3.177 \(\int \frac{\text{csch}^6(x)}{a+b \cosh (x)} \, dx\)

Optimal. Leaf size=159 \[ \frac{\text{csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}+\frac{\text{csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{15 \left (a^2-b^2\right )^2}+\frac{\text{csch}(x) \left (15 b^5-a \left (-26 a^2 b^2+8 a^4+33 b^4\right ) \cosh (x)\right )}{15 \left (a^2-b^2\right )^3}+\frac{2 b^6 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}} \]

[Out]

(2*b^6*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)) + ((15*b^5 - a*(8*a^4 - 26*
a^2*b^2 + 33*b^4)*Cosh[x])*Csch[x])/(15*(a^2 - b^2)^3) + ((5*b^3 + a*(4*a^2 - 9*b^2)*Cosh[x])*Csch[x]^3)/(15*(
a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^5)/(5*(a^2 - b^2))

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Rubi [A]  time = 0.476846, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2696, 2866, 12, 2659, 208} \[ \frac{\text{csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}+\frac{\text{csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{15 \left (a^2-b^2\right )^2}+\frac{\text{csch}(x) \left (15 b^5-a \left (-26 a^2 b^2+8 a^4+33 b^4\right ) \cosh (x)\right )}{15 \left (a^2-b^2\right )^3}+\frac{2 b^6 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^6/(a + b*Cosh[x]),x]

[Out]

(2*b^6*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)) + ((15*b^5 - a*(8*a^4 - 26*
a^2*b^2 + 33*b^4)*Cosh[x])*Csch[x])/(15*(a^2 - b^2)^3) + ((5*b^3 + a*(4*a^2 - 9*b^2)*Cosh[x])*Csch[x]^3)/(15*(
a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^5)/(5*(a^2 - b^2))

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 2659

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\text{csch}^6(x)}{a+b \cosh (x)} \, dx &=\frac{(b-a \cosh (x)) \text{csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac{\int \frac{\left (-4 a^2+5 b^2-4 a b \cosh (x)\right ) \text{csch}^4(x)}{a+b \cosh (x)} \, dx}{5 \left (a^2-b^2\right )}\\ &=\frac{\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text{csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac{\int \frac{\left (8 a^4-18 a^2 b^2+15 b^4+2 a b \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text{csch}^2(x)}{a+b \cosh (x)} \, dx}{15 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text{csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac{\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text{csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac{\int \frac{15 b^6}{a+b \cosh (x)} \, dx}{15 \left (a^2-b^2\right )^3}\\ &=\frac{\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text{csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac{\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text{csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac{b^6 \int \frac{1}{a+b \cosh (x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=\frac{\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text{csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac{\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text{csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac{\left (2 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^3}\\ &=\frac{2 b^6 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}+\frac{\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text{csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac{\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text{csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^5(x)}{5 \left (a^2-b^2\right )}\\ \end{align*}

Mathematica [A]  time = 1.71403, size = 201, normalized size = 1.26 \[ \frac{1}{480} \left (-\frac{2 \left (64 a^2-183 a b+149 b^2\right ) \tanh \left (\frac{x}{2}\right )}{(a-b)^3}-\frac{2 \left (64 a^2+183 a b+149 b^2\right ) \coth \left (\frac{x}{2}\right )}{(a+b)^3}+\frac{960 b^6 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}-\frac{96 \sinh ^6\left (\frac{x}{2}\right ) \text{csch}^5(x)}{a-b}-\frac{8 (19 a-29 b) \sinh ^4\left (\frac{x}{2}\right ) \text{csch}^3(x)}{(a-b)^2}-\frac{3 \sinh (x) \text{csch}^6\left (\frac{x}{2}\right )}{2 (a+b)}+\frac{(19 a+29 b) \sinh (x) \text{csch}^4\left (\frac{x}{2}\right )}{2 (a+b)^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^6/(a + b*Cosh[x]),x]

[Out]

((960*b^6*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - (2*(64*a^2 + 183*a*b + 149*b^2)*C
oth[x/2])/(a + b)^3 - (8*(19*a - 29*b)*Csch[x]^3*Sinh[x/2]^4)/(a - b)^2 - (96*Csch[x]^5*Sinh[x/2]^6)/(a - b) +
((19*a + 29*b)*Csch[x/2]^4*Sinh[x])/(2*(a + b)^2) - (3*Csch[x/2]^6*Sinh[x])/(2*(a + b)) - (2*(64*a^2 - 183*a*
b + 149*b^2)*Tanh[x/2])/(a - b)^3)/480

