3.18 \(\int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx\)

Optimal. Leaf size=89 \[ -\frac {\left (a^2+3 a b+3 b^2\right ) \coth (x)}{(a+b)^3}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\coth ^5(x)}{5 (a+b)}+\frac {(2 a+3 b) \coth ^3(x)}{3 (a+b)^2} \]

[Out]

-(a^2+3*a*b+3*b^2)*coth(x)/(a+b)^3+1/3*(2*a+3*b)*coth(x)^3/(a+b)^2-1/5*coth(x)^5/(a+b)-b^3*arctanh(a^(1/2)*tan
h(x)/(a+b)^(1/2))/(a+b)^(7/2)/a^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3191, 390, 208} \[ -\frac {\left (a^2+3 a b+3 b^2\right ) \coth (x)}{(a+b)^3}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\coth ^5(x)}{5 (a+b)}+\frac {(2 a+3 b) \coth ^3(x)}{3 (a+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^6/(a + b*Cosh[x]^2),x]

[Out]

-((b^3*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2))) - ((a^2 + 3*a*b + 3*b^2)*Coth[x])/(a +
b)^3 + ((2*a + 3*b)*Coth[x]^3)/(3*(a + b)^2) - Coth[x]^5/(5*(a + b))

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]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3191

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.37, size = 92, normalized size = 1.03 \[ -\frac {\coth (x) \left (-\left (4 a^2+13 a b+9 b^2\right ) \text {csch}^2(x)+8 a^2+3 (a+b)^2 \text {csch}^4(x)+26 a b+33 b^2\right )}{15 (a+b)^3}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^6/(a + b*Cosh[x]^2),x]

[Out]

-((b^3*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2))) - (Coth[x]*(8*a^2 + 26*a*b + 33*b^2 -
(4*a^2 + 13*a*b + 9*b^2)*Csch[x]^2 + 3*(a + b)^2*Csch[x]^4))/(15*(a + b)^3)

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fricas [B]  time = 0.53, size = 4977, normalized size = 55.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^6/(a+b*cosh(x)^2),x, algorithm="fricas")

[Out]

