Integrand size = 13, antiderivative size = 150 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\frac {a \left (3 a^4-6 a^2 b^2-b^4\right ) \arctan (\sinh (x))}{8 \left (a^2+b^2\right )^3}+\frac {a^4 b \log (\cosh (x))}{\left (a^2+b^2\right )^3}-\frac {a^4 b \log (b+a \sinh (x))}{\left (a^2+b^2\right )^3}-\frac {a^2 b \text {sech}^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {\text {sech}^4(x) (b-a \sinh (x))}{4 \left (a^2+b^2\right )}+\frac {a \left (3 a^2-b^2\right ) \text {sech}(x) \tanh (x)}{8 \left (a^2+b^2\right )^2} \] Output:
1/8*a*(3*a^4-6*a^2*b^2-b^4)*arctan(sinh(x))/(a^2+b^2)^3+a^4*b*ln(cosh(x))/ (a^2+b^2)^3-a^4*b*ln(b+a*sinh(x))/(a^2+b^2)^3-1/2*a^2*b*sech(x)^2/(a^2+b^2 )^2-sech(x)^4*(b-a*sinh(x))/(4*a^2+4*b^2)+1/8*a*(3*a^2-b^2)*sech(x)*tanh(x )/(a^2+b^2)^2
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.09 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\frac {a (i a-b)^3 (3 a-i b) \log (i-\sinh (x))-a (3 a+i b) (i a+b)^3 \log (i+\sinh (x))-16 a^4 b \log (b+a \sinh (x))-8 a^2 b \left (a^2+b^2\right ) \text {sech}^2(x)-4 b \left (a^2+b^2\right )^2 \text {sech}^4(x)+2 a \left (3 a^4+2 a^2 b^2-b^4\right ) \text {sech}(x) \tanh (x)+4 a \left (a^2+b^2\right )^2 \text {sech}^3(x) \tanh (x)}{16 \left (a^2+b^2\right )^3} \] Input:
Integrate[Sech[x]^5/(a + b*Csch[x]),x]
Output:
(a*(I*a - b)^3*(3*a - I*b)*Log[I - Sinh[x]] - a*(3*a + I*b)*(I*a + b)^3*Lo g[I + Sinh[x]] - 16*a^4*b*Log[b + a*Sinh[x]] - 8*a^2*b*(a^2 + b^2)*Sech[x] ^2 - 4*b*(a^2 + b^2)^2*Sech[x]^4 + 2*a*(3*a^4 + 2*a^2*b^2 - b^4)*Sech[x]*T anh[x] + 4*a*(a^2 + b^2)^2*Sech[x]^3*Tanh[x])/(16*(a^2 + b^2)^3)
Time = 0.56 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.39, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 4360, 26, 26, 3042, 26, 3316, 26, 27, 593, 25, 686, 25, 657, 2009}
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.
| \(\displaystyle \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (i x)^5 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {i \tanh (x) \text {sech}^4(x)}{i a \sinh (x)+i b}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int -\frac {i \text {sech}^4(x) \tanh (x)}{b+a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\tanh (x) \text {sech}^4(x)}{a \sinh (x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i x)}{\cos (i x)^5 (b-i a \sin (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i x)}{\cos (i x)^5 (b-i a \sin (i x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle i a^5 \int -\frac {i \sinh (x)}{(b+a \sinh (x)) \left (\sinh ^2(x) a^2+a^2\right )^3}d(a \sinh (x))\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle a^5 \int \frac {\sinh (x)}{(b+a \sinh (x)) \left (\sinh ^2(x) a^2+a^2\right )^3}d(a \sinh (x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a^4 \int \frac {a \sinh (x)}{(b+a \sinh (x)) \left (\sinh ^2(x) a^2+a^2\right )^3}d(a \sinh (x))\) |
| \(\Big \downarrow \) 593 |
