Integrand size = 13, antiderivative size = 146 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=-\frac {2 b^6 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (15 b^5+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^3} \] Output:
-2*b^6*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(7/2)+sech(x)^ 5*(b+a*sinh(x))/(5*a^2+5*b^2)+1/15*sech(x)^3*(5*b^3+a*(4*a^2+9*b^2)*sinh(x ))/(a^2+b^2)^2+1/15*sech(x)*(15*b^5+a*(8*a^4+26*a^2*b^2+33*b^4)*sinh(x))/( a^2+b^2)^3
Time = 0.35 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\frac {\frac {30 b^6 \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+15 b^5 \text {sech}(x)+3 \left (a^2+b^2\right )^2 \text {sech}^5(x) (b+a \sinh (x))+\left (a^2+b^2\right ) \text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \tanh (x)}{15 \left (a^2+b^2\right )^3} \] Input:
Integrate[Sech[x]^6/(a + b*Sinh[x]),x]
Output:
((30*b^6*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + 15 *b^5*Sech[x] + 3*(a^2 + b^2)^2*Sech[x]^5*(b + a*Sinh[x]) + (a^2 + b^2)*Sec h[x]^3*(5*b^3 + a*(4*a^2 + 9*b^2)*Sinh[x]) + a*(8*a^4 + 26*a^2*b^2 + 33*b^ 4)*Tanh[x])/(15*(a^2 + b^2)^3)
Time = 1.01 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.21, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3175, 25, 3042, 3345, 25, 3042, 3345, 27, 3042, 3139, 1083, 219}
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.
| \(\displaystyle \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (i x)^6 (a-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 3175 |
| \(\displaystyle \frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}-\frac {\int -\frac {\text {sech}^4(x) \left (4 a^2+4 b \sinh (x) a+5 b^2\right )}{a+b \sinh (x)}dx}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\text {sech}^4(x) \left (4 a^2+4 b \sinh (x) a+5 b^2\right )}{a+b \sinh (x)}dx}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}+\frac {\int \frac {4 a^2-4 i b \sin (i x) a+5 b^2}{\cos (i x)^4 (a-i b \sin (i x))}dx}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}-\frac {\int -\frac {\text {sech}^2(x) \left (8 a^4+18 b^2 a^2+2 b \left (4 a^2+9 b^2\right ) \sinh (x) a+15 b^4\right )}{a+b \sinh (x)}dx}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\text {sech}^2(x) \left (8 a^4+18 b^2 a^2+2 b \left (4 a^2+9 b^2\right ) \sinh (x) a+15 b^4\right )}{a+b \sinh (x)}dx}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}+\frac {\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}+\frac {\int \frac {8 a^4+18 b^2 a^2-2 i b \left (4 a^2+9 b^2\right ) \sin (i x) a+15 b^4}{\cos (i x)^2 (a-i b \sin (i x))}dx}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\frac {\text {sech}(x) \left (a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)+15 b^5\right )}{a^2+b^2}-\frac {\int -\frac {15 b^6}{a+b \sinh (x)}dx}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {15 b^6 \int \frac {1}{a+b \sinh (x)}dx}{a^2+b^2}+\frac {\text {sech}(x) \left (a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)+15 b^5\right )}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}+\frac {\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}+\frac {\frac {\text {sech}(x) \left (a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)+15 b^5\right )}{a^2+b^2}+\frac {15 b^6 \int \frac {1}{a-i b \sin (i x)}dx}{a^2+b^2}}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {30 b^6 \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a^2+b^2}+\frac {\text {sech}(x) \left (a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)+15 b^5\right )}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {\text {sech}(x) \left (a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)+15 b^5\right )}{a^2+b^2}-\frac {60 b^6 \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}+\frac {\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{3 \left (a^2+b^2\right )}+\frac {\frac {\text {sech}(x) \left (a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)+15 b^5\right )}{a^2+b^2}-\frac {30 b^6 \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}}{3 \left (a^2+b^2\right )}}{5 \left (a^2+b^2\right )}\) |
Input:
Int[Sech[x]^6/(a + b*Sinh[x]),x]
Output:
(Sech[x]^5*(b + a*Sinh[x]))/(5*(a^2 + b^2)) + ((Sech[x]^3*(5*b^3 + a*(4*a^ 2 + 9*b^2)*Sinh[x]))/(3*(a^2 + b^2)) + ((-30*b^6*ArcTanh[(2*b - 2*a*Tanh[x /2])/(2*Sqrt[a^2 + b^2])])/(a^2 + b^2)^(3/2) + (Sech[x]*(15*b^5 + a*(8*a^4 + 26*a^2*b^2 + 33*b^4)*Sinh[x]))/(a^2 + b^2))/(3*(a^2 + b^2)))/(5*(a^2 + b^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[(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_) + (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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ (m + 1)*((b - a*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2* (a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m* (a^2*(p + 2) - b^2*(m + p + 2) + a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; F reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegersQ [2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(349\) vs. \(2(137)=274\).