[-1/30*(60*(a^2*b^2 + a*b^3)*cosh(x)^8 + 480*(a^2*b^2 + a*b^3)*cosh(x)*sinh(x)^7 + 60*(a^2*b^2 + a*b^3)*sinh(x
)^8 - 120*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^6 - 120*(a^3*b + 4*a^2*b^2 + 3*a*b^3 - 14*(a^2*b^2 + a*b^3)*co
sh(x)^2)*sinh(x)^6 + 240*(14*(a^2*b^2 + a*b^3)*cosh(x)^3 - 3*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x))*sinh(x)^5
+ 40*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^4 + 40*(105*(a^2*b^2 + a*b^3)*cosh(x)^4 + 8*a^4 + 31*a
^3*b + 47*a^2*b^2 + 24*a*b^3 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^2)*sinh(x)^4 + 32*a^4 + 136*a^3*b + 23
6*a^2*b^2 + 132*a*b^3 + 160*(21*(a^2*b^2 + a*b^3)*cosh(x)^5 - 15*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^3 + (8*
a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x))*sinh(x)^3 - 40*(4*a^4 + 17*a^3*b + 28*a^2*b^2 + 15*a*b^3)*cos
h(x)^2 + 40*(42*(a^2*b^2 + a*b^3)*cosh(x)^6 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^4 - 4*a^4 - 17*a^3*b -
28*a^2*b^2 - 15*a*b^3 + 6*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^2)*sinh(x)^2 - 15*(b^3*cosh(x)^10
+ 10*b^3*cosh(x)*sinh(x)^9 + b^3*sinh(x)^10 - 5*b^3*cosh(x)^8 + 10*b^3*cosh(x)^6 + 5*(9*b^3*cosh(x)^2 - b^3)*
sinh(x)^8 + 40*(3*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^7 - 10*b^3*cosh(x)^4 + 10*(21*b^3*cosh(x)^4 - 14*b^3*co
sh(x)^2 + b^3)*sinh(x)^6 + 4*(63*b^3*cosh(x)^5 - 70*b^3*cosh(x)^3 + 15*b^3*cosh(x))*sinh(x)^5 + 5*b^3*cosh(x)^
2 + 10*(21*b^3*cosh(x)^6 - 35*b^3*cosh(x)^4 + 15*b^3*cosh(x)^2 - b^3)*sinh(x)^4 + 40*(3*b^3*cosh(x)^7 - 7*b^3*
cosh(x)^5 + 5*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^3 - b^3 + 5*(9*b^3*cosh(x)^8 - 28*b^3*cosh(x)^6 + 30*b^3*co
sh(x)^4 - 12*b^3*cosh(x)^2 + b^3)*sinh(x)^2 + 10*(b^3*cosh(x)^9 - 4*b^3*cosh(x)^7 + 6*b^3*cosh(x)^5 - 4*b^3*co
sh(x)^3 + b^3*cosh(x))*sinh(x))*sqrt(a^2 + a*b)*log((b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 +
2*(2*a*b + b^2)*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + 2*a*b + b^2)*sinh(x)^2 + 8*a^2 + 8*a*b + b^2 + 4*(b^2*cosh(x
)^3 + (2*a*b + b^2)*cosh(x))*sinh(x) + 4*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 2*a + b)*sqrt(a^2
+ a*b))/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a +
b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) + 80*(6*(a^2*b^2 + a*b^3)*cosh(x)^7 - 9*(a^3*
b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^5 + 2*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^3 - (4*a^4 + 17*a^3*
b + 28*a^2*b^2 + 15*a*b^3)*cosh(x))*sinh(x))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^10 + 10*
(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)*sinh(x)^9 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a
*b^4)*sinh(x)^10 - 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^8 - 5*(a^5 + 4*a^4*b + 6*a^3*b^2
+ 4*a^2*b^3 + a*b^4 - 9*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^5 + 4*
a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*si
nh(x)^7 + 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 + 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2
*b^3 + a*b^4 + 21*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 - 14*(a^5 + 4*a^4*b + 6*a^3*b^2 +
4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 - 70
*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 + 15*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4
)*cosh(x))*sinh(x)^5 - a^5 - 4*a^4*b - 6*a^3*b^2 - 4*a^2*b^3 - a*b^4 - 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b
^3 + a*b^4)*cosh(x)^4 + 10*(21*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 - a^5 - 4*a^4*b - 6*a
^3*b^2 - 4*a^2*b^3 - a*b^4 - 35*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 + 15*(a^5 + 4*a^4*b
+ 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*
cosh(x)^7 - 7*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 + 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2
*b^3 + a*b^4)*cosh(x)^3 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sinh(x)^3 + 5*(a^5 + 4*a^4*
b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^2 + 5*(9*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^8
- 28*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 + a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4
+ 30*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 - 12*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 +
a*b^4)*cosh(x)^2)*sinh(x)^2 + 10*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^9 - 4*(a^5 + 4*a^4*b
