3.199 \(\int \frac{\text{sech}^6(x)}{a+b \sinh (x)} \, dx\)

Optimal. Leaf size=146 \[ -\frac{2 b^6 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac{\text{sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}+\frac{\text{sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{15 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (a \left (26 a^2 b^2+8 a^4+33 b^4\right ) \sinh (x)+15 b^5\right )}{15 \left (a^2+b^2\right )^3} \]

[Out]

(-2*b^6*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (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]))/(15*(a^2 + b^2)^2) + (Sech[x]*(15*b^5 + a*(8*a^4 + 26*
a^2*b^2 + 33*b^4)*Sinh[x]))/(15*(a^2 + b^2)^3)

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Rubi [A]  time = 0.41731, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2696, 2866, 12, 2660, 618, 206} \[ -\frac{2 b^6 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac{\text{sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}+\frac{\text{sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{15 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (a \left (26 a^2 b^2+8 a^4+33 b^4\right ) \sinh (x)+15 b^5\right )}{15 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^6/(a + b*Sinh[x]),x]

[Out]

(-2*b^6*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (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]))/(15*(a^2 + b^2)^2) + (Sech[x]*(15*b^5 + a*(8*a^4 + 26*
a^2*b^2 + 33*b^4)*Sinh[x]))/(15*(a^2 + b^2)^3)

Rule 2696

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[((g*Co
s[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] + Dist[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] /; FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] &&
IntegersQ[2*m, 2*p]

Rule 2866

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((g*Cos[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] + Dist[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] && LtQ[p, -1] && IntegerQ[2*m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(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]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\text{sech}^6(x)}{a+b \sinh (x)} \, dx &=\frac{\text{sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}-\frac{\int \frac{\text{sech}^4(x) \left (-4 a^2-5 b^2-4 a b \sinh (x)\right )}{a+b \sinh (x)} \, dx}{5 \left (a^2+b^2\right )}\\ &=\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{\int \frac{\text{sech}^2(x) \left (8 a^4+18 a^2 b^2+15 b^4+2 a b \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{15 \left (a^2+b^2\right )^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}-\frac{\int -\frac{15 b^6}{a+b \sinh (x)} \, dx}{15 \left (a^2+b^2\right )^3}\\ &=\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}+\frac{b^6 \int \frac{1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\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}+\frac{\left (2 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^3}\\ &=\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}-\frac{\left (4 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^3}\\ &=-\frac{2 b^6 \tanh ^{-1}\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}\\ \end{align*}

Mathematica [A]  time = 0.444767, size = 146, normalized size = 1. \[ \frac{a \left (26 a^2 b^2+8 a^4+33 b^4\right ) \tanh (x)+\frac{30 b^6 \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+3 \left (a^2+b^2\right )^2 \text{sech}^5(x) (a \sinh (x)+b)+\left (a^2+b^2\right ) \text{sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )+15 b^5 \text{sech}(x)}{15 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^6/(a + b*Sinh[x]),x]

[Out]

((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)*Sech[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)

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Maple [B]  time = 0.048, size = 350, normalized size = 2.4 \begin{align*} -2\,{\frac{1}{ \left ({a}^{6}+3\,{a}^{4}{b}^{2}+3\,{a}^{2}{b}^{4}+{b}^{6} \right ) \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{5}} \left ( \left ( -{a}^{5}-3\,{a}^{3}{b}^{2}-3\,a{b}^{4} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{9}+ \left ( -{a}^{4}b-3\,{a}^{2}{b}^{3}-3\,{b}^{5} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{8}+ \left ( -4/3\,{a}^{5}-16/3\,{a}^{3}{b}^{2}-8\,a{b}^{4} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{7}+ \left ( -2\,{a}^{2}{b}^{3}-6\,{b}^{5} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{6}+ \left ( -{\frac{58\,{a}^{5}}{15}}-{\frac{166\,{a}^{3}{b}^{2}}{15}}-{\frac{66\,a{b}^{4}}{5}} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{5}+ \left ( -2\,{a}^{4}b-16/3\,{a}^{2}{b}^{3}-{\frac{28\,{b}^{5}}{3}} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+ \left ( -4/3\,{a}^{5}-16/3\,{a}^{3}{b}^{2}-8\,a{b}^{4} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{3}+ \left ( -2/3\,{a}^{2}{b}^{3}-14/3\,{b}^{5} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+ \left ( -{a}^{5}-3\,{a}^{3}{b}^{2}-3\,a{b}^{4} \right ) \tanh \left ( x/2 \right ) -1/5\,{a}^{4}b-{\frac{11\,{a}^{2}{b}^{3}}{15}}-{\frac{23\,{b}^{5}}{15}} \right ) }+2\,{\frac{{b}^{6}}{ \left ({a}^{6}+3\,{a}^{4}{b}^{2}+3\,{a}^{2}{b}^{4}+{b}^{6} \right ) \sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^6/(a+b*sinh(x)),x)