Time = 150.02 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.40
| method | result | size |
| default | \(\frac {2 b^{6} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \sqrt {a^{2}+b^{2}}}-\frac {2 \left (\left (-a^{5}-3 a^{3} b^{2}-3 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{9}+\left (-a^{4} b -3 a^{2} b^{3}-3 b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{8}+\left (-\frac {4}{3} a^{5}-\frac {16}{3} a^{3} b^{2}-8 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (-2 a^{2} b^{3}-6 b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (-\frac {58}{15} a^{5}-\frac {166}{15} a^{3} b^{2}-\frac {66}{5} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-2 a^{4} b -\frac {16}{3} a^{2} b^{3}-\frac {28}{3} b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (-\frac {4}{3} a^{5}-\frac {16}{3} a^{3} b^{2}-8 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-\frac {2}{3} a^{2} b^{3}-\frac {14}{3} b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (-a^{5}-3 a^{3} b^{2}-3 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )-\frac {a^{4} b}{5}-\frac {11 a^{2} b^{3}}{15}-\frac {23 b^{5}}{15}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{5}}\) | \(350\) |
| risch | \(-\frac {2 \left (-15 b^{5} {\mathrm e}^{9 x}+15 a \,b^{4} {\mathrm e}^{8 x}-20 a^{2} b^{3} {\mathrm e}^{7 x}-80 b^{5} {\mathrm e}^{7 x}+30 a^{3} b^{2} {\mathrm e}^{6 x}+90 a \,b^{4} {\mathrm e}^{6 x}-48 a^{4} b \,{\mathrm e}^{5 x}-136 a^{2} b^{3} {\mathrm e}^{5 x}-178 \,{\mathrm e}^{5 x} b^{5}+80 a^{5} {\mathrm e}^{4 x}+230 a^{3} b^{2} {\mathrm e}^{4 x}+240 a \,{\mathrm e}^{4 x} b^{4}-20 \,{\mathrm e}^{3 x} a^{2} b^{3}-80 \,{\mathrm e}^{3 x} b^{5}+40 a^{5} {\mathrm e}^{2 x}+130 \,{\mathrm e}^{2 x} a^{3} b^{2}+150 \,{\mathrm e}^{2 x} a \,b^{4}-15 \,{\mathrm e}^{x} b^{5}+8 a^{5}+26 a^{3} b^{2}+33 a \,b^{4}\right )}{15 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 x}+1\right )^{5} \left (a^{2}+b^{2}\right )}+\frac {b^{6} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {b^{6} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\) | \(390\) |
Input:
int(sech(x)^6/(a+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
2*b^6/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(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^6+3*a^4*b^2+3*a^2*b^4+b^6)*((-a^5-3*a^3* b^2-3*a*b^4)*tanh(1/2*x)^9+(-a^4*b-3*a^2*b^3-3*b^5)*tanh(1/2*x)^8+(-4/3*a^ 5-16/3*a^3*b^2-8*a*b^4)*tanh(1/2*x)^7+(-2*a^2*b^3-6*b^5)*tanh(1/2*x)^6+(-5 8/15*a^5-166/15*a^3*b^2-66/5*a*b^4)*tanh(1/2*x)^5+(-2*a^4*b-16/3*a^2*b^3-2 8/3*b^5)*tanh(1/2*x)^4+(-4/3*a^5-16/3*a^3*b^2-8*a*b^4)*tanh(1/2*x)^3+(-2/3 *a^2*b^3-14/3*b^5)*tanh(1/2*x)^2+(-a^5-3*a^3*b^2-3*a*b^4)*tanh(1/2*x)-1/5* a^4*b-11/15*a^2*b^3-23/15*b^5)/(1+tanh(1/2*x)^2)^5
Leaf count of result is larger than twice the leaf count of optimal. 3175 vs. \(2 (136) = 272\).
Time = 0.11 (sec) , antiderivative size = 3175, normalized size of antiderivative = 21.75 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \] Input:
integrate(sech(x)^6/(a+b*sinh(x)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\int \frac {\operatorname {sech}^{6}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \] Input:
integrate(sech(x)**6/(a+b*sinh(x)),x)
Output:
Integral(sech(x)**6/(a + b*sinh(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (136) = 272\).