+ 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^7 + 6*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 - 4*
(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*co
sh(x))*sinh(x)), -1/15*(30*(a^2*b^2 + a*b^3)*cosh(x)^8 + 240*(a^2*b^2 + a*b^3)*cosh(x)*sinh(x)^7 + 30*(a^2*b^2
+ a*b^3)*sinh(x)^8 - 60*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^6 - 60*(a^3*b + 4*a^2*b^2 + 3*a*b^3 - 14*(a^2*b
^2 + a*b^3)*cosh(x)^2)*sinh(x)^6 + 120*(14*(a^2*b^2 + a*b^3)*cosh(x)^3 - 3*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(
x))*sinh(x)^5 + 20*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^4 + 20*(105*(a^2*b^2 + a*b^3)*cosh(x)^4
+ 8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^2)*sinh(x)^4 + 16*a^4 +
68*a^3*b + 118*a^2*b^2 + 66*a*b^3 + 80*(21*(a^2*b^2 + a*b^3)*cosh(x)^5 - 15*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh
(x)^3 + (8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x))*sinh(x)^3 - 20*(4*a^4 + 17*a^3*b + 28*a^2*b^2 + 15
*a*b^3)*cosh(x)^2 + 20*(42*(a^2*b^2 + a*b^3)*cosh(x)^6 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^4 - 4*a^4 -
17*a^3*b - 28*a^2*b^2 - 15*a*b^3 + 6*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^2)*sinh(x)^2 + 15*(b^3
*cosh(x)^10 + 10*b^3*cosh(x)*sinh(x)^9 + b^3*sinh(x)^10 - 5*b^3*cosh(x)^8 + 10*b^3*cosh(x)^6 + 5*(9*b^3*cosh(x
)^2 - b^3)*sinh(x)^8 + 40*(3*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^7 - 10*b^3*cosh(x)^4 + 10*(21*b^3*cosh(x)^4
- 14*b^3*cosh(x)^2 + b^3)*sinh(x)^6 + 4*(63*b^3*cosh(x)^5 - 70*b^3*cosh(x)^3 + 15*b^3*cosh(x))*sinh(x)^5 + 5*b
^3*cosh(x)^2 + 10*(21*b^3*cosh(x)^6 - 35*b^3*cosh(x)^4 + 15*b^3*cosh(x)^2 - b^3)*sinh(x)^4 + 40*(3*b^3*cosh(x)
^7 - 7*b^3*cosh(x)^5 + 5*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^3 - b^3 + 5*(9*b^3*cosh(x)^8 - 28*b^3*cosh(x)^6
+ 30*b^3*cosh(x)^4 - 12*b^3*cosh(x)^2 + b^3)*sinh(x)^2 + 10*(b^3*cosh(x)^9 - 4*b^3*cosh(x)^7 + 6*b^3*cosh(x)^5
- 4*b^3*cosh(x)^3 + b^3*cosh(x))*sinh(x))*sqrt(-a^2 - a*b)*arctan(1/2*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*
sinh(x)^2 + 2*a + b)*sqrt(-a^2 - a*b)/(a^2 + a*b)) + 40*(6*(a^2*b^2 + a*b^3)*cosh(x)^7 - 9*(a^3*b + 4*a^2*b^2
+ 3*a*b^3)*cosh(x)^5 + 2*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^3 - (4*a^4 + 17*a^3*b + 28*a^2*b^2
+ 15*a*b^3)*cosh(x))*sinh(x))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^10 + 10*(a^5 + 4*a^4*b
+ 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)*sinh(x)^9 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*sinh(x)^
10 - 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^8 - 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 +
a*b^4 - 9*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^5 + 4*a^4*b + 6*a^3*
b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sinh(x)^7 + 10*(
a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 + 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4 +
21*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 - 14*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*
b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 - 70*(a^5 + 4*a^4*
b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 + 15*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sin
h(x)^5 - a^5 - 4*a^4*b - 6*a^3*b^2 - 4*a^2*b^3 - a*b^4 - 10*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*co
sh(x)^4 + 10*(21*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 - a^5 - 4*a^4*b - 6*a^3*b^2 - 4*a^2
*b^3 - a*b^4 - 35*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 + 15*(a^5 + 4*a^4*b + 6*a^3*b^2 +
4*a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^7 - 7*
(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 + 5*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*
cosh(x)^3 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sinh(x)^3 + 5*(a^5 + 4*a^4*b + 6*a^3*b^2
+ 4*a^2*b^3 + a*b^4)*cosh(x)^2 + 5*(9*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^8 - 28*(a^5 + 4*
a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^6 + a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4 + 30*(a^5 + 4
*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^4 - 12*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)
^2)*sinh(x)^2 + 10*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^9 - 4*(a^5 + 4*a^4*b + 6*a^3*b^2 +
4*a^2*b^3 + a*b^4)*cosh(x)^7 + 6*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^5 - 4*(a^5 + 4*a^4*b
+ 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x)^3 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cosh(x))*sinh(x)
)]