| \(\displaystyle a^4 \left (\frac {\int -\frac {b-3 a \sinh (x)}{(b+a \sinh (x)) \left (\sinh ^2(x) a^2+a^2\right )^2}d(a \sinh (x))}{4 \left (a^2+b^2\right )}-\frac {b-a \sinh (x)}{4 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a^4 \left (-\frac {\int \frac {b-3 a \sinh (x)}{(b+a \sinh (x)) \left (\sinh ^2(x) a^2+a^2\right )^2}d(a \sinh (x))}{4 \left (a^2+b^2\right )}-\frac {b-a \sinh (x)}{4 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )^2}\right )\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle a^4 \left (-\frac {\frac {4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)}{2 a^2 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )}-\frac {\int -\frac {b \left (5 a^2+b^2\right )-a \left (3 a^2-b^2\right ) \sinh (x)}{(b+a \sinh (x)) \left (\sinh ^2(x) a^2+a^2\right )}d(a \sinh (x))}{2 a^2 \left (a^2+b^2\right )}}{4 \left (a^2+b^2\right )}-\frac {b-a \sinh (x)}{4 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a^4 \left (-\frac {\frac {\int \frac {b \left (5 a^2+b^2\right )-a \left (3 a^2-b^2\right ) \sinh (x)}{(b+a \sinh (x)) \left (\sinh ^2(x) a^2+a^2\right )}d(a \sinh (x))}{2 a^2 \left (a^2+b^2\right )}+\frac {4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)}{2 a^2 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )}}{4 \left (a^2+b^2\right )}-\frac {b-a \sinh (x)}{4 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )^2}\right )\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle a^4 \left (-\frac {\frac {\int \left (\frac {8 b a^2}{\left (a^2+b^2\right ) (b+a \sinh (x))}+\frac {-3 a^4-8 b \sinh (x) a^3+6 b^2 a^2+b^4}{\left (a^2+b^2\right ) \left (\sinh ^2(x) a^2+a^2\right )}\right )d(a \sinh (x))}{2 a^2 \left (a^2+b^2\right )}+\frac {4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)}{2 a^2 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )}}{4 \left (a^2+b^2\right )}-\frac {b-a \sinh (x)}{4 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \left (-\frac {b-a \sinh (x)}{4 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )^2}-\frac {\frac {4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)}{2 a^2 \left (a^2+b^2\right ) \left (a^2 \sinh ^2(x)+a^2\right )}+\frac {-\frac {4 a^2 b \log \left (a^2 \sinh ^2(x)+a^2\right )}{a^2+b^2}+\frac {8 a^2 b \log (a \sinh (x)+b)}{a^2+b^2}-\frac {\left (3 a^4-6 a^2 b^2-b^4\right ) \arctan (\sinh (x))}{a \left (a^2+b^2\right )}}{2 a^2 \left (a^2+b^2\right )}}{4 \left (a^2+b^2\right )}\right )\) |
Input:
Int[Sech[x]^5/(a + b*Csch[x]),x]
Output:
a^4*(-1/4*(b - a*Sinh[x])/((a^2 + b^2)*(a^2 + a^2*Sinh[x]^2)^2) - ((-(((3* a^4 - 6*a^2*b^2 - b^4)*ArcTan[Sinh[x]])/(a*(a^2 + b^2))) + (8*a^2*b*Log[b + a*Sinh[x]])/(a^2 + b^2) - (4*a^2*b*Log[a^2 + a^2*Sinh[x]^2])/(a^2 + b^2) )/(2*a^2*(a^2 + b^2)) + (4*a^2*b - a*(3*a^2 - b^2)*Sinh[x])/(2*a^2*(a^2 + b^2)*(a^2 + a^2*Sinh[x]^2)))/(4*(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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^(n + 1)*(c - d*x)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + a*d^2))), x] - Simp[d/(2*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b* x^2)^(p + 1)*(c*n - d*(n + 2*p + 4)*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(315\) vs. \(2(146)=292\).