Time = 0.13 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.00 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\frac {b^{6} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, b^{5} e^{\left (-x\right )} + 15 \, a b^{4} e^{\left (-8 \, x\right )} + 15 \, b^{5} e^{\left (-9 \, x\right )} + 8 \, a^{5} + 26 \, a^{3} b^{2} + 33 \, a b^{4} + 10 \, {\left (4 \, a^{5} + 13 \, a^{3} b^{2} + 15 \, a b^{4}\right )} e^{\left (-2 \, x\right )} + 20 \, {\left (a^{2} b^{3} + 4 \, b^{5}\right )} e^{\left (-3 \, x\right )} + 10 \, {\left (8 \, a^{5} + 23 \, a^{3} b^{2} + 24 \, a b^{4}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (24 \, a^{4} b + 68 \, a^{2} b^{3} + 89 \, b^{5}\right )} e^{\left (-5 \, x\right )} + 30 \, {\left (a^{3} b^{2} + 3 \, a b^{4}\right )} e^{\left (-6 \, x\right )} + 20 \, {\left (a^{2} b^{3} + 4 \, b^{5}\right )} e^{\left (-7 \, x\right )}\right )}}{15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6} + 5 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-4 \, x\right )} + 10 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-6 \, x\right )} + 5 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-8 \, x\right )} + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-10 \, x\right )}\right )}} \] Input:
integrate(sech(x)^6/(a+b*sinh(x)),x, algorithm="maxima")
Output:
b^6*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2))) /((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2/15*(15*b^5*e^(- x) + 15*a*b^4*e^(-8*x) + 15*b^5*e^(-9*x) + 8*a^5 + 26*a^3*b^2 + 33*a*b^4 + 10*(4*a^5 + 13*a^3*b^2 + 15*a*b^4)*e^(-2*x) + 20*(a^2*b^3 + 4*b^5)*e^(-3* x) + 10*(8*a^5 + 23*a^3*b^2 + 24*a*b^4)*e^(-4*x) + 2*(24*a^4*b + 68*a^2*b^ 3 + 89*b^5)*e^(-5*x) + 30*(a^3*b^2 + 3*a*b^4)*e^(-6*x) + 20*(a^2*b^3 + 4*b ^5)*e^(-7*x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + 5*(a^6 + 3*a^4*b^2 + 3* a^2*b^4 + b^6)*e^(-2*x) + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*e^(-4*x) + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*e^(-6*x) + 5*(a^6 + 3*a^4*b^2 + 3 *a^2*b^4 + b^6)*e^(-8*x) + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*e^(-10*x))
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (136) = 272\).
Time = 0.13 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.21 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\frac {b^{6} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, b^{5} e^{\left (9 \, x\right )} - 15 \, a b^{4} e^{\left (8 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (7 \, x\right )} + 80 \, b^{5} e^{\left (7 \, x\right )} - 30 \, a^{3} b^{2} e^{\left (6 \, x\right )} - 90 \, a b^{4} e^{\left (6 \, x\right )} + 48 \, a^{4} b e^{\left (5 \, x\right )} + 136 \, a^{2} b^{3} e^{\left (5 \, x\right )} + 178 \, b^{5} e^{\left (5 \, x\right )} - 80 \, a^{5} e^{\left (4 \, x\right )} - 230 \, a^{3} b^{2} e^{\left (4 \, x\right )} - 240 \, a b^{4} e^{\left (4 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (3 \, x\right )} + 80 \, b^{5} e^{\left (3 \, x\right )} - 40 \, a^{5} e^{\left (2 \, x\right )} - 130 \, a^{3} b^{2} e^{\left (2 \, x\right )} - 150 \, a b^{4} e^{\left (2 \, x\right )} + 15 \, b^{5} e^{x} - 8 \, a^{5} - 26 \, a^{3} b^{2} - 33 \, a b^{4}\right )}}{15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \] Input:
integrate(sech(x)^6/(a+b*sinh(x)),x, algorithm="giac")
Output:
b^6*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^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2/15* (15*b^5*e^(9*x) - 15*a*b^4*e^(8*x) + 20*a^2*b^3*e^(7*x) + 80*b^5*e^(7*x) - 30*a^3*b^2*e^(6*x) - 90*a*b^4*e^(6*x) + 48*a^4*b*e^(5*x) + 136*a^2*b^3*e^ (5*x) + 178*b^5*e^(5*x) - 80*a^5*e^(4*x) - 230*a^3*b^2*e^(4*x) - 240*a*b^4 *e^(4*x) + 20*a^2*b^3*e^(3*x) + 80*b^5*e^(3*x) - 40*a^5*e^(2*x) - 130*a^3* b^2*e^(2*x) - 150*a*b^4*e^(2*x) + 15*b^5*e^x - 8*a^5 - 26*a^3*b^2 - 33*a*b ^4)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(e^(2*x) + 1)^5)
Time = 3.06 (sec) , antiderivative size = 1010, normalized size of antiderivative = 6.92 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx =\text {Too large to display} \] Input:
int(1/(cosh(x)^6*(a + b*sinh(x))),x)
Output:
((2*b^5*exp(x))/(a^2 + b^2)^3 - (2*a*b^4)/(a^2 + b^2)^3)/(exp(2*x) + 1) - ((8*(3*a*b^2 + 4*a^3))/(3*(a^2 + b^2)^2) - (8*exp(x)*(12*a^2*b + 7*b^3))/( 15*(a^2 + b^2)^2))/(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - ((4*(a*b^4 + a^3*b^2))/(a^2 + b^2)^3 - (8*exp(x)*(b^5 + a^2*b^3))/(3*(a^2 + b^2)^3))/( 2*exp(2*x) + exp(4*x) + 1) - ((32*a)/(5*(a^2 + b^2)) - (32*b*exp(x))/(5*(a ^2 + b^2)))/(5*exp(2*x) + 10*exp(4*x) + 10*exp(6*x) + 5*exp(8*x) + exp(10* x) + 1) + ((16*(a*b^2 + a^3))/(a^2 + b^2)^2 - (64*exp(x)*(a^2*b + b^3))/(5 *(a^2 + b^2)^2))/(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1) - ( 2*atan((exp(x)*((2*b^4)/((b^12)^(1/2)*(a^2 + b^2)^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*a*(a^7*(b^12)^(1/2) + 3*a^3*b^4*(b^12)^(1/2) + 3*a^5*b ^2*(b^12)^(1/2) + a*b^6*(b^12)^(1/2)))/(b^8*(-(a^2 + b^2)^7)^(1/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(- a^14 - b^14 - 7*a^2*b^12 - 21*a^4*b^10 - 3 5*a^6*b^8 - 35*a^8*b^6 - 21*a^10*b^4 - 7*a^12*b^2)^(1/2))) - (2*a*(b^7*(b^ 12)^(1/2) + 3*a^2*b^5*(b^12)^(1/2) + 3*a^4*b^3*(b^12)^(1/2) + a^6*b*(b^12) ^(1/2)))/(b^8*(-(a^2 + b^2)^7)^(1/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*( - a^14 - b^14 - 7*a^2*b^12 - 21*a^4*b^10 - 35*a^6*b^8 - 35*a^8*b^6 - 21*a^ 10*b^4 - 7*a^12*b^2)^(1/2)))*((b^7*(- a^14 - b^14 - 7*a^2*b^12 - 21*a^4*b^ 10 - 35*a^6*b^8 - 35*a^8*b^6 - 21*a^10*b^4 - 7*a^12*b^2)^(1/2))/2 + (3*a^2 *b^5*(- a^14 - b^14 - 7*a^2*b^12 - 21*a^4*b^10 - 35*a^6*b^8 - 35*a^8*b^6 - 21*a^10*b^4 - 7*a^12*b^2)^(1/2))/2 + (3*a^4*b^3*(- a^14 - b^14 - 7*a^2...