[Out]

-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+(-58/15*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^3-28/3*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^3-23/1
5*b^5)/(tanh(1/2*x)^2+1)^5+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))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^6/(a+b*sinh(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.39346, size = 7626, normalized size = 52.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^6/(a+b*sinh(x)),x, algorithm="fricas")

[Out]

1/15*(30*(a^2*b^5 + b^7)*cosh(x)^9 + 30*(a^2*b^5 + b^7)*sinh(x)^9 - 30*(a^3*b^4 + a*b^6)*cosh(x)^8 - 30*(a^3*b
^4 + a*b^6 - 9*(a^2*b^5 + b^7)*cosh(x))*sinh(x)^8 + 40*(a^4*b^3 + 5*a^2*b^5 + 4*b^7)*cosh(x)^7 + 40*(a^4*b^3 +
5*a^2*b^5 + 4*b^7 + 27*(a^2*b^5 + b^7)*cosh(x)^2 - 6*(a^3*b^4 + a*b^6)*cosh(x))*sinh(x)^7 - 16*a^7 - 68*a^5*b
^2 - 118*a^3*b^4 - 66*a*b^6 - 60*(a^5*b^2 + 4*a^3*b^4 + 3*a*b^6)*cosh(x)^6 - 20*(3*a^5*b^2 + 12*a^3*b^4 + 9*a*
b^6 - 126*(a^2*b^5 + b^7)*cosh(x)^3 + 42*(a^3*b^4 + a*b^6)*cosh(x)^2 - 14*(a^4*b^3 + 5*a^2*b^5 + 4*b^7)*cosh(x
))*sinh(x)^6 + 4*(24*a^6*b + 92*a^4*b^3 + 157*a^2*b^5 + 89*b^7)*cosh(x)^5 + 4*(24*a^6*b + 92*a^4*b^3 + 157*a^2
*b^5 + 89*b^7 + 945*(a^2*b^5 + b^7)*cosh(x)^4 - 420*(a^3*b^4 + a*b^6)*cosh(x)^3 + 210*(a^4*b^3 + 5*a^2*b^5 + 4
*b^7)*cosh(x)^2 - 90*(a^5*b^2 + 4*a^3*b^4 + 3*a*b^6)*cosh(x))*sinh(x)^5 - 20*(8*a^7 + 31*a^5*b^2 + 47*a^3*b^4
+ 24*a*b^6)*cosh(x)^4 - 20*(8*a^7 + 31*a^5*b^2 + 47*a^3*b^4 + 24*a*b^6 - 189*(a^2*b^5 + b^7)*cosh(x)^5 + 105*(
a^3*b^4 + a*b^6)*cosh(x)^4 - 70*(a^4*b^3 + 5*a^2*b^5 + 4*b^7)*cosh(x)^3 + 45*(a^5*b^2 + 4*a^3*b^4 + 3*a*b^6)*c
osh(x)^2 - (24*a^6*b + 92*a^4*b^3 + 157*a^2*b^5 + 89*b^7)*cosh(x))*sinh(x)^4 + 40*(a^4*b^3 + 5*a^2*b^5 + 4*b^7
)*cosh(x)^3 + 40*(a^4*b^3 + 5*a^2*b^5 + 4*b^7 + 63*(a^2*b^5 + b^7)*cosh(x)^6 - 42*(a^3*b^4 + a*b^6)*cosh(x)^5
+ 35*(a^4*b^3 + 5*a^2*b^5 + 4*b^7)*cosh(x)^4 - 30*(a^5*b^2 + 4*a^3*b^4 + 3*a*b^6)*cosh(x)^3 + (24*a^6*b + 92*a
^4*b^3 + 157*a^2*b^5 + 89*b^7)*cosh(x)^2 - 2*(8*a^7 + 31*a^5*b^2 + 47*a^3*b^4 + 24*a*b^6)*cosh(x))*sinh(x)^3 -
20*(4*a^7 + 17*a^5*b^2 + 28*a^3*b^4 + 15*a*b^6)*cosh(x)^2 + 20*(54*(a^2*b^5 + b^7)*cosh(x)^7 - 4*a^7 - 17*a^5
*b^2 - 28*a^3*b^4 - 15*a*b^6 - 42*(a^3*b^4 + a*b^6)*cosh(x)^6 + 42*(a^4*b^3 + 5*a^2*b^5 + 4*b^7)*cosh(x)^5 - 4
5*(a^5*b^2 + 4*a^3*b^4 + 3*a*b^6)*cosh(x)^4 + 2*(24*a^6*b + 92*a^4*b^3 + 157*a^2*b^5 + 89*b^7)*cosh(x)^3 - 6*(
8*a^7 + 31*a^5*b^2 + 47*a^3*b^4 + 24*a*b^6)*cosh(x)^2 + 6*(a^4*b^3 + 5*a^2*b^5 + 4*b^7)*cosh(x))*sinh(x)^2 + 1
5*(b^6*cosh(x)^10 + 10*b^6*cosh(x)*sinh(x)^9 + b^6*sinh(x)^10 + 5*b^6*cosh(x)^8 + 10*b^6*cosh(x)^6 + 10*b^6*co
sh(x)^4 + 5*(9*b^6*cosh(x)^2 + b^6)*sinh(x)^8 + 5*b^6*cosh(x)^2 + 40*(3*b^6*cosh(x)^3 + b^6*cosh(x))*sinh(x)^7
+ 10*(21*b^6*cosh(x)^4 + 14*b^6*cosh(x)^2 + b^6)*sinh(x)^6 + b^6 + 4*(63*b^6*cosh(x)^5 + 70*b^6*cosh(x)^3 + 1
5*b^6*cosh(x))*sinh(x)^5 + 10*(21*b^6*cosh(x)^6 + 35*b^6*cosh(x)^4 + 15*b^6*cosh(x)^2 + b^6)*sinh(x)^4 + 40*(3
*b^6*cosh(x)^7 + 7*b^6*cosh(x)^5 + 5*b^6*cosh(x)^3 + b^6*cosh(x))*sinh(x)^3 + 5*(9*b^6*cosh(x)^8 + 28*b^6*cosh
(x)^6 + 30*b^6*cosh(x)^4 + 12*b^6*cosh(x)^2 + b^6)*sinh(x)^2 + 10*(b^6*cosh(x)^9 + 4*b^6*cosh(x)^7 + 6*b^6*cos
h(x)^5 + 4*b^6*cosh(x)^3 + b^6*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*co