Time = 22.98 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.11
| method | result | size |
| default | \(-\frac {a^{4} b \ln \left (-b \tanh \left (\frac {x}{2}\right )^{2}+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {\frac {2 \left (\left (-\frac {5}{8} a^{5}-\frac {3}{4} a^{3} b^{2}-\frac {1}{8} b^{4} a \right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (2 a^{4} b +3 a^{2} b^{3}+b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (\frac {3}{8} a^{5}+\frac {5}{4} a^{3} b^{2}+\frac {7}{8} b^{4} a \right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (2 a^{4} b +2 a^{2} b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (-\frac {3}{8} a^{5}-\frac {5}{4} a^{3} b^{2}-\frac {7}{8} b^{4} a \right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (2 a^{4} b +3 a^{2} b^{3}+b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {5}{8} a^{5}+\frac {3}{4} a^{3} b^{2}+\frac {1}{8} b^{4} a \right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{4}}+\frac {a \left (4 a^{3} b \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )+\left (3 a^{4}-6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{4}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}\) | \(316\) |
| risch | \(\frac {\left (3 a^{3} {\mathrm e}^{6 x}-a \,b^{2} {\mathrm e}^{6 x}-8 a^{2} b \,{\mathrm e}^{5 x}+11 a^{3} {\mathrm e}^{4 x}+7 a \,b^{2} {\mathrm e}^{4 x}-32 a^{2} b \,{\mathrm e}^{3 x}-16 b^{3} {\mathrm e}^{3 x}-11 a^{3} {\mathrm e}^{2 x}-7 a \,b^{2} {\mathrm e}^{2 x}-8 b \,{\mathrm e}^{x} a^{2}-3 a^{3}+a \,b^{2}\right ) {\mathrm e}^{x}}{4 \left (a^{2}+b^{2}\right )^{2} \left ({\mathrm e}^{2 x}+1\right )^{4}}-\frac {3 i a^{5} \ln \left ({\mathrm e}^{x}-i\right )}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 i a^{3} \ln \left ({\mathrm e}^{x}-i\right ) b^{2}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {i a \ln \left ({\mathrm e}^{x}-i\right ) b^{4}}{8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}}+\frac {a^{4} \ln \left ({\mathrm e}^{x}-i\right ) b}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 i a^{5} \ln \left ({\mathrm e}^{x}+i\right )}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 i a^{3} \ln \left ({\mathrm e}^{x}+i\right ) b^{2}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {i a \ln \left ({\mathrm e}^{x}+i\right ) b^{4}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {a^{4} \ln \left ({\mathrm e}^{x}+i\right ) b}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {a^{4} b \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(476\) |
Input:
int(sech(x)^5/(a+b*csch(x)),x,method=_RETURNVERBOSE)
Output:
-a^4*b/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*ln(-b*tanh(1/2*x)^2+2*a*tanh(1/2*x)+b )+2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*(((-5/8*a^5-3/4*a^3*b^2-1/8*b^4*a)*tanh( 1/2*x)^7+(2*a^4*b+3*a^2*b^3+b^5)*tanh(1/2*x)^6+(3/8*a^5+5/4*a^3*b^2+7/8*b^ 4*a)*tanh(1/2*x)^5+(2*a^4*b+2*a^2*b^3)*tanh(1/2*x)^4+(-3/8*a^5-5/4*a^3*b^2 -7/8*b^4*a)*tanh(1/2*x)^3+(2*a^4*b+3*a^2*b^3+b^5)*tanh(1/2*x)^2+(5/8*a^5+3 /4*a^3*b^2+1/8*b^4*a)*tanh(1/2*x))/(tanh(1/2*x)^2+1)^4+1/8*a*(4*a^3*b*ln(t anh(1/2*x)^2+1)+(3*a^4-6*a^2*b^2-b^4)*arctan(tanh(1/2*x))))
Leaf count of result is larger than twice the leaf count of optimal. 2778 vs. \(2 (143) = 286\).
