Integrand size = 13, antiderivative size = 135 \[ \int \frac {\text {sech}^5(x)}{a+b \sinh (x)} \, dx=\frac {a \left (3 a^4+10 a^2 b^2+15 b^4\right ) \arctan (\sinh (x))}{8 \left (a^2+b^2\right )^3}-\frac {b^5 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac {b^5 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {\text {sech}^4(x) (b+a \sinh (x))}{4 \left (a^2+b^2\right )}+\frac {\text {sech}^2(x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2} \]
1/8*a*(3*a^4+10*a^2*b^2+15*b^4)*arctan(sinh(x))/(a^2+b^2)^3-b^5*ln(cosh(x) )/(a^2+b^2)^3+b^5*ln(a+b*sinh(x))/(a^2+b^2)^3+1/4*sech(x)^4*(b+a*sinh(x))/ (a^2+b^2)+1/8*sech(x)^2*(4*b^3+a*(3*a^2+7*b^2)*sinh(x))/(a^2+b^2)^2
Leaf count is larger than twice the leaf count of optimal. \(284\) vs. \(2(135)=270\).
Time = 0.32 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.10 \[ \int \frac {\text {sech}^5(x)}{a+b \sinh (x)} \, dx=\frac {-\left (\left (8 b^6+3 a^5 \sqrt {-b^2}+15 a b^4 \sqrt {-b^2}-10 a^3 \left (-b^2\right )^{3/2}\right ) \log \left (\sqrt {-b^2}-b \sinh (x)\right )\right )+16 b^6 \log (a+b \sinh (x))-8 b^6 \log \left (\sqrt {-b^2}+b \sinh (x)\right )+3 a^5 \sqrt {-b^2} \log \left (\sqrt {-b^2}+b \sinh (x)\right )+15 a b^4 \sqrt {-b^2} \log \left (\sqrt {-b^2}+b \sinh (x)\right )-10 a^3 \left (-b^2\right )^{3/2} \log \left (\sqrt {-b^2}+b \sinh (x)\right )+8 b^4 \left (a^2+b^2\right ) \text {sech}^2(x)+4 b^2 \left (a^2+b^2\right )^2 \text {sech}^4(x)+2 a b \left (3 a^4+10 a^2 b^2+7 b^4\right ) \text {sech}(x) \tanh (x)+4 a b \left (a^2+b^2\right )^2 \text {sech}^3(x) \tanh (x)}{16 b \left (a^2+b^2\right )^3} \]
(-((8*b^6 + 3*a^5*Sqrt[-b^2] + 15*a*b^4*Sqrt[-b^2] - 10*a^3*(-b^2)^(3/2))* Log[Sqrt[-b^2] - b*Sinh[x]]) + 16*b^6*Log[a + b*Sinh[x]] - 8*b^6*Log[Sqrt[ -b^2] + b*Sinh[x]] + 3*a^5*Sqrt[-b^2]*Log[Sqrt[-b^2] + b*Sinh[x]] + 15*a*b ^4*Sqrt[-b^2]*Log[Sqrt[-b^2] + b*Sinh[x]] - 10*a^3*(-b^2)^(3/2)*Log[Sqrt[- b^2] + b*Sinh[x]] + 8*b^4*(a^2 + b^2)*Sech[x]^2 + 4*b^2*(a^2 + b^2)^2*Sech [x]^4 + 2*a*b*(3*a^4 + 10*a^2*b^2 + 7*b^4)*Sech[x]*Tanh[x] + 4*a*b*(a^2 + b^2)^2*Sech[x]^3*Tanh[x])/(16*b*(a^2 + b^2)^3)
Time = 0.44 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.59, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3042, 3147, 25, 496, 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.