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Maple [A]  time = 0.026, size = 213, normalized size = 1.3 \begin{align*} -{\frac{1}{32\, \left ( a-b \right ) ^{3}} \left ({\frac{{a}^{2}}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{2\,ab}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}+{\frac{{b}^{2}}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{5\,{a}^{2}}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+4\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}ab-{\frac{7\,{b}^{2}}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+10\,{a}^{2}\tanh \left ( x/2 \right ) -28\,ab\tanh \left ( x/2 \right ) +22\,{b}^{2}\tanh \left ( x/2 \right ) \right ) }+2\,{\frac{{b}^{6}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-{\frac{1}{160\,a+160\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-5}}-{\frac{-5\,a-7\,b}{96\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{10\,{a}^{2}+28\,ab+22\,{b}^{2}}{32\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^6/(a+b*cosh(x)),x)

[Out]

-1/32/(a-b)^3*(1/5*tanh(1/2*x)^5*a^2-2/5*tanh(1/2*x)^5*a*b+1/5*b^2*tanh(1/2*x)^5-5/3*tanh(1/2*x)^3*a^2+4*tanh(
1/2*x)^3*a*b-7/3*tanh(1/2*x)^3*b^2+10*a^2*tanh(1/2*x)-28*a*b*tanh(1/2*x)+22*b^2*tanh(1/2*x))+2/(a-b)^3/(a+b)^3
*b^6/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))-1/160/(a+b)/tanh(1/2*x)^5-1/96*(-5*a-7
*b)/(a+b)^2/tanh(1/2*x)^3-1/32/(a+b)^3*(10*a^2+28*a*b+22*b^2)/tanh(1/2*x)

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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(csch(x)^6/(a+b*cosh(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.78489, size = 15046, normalized size = 94.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^6/(a+b*cosh(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)*
cosh(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 -
45*(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 -
15*(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*c
osh(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 +
15*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*cos
h(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*co
sh(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*c
osh(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)*c
osh(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)*cos
h(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)*
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)^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)), 2/15*(15*(a^2*b^5 - b^7)*cosh(x)^9 + 15*(a^2*b^5 - b^7)*sinh(x)^9 - 15
*(a^3*b^4 - a*b^6)*cosh(x)^8 - 15*(a^3*b^4 - a*b^6 - 9*(a^2*b^5 - b^7)*cosh(x))*sinh(x)^8 + 20*(a^4*b^3 - 5*a^
2*b^5 + 4*b^7)*cosh(x)^7 + 20*(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 - 8*a^7 + 34*a^5*b^2 - 59*a^3*b^4 + 33*a*b^6 - 30*(a^5*b^2 - 4*a^3*b^4 + 3*a*b^6)*cosh(x)
^6 - 10*(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 - 1
4*(a^4*b^3 - 5*a^2*b^5 + 4*b^7)*cosh(x))*sinh(x)^6 + 2*(24*a^6*b - 92*a^4*b^3 + 157*a^2*b^5 - 89*b^7)*cosh(x)^
5 + 2*(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)*co
sh(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
- 10*(8*a^7 - 31*a^5*b^2 + 47*a^3*b^4 - 24*a*b^6)*cosh(x)^4 - 10*(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)*cosh(x)^2 - (24*a^6*b - 92*a^4*b^3 + 157*a^2*b^5 - 89*b^7)*cosh(x))*sinh(
x)^4 + 20*(a^4*b^3 - 5*a^2*b^5 + 4*b^7)*cosh(x)^3 + 20*(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 + 10*(4*a^7 - 17*a^5*b^2 + 28*a^3*b^4 - 15*a*b^6)*cosh(x)^2 + 10*(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 - 45*(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 - 15*(b^6*cosh(x)^10 + 10*b^6*cosh(x)*sinh(x)^9 + b^6*sinh(x)^10 - 5*b^6*co
sh(x)^8 + 10*b^6*cosh(x)^6 - 10*b^6*cosh(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*(6
3*b^6*cosh(x)^5 - 70*b^6*cosh(x)^3 + 15*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*c
osh(x)^9 - 4*b^6*cosh(x)^7 + 6*b^6*cosh(x)^5 - 4*b^6*cosh(x)^3 + b^6*cosh(x))*sinh(x))*sqrt(-a^2 + b^2)*arctan
(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a)/(a^2 - b^2)) + 15*(a^2*b^5 - b^7)*cosh(x) + 5*(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)*cosh(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)*c
osh(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)*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)^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(csch(x)**6/(a+b*cosh(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.19861, size = 409, normalized size = 2.57 \begin{align*} \frac{2 \, b^{6} \arctan \left (\frac{b e^{x} + 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 (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(csch(x)^6/(a+b*cosh(x)),x, algorithm="giac")

[Out]

2*b^6*arctan((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^(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)