Time = 0.14 (sec) , antiderivative size = 2778, normalized size of antiderivative = 18.52 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \] Input:
integrate(sech(x)^5/(a+b*csch(x)),x, algorithm="fricas")
Output:
1/4*((3*a^5 + 2*a^3*b^2 - a*b^4)*cosh(x)^7 + (3*a^5 + 2*a^3*b^2 - a*b^4)*s inh(x)^7 - 8*(a^4*b + a^2*b^3)*cosh(x)^6 - (8*a^4*b + 8*a^2*b^3 - 7*(3*a^5 + 2*a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^6 + (11*a^5 + 18*a^3*b^2 + 7*a*b^4) *cosh(x)^5 + (11*a^5 + 18*a^3*b^2 + 7*a*b^4 + 21*(3*a^5 + 2*a^3*b^2 - a*b^ 4)*cosh(x)^2 - 48*(a^4*b + a^2*b^3)*cosh(x))*sinh(x)^5 - 16*(2*a^4*b + 3*a ^2*b^3 + b^5)*cosh(x)^4 - (32*a^4*b + 48*a^2*b^3 + 16*b^5 - 35*(3*a^5 + 2* a^3*b^2 - a*b^4)*cosh(x)^3 + 120*(a^4*b + a^2*b^3)*cosh(x)^2 - 5*(11*a^5 + 18*a^3*b^2 + 7*a*b^4)*cosh(x))*sinh(x)^4 - (11*a^5 + 18*a^3*b^2 + 7*a*b^4 )*cosh(x)^3 - (11*a^5 + 18*a^3*b^2 + 7*a*b^4 - 35*(3*a^5 + 2*a^3*b^2 - a*b ^4)*cosh(x)^4 + 160*(a^4*b + a^2*b^3)*cosh(x)^3 - 10*(11*a^5 + 18*a^3*b^2 + 7*a*b^4)*cosh(x)^2 + 64*(2*a^4*b + 3*a^2*b^3 + b^5)*cosh(x))*sinh(x)^3 - 8*(a^4*b + a^2*b^3)*cosh(x)^2 + (21*(3*a^5 + 2*a^3*b^2 - a*b^4)*cosh(x)^5 - 8*a^4*b - 8*a^2*b^3 - 120*(a^4*b + a^2*b^3)*cosh(x)^4 + 10*(11*a^5 + 18 *a^3*b^2 + 7*a*b^4)*cosh(x)^3 - 96*(2*a^4*b + 3*a^2*b^3 + b^5)*cosh(x)^2 - 3*(11*a^5 + 18*a^3*b^2 + 7*a*b^4)*cosh(x))*sinh(x)^2 + ((3*a^5 - 6*a^3*b^ 2 - a*b^4)*cosh(x)^8 + 8*(3*a^5 - 6*a^3*b^2 - a*b^4)*cosh(x)*sinh(x)^7 + ( 3*a^5 - 6*a^3*b^2 - a*b^4)*sinh(x)^8 + 4*(3*a^5 - 6*a^3*b^2 - a*b^4)*cosh( x)^6 + 4*(3*a^5 - 6*a^3*b^2 - a*b^4 + 7*(3*a^5 - 6*a^3*b^2 - a*b^4)*cosh(x )^2)*sinh(x)^6 + 8*(7*(3*a^5 - 6*a^3*b^2 - a*b^4)*cosh(x)^3 + 3*(3*a^5 - 6 *a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^5 + 3*a^5 - 6*a^3*b^2 - a*b^4 + 6*(3...
\[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\operatorname {sech}^{5}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \] Input:
integrate(sech(x)**5/(a+b*csch(x)),x)
Output:
Integral(sech(x)**5/(a + b*csch(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (143) = 286\).
Time = 0.13 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.32 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {a^{4} b \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {a^{4} b \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right )} \arctan \left (e^{\left (-x\right )}\right )}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {8 \, a^{2} b e^{\left (-2 \, x\right )} + 8 \, a^{2} b e^{\left (-6 \, x\right )} - {\left (3 \, a^{3} - a b^{2}\right )} e^{\left (-x\right )} - {\left (11 \, a^{3} + 7 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (2 \, a^{2} b + b^{3}\right )} e^{\left (-4 \, x\right )} + {\left (11 \, a^{3} + 7 \, a b^{2}\right )} e^{\left (-5 \, x\right )} + {\left (3 \, a^{3} - a b^{2}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} \] Input:
integrate(sech(x)^5/(a+b*csch(x)),x, algorithm="maxima")
Output:
-a^4*b*log(-2*b*e^(-x) + a*e^(-2*x) - a)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^ 6) + a^4*b*log(e^(-2*x) + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/4*(3* a^5 - 6*a^3*b^2 - a*b^4)*arctan(e^(-x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 ) - 1/4*(8*a^2*b*e^(-2*x) + 8*a^2*b*e^(-6*x) - (3*a^3 - a*b^2)*e^(-x) - (1 1*a^3 + 7*a*b^2)*e^(-3*x) + 16*(2*a^2*b + b^3)*e^(-4*x) + (11*a^3 + 7*a*b^ 2)*e^(-5*x) + (3*a^3 - a*b^2)*e^(-7*x))/(a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*e^(-2*x) + 6*(a^4 + 2*a^2*b^2 + b^4)*e^(-4*x) + 4*(a^4 + 2*a^2*b^2 + b^4)*e^(-6*x) + (a^4 + 2*a^2*b^2 + b^4)*e^(-8*x))
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (143) = 286\).