| \(\displaystyle \int \frac {\text {sech}^5(x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (i x)^5 (a-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle -b^5 \int -\frac {1}{(a+b \sinh (x)) \left (\sinh ^2(x) b^2+b^2\right )^3}d(b \sinh (x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b^5 \int \frac {1}{(a+b \sinh (x)) \left (\sinh ^2(x) b^2+b^2\right )^3}d(b \sinh (x))\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle -b^5 \left (\frac {\int -\frac {3 a^2+3 b \sinh (x) a+4 b^2}{(a+b \sinh (x)) \left (\sinh ^2(x) b^2+b^2\right )^2}d(b \sinh (x))}{4 b^2 \left (a^2+b^2\right )}-\frac {a b \sinh (x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b^5 \left (-\frac {\int \frac {3 a^2+3 b \sinh (x) a+4 b^2}{(a+b \sinh (x)) \left (\sinh ^2(x) b^2+b^2\right )^2}d(b \sinh (x))}{4 b^2 \left (a^2+b^2\right )}-\frac {a b \sinh (x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )^2}\right )\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle -b^5 \left (-\frac {\frac {a b \left (3 a^2+7 b^2\right ) \sinh (x)+4 b^4}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )}-\frac {\int -\frac {3 a^4+7 b^2 a^2+b \left (3 a^2+7 b^2\right ) \sinh (x) a+8 b^4}{(a+b \sinh (x)) \left (\sinh ^2(x) b^2+b^2\right )}d(b \sinh (x))}{2 b^2 \left (a^2+b^2\right )}}{4 b^2 \left (a^2+b^2\right )}-\frac {a b \sinh (x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b^5 \left (-\frac {\frac {\int \frac {3 a^4+7 b^2 a^2+b \left (3 a^2+7 b^2\right ) \sinh (x) a+8 b^4}{(a+b \sinh (x)) \left (\sinh ^2(x) b^2+b^2\right )}d(b \sinh (x))}{2 b^2 \left (a^2+b^2\right )}+\frac {a b \left (3 a^2+7 b^2\right ) \sinh (x)+4 b^4}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )}}{4 b^2 \left (a^2+b^2\right )}-\frac {a b \sinh (x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )^2}\right )\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle -b^5 \left (-\frac {\frac {\int \left (\frac {8 b^4}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {3 a^5+10 b^2 a^3+15 b^4 a-8 b^5 \sinh (x)}{\left (a^2+b^2\right ) \left (\sinh ^2(x) b^2+b^2\right )}\right )d(b \sinh (x))}{2 b^2 \left (a^2+b^2\right )}+\frac {a b \left (3 a^2+7 b^2\right ) \sinh (x)+4 b^4}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )}}{4 b^2 \left (a^2+b^2\right )}-\frac {a b \sinh (x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -b^5 \left (-\frac {a b \sinh (x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )^2}-\frac {\frac {a b \left (3 a^2+7 b^2\right ) \sinh (x)+4 b^4}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(x)+b^2\right )}+\frac {-\frac {4 b^4 \log \left (b^2 \sinh ^2(x)+b^2\right )}{a^2+b^2}+\frac {8 b^4 \log (a+b \sinh (x))}{a^2+b^2}+\frac {a \left (3 a^4+10 a^2 b^2+15 b^4\right ) \arctan (\sinh (x))}{b \left (a^2+b^2\right )}}{2 b^2 \left (a^2+b^2\right )}}{4 b^2 \left (a^2+b^2\right )}\right )\) |
-(b^5*(-1/4*(b^2 + a*b*Sinh[x])/(b^2*(a^2 + b^2)*(b^2 + b^2*Sinh[x]^2)^2) - (((a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*ArcTan[Sinh[x]])/(b*(a^2 + b^2)) + (8 *b^4*Log[a + b*Sinh[x]])/(a^2 + b^2) - (4*b^4*Log[b^2 + b^2*Sinh[x]^2])/(a ^2 + b^2))/(2*b^2*(a^2 + b^2)) + (4*b^4 + a*b*(3*a^2 + 7*b^2)*Sinh[x])/(2* b^2*(a^2 + b^2)*(b^2 + b^2*Sinh[x]^2)))/(4*b^2*(a^2 + b^2))))
3.2.98.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
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 _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(312\) vs. \(2(129)=258\).
