{2}}}\right ) b^{5}+3 e^{8 x} a^{6}-3 e^{8 x} a^{4} b^{2}-8 e^{7 x} a^{5} b +8 e^{7 x} a^{3} b^{3}+24 e^{6 x} a^{6}-24 e^{6 x} a^{2} b^{4}-72 e^{5 x} a^{5} b -24 e^{5 x} a^{3} b^{3}+96 e^{5 x} a \,b^{5}+72 e^{4 x} a^{6} x +24 e^{4 x} a^{4} b^{2} x +96 e^{4 x} a^{2} b^{4} x -192 e^{4 x} b^{6} x +72 e^{3 x} a^{5} b +24 e^{3 x} a^{3} b^{3}-96 e^{3 x} a \,b^{5}-24 e^{2 x} a^{6}+24 e^{2 x} a^{2} b^{4}+8 e^{x} a^{5} b -8 e^{x} a^{3} b^{3}-3 a^{6}+3 a^{4} b^{2}}{192 e^{4 x} a^{5} \left (a^{2}-b^{2}\right )} \] Input:
int(cosh(x)^4/(a+b*sech(x)),x)
Output:
( - 384*e**(4*x)*sqrt(a**2 - b**2)*atan((e**x*a + b)/sqrt(a**2 - b**2))*b* *5 + 3*e**(8*x)*a**6 - 3*e**(8*x)*a**4*b**2 - 8*e**(7*x)*a**5*b + 8*e**(7* x)*a**3*b**3 + 24*e**(6*x)*a**6 - 24*e**(6*x)*a**2*b**4 - 72*e**(5*x)*a**5 *b - 24*e**(5*x)*a**3*b**3 + 96*e**(5*x)*a*b**5 + 72*e**(4*x)*a**6*x + 24* e**(4*x)*a**4*b**2*x + 96*e**(4*x)*a**2*b**4*x - 192*e**(4*x)*b**6*x + 72* e**(3*x)*a**5*b + 24*e**(3*x)*a**3*b**3 - 96*e**(3*x)*a*b**5 - 24*e**(2*x) *a**6 + 24*e**(2*x)*a**2*b**4 + 8*e**x*a**5*b - 8*e**x*a**3*b**3 - 3*a**6 + 3*a**4*b**2)/(192*e**(4*x)*a**5*(a**2 - b**2))