Time = 0.13 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.49 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {a^{5} b \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} + \frac {a^{4} b \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right )}}{16 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {3 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 3 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 2 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 32 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 8 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 20 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )} + 24 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )} + 96 \, a^{4} b + 64 \, a^{2} b^{3} + 16 \, b^{5}}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} \] Input:
integrate(sech(x)^5/(a+b*csch(x)),x, algorithm="giac")
Output:
-a^5*b*log(abs(-a*(e^(-x) - e^x) + 2*b))/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a* b^6) + 1/2*a^4*b*log((e^(-x) - e^x)^2 + 4)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/16*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*(3*a^5 - 6*a^3*b^2 - a*b^4)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/4*(3*a^4*b*(e^(-x) - e^x)^ 4 + 3*a^5*(e^(-x) - e^x)^3 + 2*a^3*b^2*(e^(-x) - e^x)^3 - a*b^4*(e^(-x) - e^x)^3 + 32*a^4*b*(e^(-x) - e^x)^2 + 8*a^2*b^3*(e^(-x) - e^x)^2 + 20*a^5*( e^(-x) - e^x) + 24*a^3*b^2*(e^(-x) - e^x) + 4*a*b^4*(e^(-x) - e^x) + 96*a^ 4*b + 64*a^2*b^3 + 16*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*((e^(-x) - e^x)^2 + 4)^2)
Time = 6.87 (sec) , antiderivative size = 513, normalized size of antiderivative = 3.42 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {8\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}-\frac {6\,{\mathrm {e}}^x\,\left (a^3+a\,b^2\right )}{{\left (a^2+b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {2\,\left (a^2\,b+2\,b^3\right )}{{\left (a^2+b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (a^3+5\,a\,b^2\right )}{2\,{\left (a^2+b^2\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\frac {4\,b}{a^2+b^2}-\frac {4\,a\,{\mathrm {e}}^x}{a^2+b^2}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {\frac {2\,\left (a^4\,b+a^2\,b^3\right )}{{\left (a^2+b^2\right )}^3}-\frac {{\mathrm {e}}^x\,\left (3\,a^5+2\,a^3\,b^2-a\,b^4\right )}{4\,{\left (a^2+b^2\right )}^3}}{{\mathrm {e}}^{2\,x}+1}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (a\,b-a^2\,3{}\mathrm {i}\right )}{8\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (-3\,a^2+a\,b\,1{}\mathrm {i}\right )}{8\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {a^4\,b\,\ln \left (9\,a^{10}\,{\mathrm {e}}^{2\,x}-9\,a^{10}-a^2\,b^8-12\,a^4\,b^6-30\,a^6\,b^4-220\,a^8\,b^2+a^2\,b^8\,{\mathrm {e}}^{2\,x}+12\,a^4\,b^6\,{\mathrm {e}}^{2\,x}+30\,a^6\,b^4\,{\mathrm {e}}^{2\,x}+220\,a^8\,b^2\,{\mathrm {e}}^{2\,x}+2\,a\,b^9\,{\mathrm {e}}^x+18\,a^9\,b\,{\mathrm {e}}^x+24\,a^3\,b^7\,{\mathrm {e}}^x+60\,a^5\,b^5\,{\mathrm {e}}^x+440\,a^7\,b^3\,{\mathrm {e}}^x\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6} \] Input:
int(1/(cosh(x)^5*(a + b/sinh(x))),x)
Output:
((8*(a^2*b + b^3))/(a^2 + b^2)^2 - (6*exp(x)*(a*b^2 + a^3))/(a^2 + b^2)^2) /(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - ((2*(a^2*b + 2*b^3))/(a^2 + b^ 2)^2 - (exp(x)*(5*a*b^2 + a^3))/(2*(a^2 + b^2)^2))/(2*exp(2*x) + exp(4*x) + 1) - ((4*b)/(a^2 + b^2) - (4*a*exp(x))/(a^2 + b^2))/(4*exp(2*x) + 6*exp( 4*x) + 4*exp(6*x) + exp(8*x) + 1) - ((2*(a^4*b + a^2*b^3))/(a^2 + b^2)^3 - (exp(x)*(3*a^5 - a*b^4 + 2*a^3*b^2))/(4*(a^2 + b^2)^3))/(exp(2*x) + 1) + (log(exp(x) + 1i)*(a*b - a^2*3i))/(8*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) + (log(exp(x)*1i + 1)*(a*b*1i - 3*a^2))/(8*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) - (a^4*b*log(9*a^10*exp(2*x) - 9*a^10 - a^2*b^8 - 12*a^4*b^6 - 30*a^ 6*b^4 - 220*a^8*b^2 + a^2*b^8*exp(2*x) + 12*a^4*b^6*exp(2*x) + 30*a^6*b^4* exp(2*x) + 220*a^8*b^2*exp(2*x) + 2*a*b^9*exp(x) + 18*a^9*b*exp(x) + 24*a^ 3*b^7*exp(x) + 60*a^5*b^5*exp(x) + 440*a^7*b^3*exp(x)))/(a^6 + b^6 + 3*a^2 *b^4 + 3*a^4*b^2)
Time = 0.22 (sec) , antiderivative size = 861, normalized size of antiderivative = 5.74 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx =\text {Too large to display} \] Input:
int(sech(x)^5/(a+b*csch(x)),x)
Output:
(3*e**(8*x)*atan(e**x)*a**5 - 6*e**(8*x)*atan(e**x)*a**3*b**2 - e**(8*x)*a tan(e**x)*a*b**4 + 12*e**(6*x)*atan(e**x)*a**5 - 24*e**(6*x)*atan(e**x)*a* *3*b**2 - 4*e**(6*x)*atan(e**x)*a*b**4 + 18*e**(4*x)*atan(e**x)*a**5 - 36* e**(4*x)*atan(e**x)*a**3*b**2 - 6*e**(4*x)*atan(e**x)*a*b**4 + 12*e**(2*x) *atan(e**x)*a**5 - 24*e**(2*x)*atan(e**x)*a**3*b**2 - 4*e**(2*x)*atan(e**x )*a*b**4 + 3*atan(e**x)*a**5 - 6*atan(e**x)*a**3*b**2 - atan(e**x)*a*b**4 + 4*e**(8*x)*log(e**(2*x) + 1)*a**4*b - 4*e**(8*x)*log(e**(2*x)*a + 2*e**x *b - a)*a**4*b + 2*e**(8*x)*a**4*b + 2*e**(8*x)*a**2*b**3 + 3*e**(7*x)*a** 5 + 2*e**(7*x)*a**3*b**2 - e**(7*x)*a*b**4 + 16*e**(6*x)*log(e**(2*x) + 1) *a**4*b - 16*e**(6*x)*log(e**(2*x)*a + 2*e**x*b - a)*a**4*b + 11*e**(5*x)* a**5 + 18*e**(5*x)*a**3*b**2 + 7*e**(5*x)*a*b**4 + 24*e**(4*x)*log(e**(2*x ) + 1)*a**4*b - 24*e**(4*x)*log(e**(2*x)*a + 2*e**x*b - a)*a**4*b - 20*e** (4*x)*a**4*b - 36*e**(4*x)*a**2*b**3 - 16*e**(4*x)*b**5 - 11*e**(3*x)*a**5 - 18*e**(3*x)*a**3*b**2 - 7*e**(3*x)*a*b**4 + 16*e**(2*x)*log(e**(2*x) + 1)*a**4*b - 16*e**(2*x)*log(e**(2*x)*a + 2*e**x*b - a)*a**4*b - 3*e**x*a** 5 - 2*e**x*a**3*b**2 + e**x*a*b**4 + 4*log(e**(2*x) + 1)*a**4*b - 4*log(e* *(2*x)*a + 2*e**x*b - a)*a**4*b + 2*a**4*b + 2*a**2*b**3)/(4*(e**(8*x)*a** 6 + 3*e**(8*x)*a**4*b**2 + 3*e**(8*x)*a**2*b**4 + e**(8*x)*b**6 + 4*e**(6* x)*a**6 + 12*e**(6*x)*a**4*b**2 + 12*e**(6*x)*a**2*b**4 + 4*e**(6*x)*b**6 + 6*e**(4*x)*a**6 + 18*e**(4*x)*a**4*b**2 + 18*e**(4*x)*a**2*b**4 + 6*e...