Time = 92.65 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.32
| method | result | size |
| default | \(\frac {b^{5} \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {\frac {2 \left (\left (-\frac {5}{8} a^{5}-\frac {7}{4} a^{3} b^{2}-\frac {9}{8} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (-a^{4} b -3 a^{2} b^{3}-2 b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (\frac {3}{8} a^{5}+\frac {1}{4} a^{3} b^{2}-\frac {1}{8} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-2 a^{2} b^{3}-2 b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (-\frac {3}{8} a^{5}-\frac {1}{4} a^{3} b^{2}+\frac {1}{8} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{4} b -3 a^{2} b^{3}-2 b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {5}{8} a^{5}+\frac {7}{4} a^{3} b^{2}+\frac {9}{8} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{4}}-b^{5} \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+\frac {\left (3 a^{5}+10 a^{3} b^{2}+15 a \,b^{4}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(313\) |
| risch | \(\frac {\left (3 a^{3} {\mathrm e}^{6 x}+7 \,{\mathrm e}^{6 x} a \,b^{2}+8 b^{3} {\mathrm e}^{5 x}+11 a^{3} {\mathrm e}^{4 x}+15 \,{\mathrm e}^{4 x} a \,b^{2}+16 a^{2} b \,{\mathrm e}^{3 x}+32 \,{\mathrm e}^{3 x} b^{3}-11 a^{3} {\mathrm e}^{2 x}-15 a \,{\mathrm e}^{2 x} b^{2}+8 b^{3} {\mathrm e}^{x}-3 a^{3}-7 a \,b^{2}\right ) {\mathrm e}^{x}}{4 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+{\mathrm e}^{2 x}\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right ) a^{5}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 i \ln \left ({\mathrm e}^{x}+i\right ) a^{3} b^{2}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {15 i \ln \left ({\mathrm e}^{x}+i\right ) a \,b^{4}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\ln \left ({\mathrm e}^{x}+i\right ) b^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right ) a^{5}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {5 i \ln \left ({\mathrm e}^{x}-i\right ) a^{3} b^{2}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {15 i \ln \left ({\mathrm e}^{x}-i\right ) a \,b^{4}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\ln \left ({\mathrm e}^{x}-i\right ) b^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {b^{5} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(481\) |
b^5/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*ln(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a)+2/ (a^6+3*a^4*b^2+3*a^2*b^4+b^6)*(((-5/8*a^5-7/4*a^3*b^2-9/8*a*b^4)*tanh(1/2* x)^7+(-a^4*b-3*a^2*b^3-2*b^5)*tanh(1/2*x)^6+(3/8*a^5+1/4*a^3*b^2-1/8*a*b^4 )*tanh(1/2*x)^5+(-2*a^2*b^3-2*b^5)*tanh(1/2*x)^4+(-3/8*a^5-1/4*a^3*b^2+1/8 *a*b^4)*tanh(1/2*x)^3+(-a^4*b-3*a^2*b^3-2*b^5)*tanh(1/2*x)^2+(5/8*a^5+7/4* a^3*b^2+9/8*a*b^4)*tanh(1/2*x))/(1+tanh(1/2*x)^2)^4-1/2*b^5*ln(1+tanh(1/2* x)^2)+1/8*(3*a^5+10*a^3*b^2+15*a*b^4)*arctan(tanh(1/2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 2707 vs. \(2 (129) = 258\).