Time = 0.17 (sec) , antiderivative size = 919, normalized size of antiderivative = 6.29 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\frac {6 e^{10 x} a^{3} b^{4}+6 e^{10 x} a \,b^{6}+30 e^{9 x} a^{2} b^{5}+40 e^{7 x} a^{4} b^{3}+200 e^{7 x} a^{2} b^{5}-60 e^{6 x} a^{5} b^{2}-180 e^{6 x} a^{3} b^{4}-120 e^{6 x} a \,b^{6}+96 e^{5 x} a^{6} b +368 e^{5 x} a^{4} b^{3}+628 e^{5 x} a^{2} b^{5}-620 e^{4 x} a^{5} b^{2}-880 e^{4 x} a^{3} b^{4}-420 e^{4 x} a \,b^{6}+40 e^{3 x} a^{4} b^{3}+200 e^{3 x} a^{2} b^{5}-340 e^{2 x} a^{5} b^{2}-530 e^{2 x} a^{3} b^{4}-270 e^{2 x} a \,b^{6}+30 e^{x} a^{2} b^{5}+30 e^{10 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{6} i +150 e^{8 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{6} i +300 e^{6 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{6} i +300 e^{4 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{6} i +150 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{6} i +30 e^{9 x} b^{7}+160 e^{7 x} b^{7}+356 e^{5 x} b^{7}-160 e^{4 x} a^{7}+160 e^{3 x} b^{7}-80 e^{2 x} a^{7}+30 e^{x} b^{7}-68 a^{5} b^{2}-112 a^{3} b^{4}-60 a \,b^{6}+30 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{6} i -16 a^{7}}{60 e^{10 x} a^{6} b^{2}+90 e^{10 x} a^{4} b^{4}+60 e^{10 x} a^{2} b^{6}+300 e^{8 x} a^{6} b^{2}+450 e^{8 x} a^{4} b^{4}+300 e^{8 x} a^{2} b^{6}+600 e^{6 x} a^{6} b^{2}+900 e^{6 x} a^{4} b^{4}+600 e^{6 x} a^{2} b^{6}+60 a^{6} b^{2}+90 a^{4} b^{4}+60 a^{2} b^{6}+15 a^{8}+150 e^{4 x} b^{8}+75 e^{2 x} a^{8}+75 e^{2 x} b^{8}+15 e^{10 x} a^{8}+15 e^{10 x} b^{8}+75 e^{8 x} a^{8}+75 e^{8 x} b^{8}+150 e^{6 x} a^{8}+150 e^{6 x} b^{8}+150 e^{4 x} a^{8}+600 e^{4 x} a^{6} b^{2}+15 b^{8}+900 e^{4 x} a^{4} b^{4}+600 e^{4 x} a^{2} b^{6}+300 e^{2 x} a^{6} b^{2}+450 e^{2 x} a^{4} b^{4}+300 e^{2 x} a^{2} b^{6}} \] Input:
int(sech(x)^6/(a+b*sinh(x)),x)
Output:
(2*(15*e**(10*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2) )*b**6*i + 75*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**6*i + 150*e**(6*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt (a**2 + b**2))*b**6*i + 150*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a* i)/sqrt(a**2 + b**2))*b**6*i + 75*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b* i + a*i)/sqrt(a**2 + b**2))*b**6*i + 15*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**6*i + 3*e**(10*x)*a**3*b**4 + 3*e**(10*x)*a*b* *6 + 15*e**(9*x)*a**2*b**5 + 15*e**(9*x)*b**7 + 20*e**(7*x)*a**4*b**3 + 10 0*e**(7*x)*a**2*b**5 + 80*e**(7*x)*b**7 - 30*e**(6*x)*a**5*b**2 - 90*e**(6 *x)*a**3*b**4 - 60*e**(6*x)*a*b**6 + 48*e**(5*x)*a**6*b + 184*e**(5*x)*a** 4*b**3 + 314*e**(5*x)*a**2*b**5 + 178*e**(5*x)*b**7 - 80*e**(4*x)*a**7 - 3 10*e**(4*x)*a**5*b**2 - 440*e**(4*x)*a**3*b**4 - 210*e**(4*x)*a*b**6 + 20* e**(3*x)*a**4*b**3 + 100*e**(3*x)*a**2*b**5 + 80*e**(3*x)*b**7 - 40*e**(2* x)*a**7 - 170*e**(2*x)*a**5*b**2 - 265*e**(2*x)*a**3*b**4 - 135*e**(2*x)*a *b**6 + 15*e**x*a**2*b**5 + 15*e**x*b**7 - 8*a**7 - 34*a**5*b**2 - 56*a**3 *b**4 - 30*a*b**6))/(15*(e**(10*x)*a**8 + 4*e**(10*x)*a**6*b**2 + 6*e**(10 *x)*a**4*b**4 + 4*e**(10*x)*a**2*b**6 + e**(10*x)*b**8 + 5*e**(8*x)*a**8 + 20*e**(8*x)*a**6*b**2 + 30*e**(8*x)*a**4*b**4 + 20*e**(8*x)*a**2*b**6 + 5 *e**(8*x)*b**8 + 10*e**(6*x)*a**8 + 40*e**(6*x)*a**6*b**2 + 60*e**(6*x)*a* *4*b**4 + 40*e**(6*x)*a**2*b**6 + 10*e**(6*x)*b**8 + 10*e**(4*x)*a**8 +...