sh(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)) + 30*(a^2*b^5 + b^7)*cosh(x) + 10*(27*(a^2*b
^5 + b^7)*cosh(x)^8 - 24*(a^3*b^4 + a*b^6)*cosh(x)^7 + 3*a^2*b^5 + 3*b^7 + 28*(a^4*b^3 + 5*a^2*b^5 + 4*b^7)*co
sh(x)^6 - 36*(a^5*b^2 + 4*a^3*b^4 + 3*a*b^6)*cosh(x)^5 + 2*(24*a^6*b + 92*a^4*b^3 + 157*a^2*b^5 + 89*b^7)*cosh
(x)^4 - 8*(8*a^7 + 31*a^5*b^2 + 47*a^3*b^4 + 24*a*b^6)*cosh(x)^3 + 12*(a^4*b^3 + 5*a^2*b^5 + 4*b^7)*cosh(x)^2
- 4*(4*a^7 + 17*a^5*b^2 + 28*a^3*b^4 + 15*a*b^6)*cosh(x))*sinh(x))/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 +
b^8)*cosh(x)^10 + 10*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)*sinh(x)^9 + (a^8 + 4*a^6*b^2 + 6
*a^4*b^4 + 4*a^2*b^6 + b^8)*sinh(x)^10 + 5*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^8 + 5*(a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 + 9*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^2)*sinh(
x)^8 + a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 + 40*(3*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c
osh(x)^3 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x))*sinh(x)^7 + 10*(a^8 + 4*a^6*b^2 + 6*a^4*b^
4 + 4*a^2*b^6 + b^8)*cosh(x)^6 + 10*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 + 21*(a^8 + 4*a^6*b^2 + 6*a
^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^4 + 14*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^2)*sinh(x)^6
+ 4*(63*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^5 + 70*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8)*cosh(x)^3 + 15*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x))*sinh(x)^5 + 10*(a^8 + 4*a^6*b
^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^4 + 10*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 + 21*(a^8 + 4*
a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^6 + 35*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^
4 + 15*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^8 + 4*a^6*b^2 + 6*a^4*b
^4 + 4*a^2*b^6 + b^8)*cosh(x)^7 + 7*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^5 + 5*(a^8 + 4*a^6
*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^3 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x))*sinh(
x)^3 + 5*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^2 + 5*(9*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2
*b^6 + b^8)*cosh(x)^8 + a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 + 28*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^
2*b^6 + b^8)*cosh(x)^6 + 30*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^4 + 12*(a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^2)*sinh(x)^2 + 10*((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x
)^9 + 4*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^7 + 6*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6
+ b^8)*cosh(x)^5 + 4*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*cosh(x)^3 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4
+ 4*a^2*b^6 + b^8)*cosh(x))*sinh(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**6/(a+b*sinh(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.28309, size = 436, normalized size = 2.99 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^6/(a+b*sinh(x)),x, algorithm="giac")

[Out]

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)