Time = 0.37 (sec) , antiderivative size = 2707, normalized size of antiderivative = 20.05 \[ \int \frac {\text {sech}^5(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \]
1/4*((3*a^5 + 10*a^3*b^2 + 7*a*b^4)*cosh(x)^7 + (3*a^5 + 10*a^3*b^2 + 7*a* b^4)*sinh(x)^7 + 8*(a^2*b^3 + b^5)*cosh(x)^6 + (8*a^2*b^3 + 8*b^5 + 7*(3*a ^5 + 10*a^3*b^2 + 7*a*b^4)*cosh(x))*sinh(x)^6 + (11*a^5 + 26*a^3*b^2 + 15* a*b^4)*cosh(x)^5 + (11*a^5 + 26*a^3*b^2 + 15*a*b^4 + 21*(3*a^5 + 10*a^3*b^ 2 + 7*a*b^4)*cosh(x)^2 + 48*(a^2*b^3 + b^5)*cosh(x))*sinh(x)^5 + 16*(a^4*b + 3*a^2*b^3 + 2*b^5)*cosh(x)^4 + (16*a^4*b + 48*a^2*b^3 + 32*b^5 + 35*(3* a^5 + 10*a^3*b^2 + 7*a*b^4)*cosh(x)^3 + 120*(a^2*b^3 + b^5)*cosh(x)^2 + 5* (11*a^5 + 26*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^4 - (11*a^5 + 26*a^3*b^2 + 15*a*b^4)*cosh(x)^3 - (11*a^5 + 26*a^3*b^2 + 15*a*b^4 - 35*(3*a^5 + 10* a^3*b^2 + 7*a*b^4)*cosh(x)^4 - 160*(a^2*b^3 + b^5)*cosh(x)^3 - 10*(11*a^5 + 26*a^3*b^2 + 15*a*b^4)*cosh(x)^2 - 64*(a^4*b + 3*a^2*b^3 + 2*b^5)*cosh(x ))*sinh(x)^3 + 8*(a^2*b^3 + b^5)*cosh(x)^2 + (21*(3*a^5 + 10*a^3*b^2 + 7*a *b^4)*cosh(x)^5 + 8*a^2*b^3 + 8*b^5 + 120*(a^2*b^3 + b^5)*cosh(x)^4 + 10*( 11*a^5 + 26*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 96*(a^4*b + 3*a^2*b^3 + 2*b^5) *cosh(x)^2 - 3*(11*a^5 + 26*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^2 + ((3*a ^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 8*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)* cosh(x)*sinh(x)^7 + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*sinh(x)^8 + 4*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 4*(3*a^5 + 10*a^3*b^2 + 15*a*b^4 + 7*( 3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*a^5 + 10*a^3 *b^2 + 15*a*b^4)*cosh(x)^3 + 3*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x))...
\[ \int \frac {\text {sech}^5(x)}{a+b \sinh (x)} \, dx=\int \frac {\operatorname {sech}^{5}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (129) = 258\).
Time = 0.30 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.56 \[ \int \frac {\text {sech}^5(x)}{a+b \sinh (x)} \, dx=\frac {b^{5} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {b^{5} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \arctan \left (e^{\left (-x\right )}\right )}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {8 \, b^{3} e^{\left (-2 \, x\right )} + 8 \, b^{3} e^{\left (-6 \, x\right )} + {\left (3 \, a^{3} + 7 \, a b^{2}\right )} e^{\left (-x\right )} + {\left (11 \, a^{3} + 15 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (a^{2} b + 2 \, b^{3}\right )} e^{\left (-4 \, x\right )} - {\left (11 \, a^{3} + 15 \, a b^{2}\right )} e^{\left (-5 \, x\right )} - {\left (3 \, a^{3} + 7 \, a b^{2}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} \]
b^5*log(-2*a*e^(-x) + b*e^(-2*x) - b)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - b^5*log(e^(-2*x) + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/4*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*arctan(e^(-x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/4*(8*b^3*e^(-2*x) + 8*b^3*e^(-6*x) + (3*a^3 + 7*a*b^2)*e^(-x) + (11*a ^3 + 15*a*b^2)*e^(-3*x) + 16*(a^2*b + 2*b^3)*e^(-4*x) - (11*a^3 + 15*a*b^2 )*e^(-5*x) - (3*a^3 + 7*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. 369 vs. \(2 (129) = 258\).
Time = 0.28 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.73 \[ \int \frac {\text {sech}^5(x)}{a+b \sinh (x)} \, dx=\frac {b^{6} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {b^{5} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )}}{16 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {3 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} - 3 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 10 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 7 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 8 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 32 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 20 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )} - 56 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} - 36 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )} + 16 \, a^{4} b + 64 \, a^{2} b^{3} + 96 \, b^{5}}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} \]
b^6*log(abs(-b*(e^(-x) - e^x) + 2*a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 ) - 1/2*b^5*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 + 10*a^3*b^2 + 15* a*b^4)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/4*(3*b^5*(e^(-x) - e^x)^4 - 3*a^5*(e^(-x) - e^x)^3 - 10*a^3*b^2*(e^(-x) - e^x)^3 - 7*a*b^4*(e^(-x) - e^x)^3 + 8*a^2*b^3*(e^(-x) - e^x)^2 + 32*b^5*(e^(-x) - e^x)^2 - 20*a^5*(e^ (-x) - e^x) - 56*a^3*b^2*(e^(-x) - e^x) - 36*a*b^4*(e^(-x) - e^x) + 16*a^4 *b + 64*a^2*b^3 + 96*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*((e^(-x) - e^x)^2 + 4)^2)
Time = 5.73 (sec) , antiderivative size = 548, normalized size of antiderivative = 4.06 \[ \int \frac {\text {sech}^5(x)}{a+b \sinh (x)} \, dx=\frac {\frac {2\,\left (2\,a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (3\,a\,b^2-a^3\right )}{2\,{\left (a^2+b^2\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\frac {8\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}+\frac {6\,{\mathrm {e}}^x\,\left (a^3+a\,b^2\right )}{{\left (a^2+b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}+\frac {\frac {4\,b}{a^2+b^2}+\frac {4\,a\,{\mathrm {e}}^x}{a^2+b^2}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\frac {2\,\left (a^2\,b^3+b^5\right )}{{\left (a^2+b^2\right )}^3}+\frac {{\mathrm {e}}^x\,\left (3\,a^5+10\,a^3\,b^2+7\,a\,b^4\right )}{4\,{\left (a^2+b^2\right )}^3}}{{\mathrm {e}}^{2\,x}+1}+\frac {b^5\,\ln \left (256\,b^{11}\,{\mathrm {e}}^{2\,x}-9\,a^{10}\,b-256\,b^{11}-225\,a^2\,b^9-300\,a^4\,b^7-190\,a^6\,b^5-60\,a^8\,b^3+18\,a^{11}\,{\mathrm {e}}^x+225\,a^2\,b^9\,{\mathrm {e}}^{2\,x}+300\,a^4\,b^7\,{\mathrm {e}}^{2\,x}+190\,a^6\,b^5\,{\mathrm {e}}^{2\,x}+60\,a^8\,b^3\,{\mathrm {e}}^{2\,x}+512\,a\,b^{10}\,{\mathrm {e}}^x+9\,a^{10}\,b\,{\mathrm {e}}^{2\,x}+450\,a^3\,b^8\,{\mathrm {e}}^x+600\,a^5\,b^6\,{\mathrm {e}}^x+380\,a^7\,b^4\,{\mathrm {e}}^x+120\,a^9\,b^2\,{\mathrm {e}}^x\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}-\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (-a^2\,3{}\mathrm {i}+9\,a\,b+b^2\,8{}\mathrm {i}\right )}{8\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}-\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (-3\,a^2+a\,b\,9{}\mathrm {i}+8\,b^2\right )}{8\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )} \]
((2*(2*a^2*b + b^3))/(a^2 + b^2)^2 - (exp(x)*(3*a*b^2 - a^3))/(2*(a^2 + b^ 2)^2))/(2*exp(2*x) + exp(4*x) + 1) - ((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) + ((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*(b^5 + a^2*b^3))/(a^2 + b^2)^3 + ( exp(x)*(7*a*b^4 + 3*a^5 + 10*a^3*b^2))/(4*(a^2 + b^2)^3))/(exp(2*x) + 1) + (b^5*log(256*b^11*exp(2*x) - 9*a^10*b - 256*b^11 - 225*a^2*b^9 - 300*a^4* b^7 - 190*a^6*b^5 - 60*a^8*b^3 + 18*a^11*exp(x) + 225*a^2*b^9*exp(2*x) + 3 00*a^4*b^7*exp(2*x) + 190*a^6*b^5*exp(2*x) + 60*a^8*b^3*exp(2*x) + 512*a*b ^10*exp(x) + 9*a^10*b*exp(2*x) + 450*a^3*b^8*exp(x) + 600*a^5*b^6*exp(x) + 380*a^7*b^4*exp(x) + 120*a^9*b^2*exp(x)))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4* b^2) - (log(exp(x)*1i + 1)*(9*a*b - a^2*3i + b^2*8i))/(8*(3*a*b^2 - a^2*b* 3i - a^3 + b^3*1i)) - (log(exp(x) + 1i)*(a*b*9i - 3*a^2 + 8*b^2))/(8*(a*b^ 2*3i - 3*a^2*b - a^3*1i + b^3))