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giac [B]  time = 0.59, size = 189, normalized size = 2.12 \[ -\frac {b^{3} \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {-a^{2} - a b}} - \frac {2 \, {\left (15 \, b^{2} e^{\left (8 \, x\right )} - 30 \, a b e^{\left (6 \, x\right )} - 90 \, b^{2} e^{\left (6 \, x\right )} + 80 \, a^{2} e^{\left (4 \, x\right )} + 230 \, a b e^{\left (4 \, x\right )} + 240 \, b^{2} e^{\left (4 \, x\right )} - 40 \, a^{2} e^{\left (2 \, x\right )} - 130 \, a b e^{\left (2 \, x\right )} - 150 \, b^{2} e^{\left (2 \, x\right )} + 8 \, a^{2} + 26 \, a b + 33 \, b^{2}\right )}}{15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^6/(a+b*cosh(x)^2),x, algorithm="giac")

[Out]

-b^3*arctan(1/2*(b*e^(2*x) + 2*a + b)/sqrt(-a^2 - a*b))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(-a^2 - a*b)) - 2
/15*(15*b^2*e^(8*x) - 30*a*b*e^(6*x) - 90*b^2*e^(6*x) + 80*a^2*e^(4*x) + 230*a*b*e^(4*x) + 240*b^2*e^(4*x) - 4
0*a^2*e^(2*x) - 130*a*b*e^(2*x) - 150*b^2*e^(2*x) + 8*a^2 + 26*a*b + 33*b^2)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*
(e^(2*x) - 1)^5)

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maple [B]  time = 0.15, size = 307, normalized size = 3.45 \[ -\frac {a^{2} \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{160 \left (a +b \right )^{3}}-\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right ) a b}{80 \left (a +b \right )^{3}}-\frac {b^{2} \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{160 \left (a +b \right )^{3}}+\frac {5 a^{2} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{96 \left (a +b \right )^{3}}+\frac {7 a \left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) b}{48 \left (a +b \right )^{3}}+\frac {3 b^{2} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{32 \left (a +b \right )^{3}}-\frac {5 a^{2} \tanh \left (\frac {x}{2}\right )}{16 \left (a +b \right )^{3}}-\frac {a b \tanh \left (\frac {x}{2}\right )}{\left (a +b \right )^{3}}-\frac {19 b^{2} \tanh \left (\frac {x}{2}\right )}{16 \left (a +b \right )^{3}}+\frac {b^{3} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \sqrt {a}\, \tanh \left (\frac {x}{2}\right )+\sqrt {a +b}\right )}{2 \left (a +b \right )^{\frac {7}{2}} \sqrt {a}}-\frac {b^{3} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \sqrt {a}\, \tanh \left (\frac {x}{2}\right )+\sqrt {a +b}\right )}{2 \left (a +b \right )^{\frac {7}{2}} \sqrt {a}}-\frac {1}{160 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{5}}+\frac {5 a}{96 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{3}}+\frac {3 b}{32 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {5 a^{2}}{16 \left (a +b \right )^{3} \tanh \left (\frac {x}{2}\right )}-\frac {a b}{\left (a +b \right )^{3} \tanh \left (\frac {x}{2}\right )}-\frac {19 b^{2}}{16 \left (a +b \right )^{3} \tanh \left (\frac {x}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^6/(a+b*cosh(x)^2),x)

[Out]

-1/160/(a+b)^3*a^2*tanh(1/2*x)^5-1/80/(a+b)^3*tanh(1/2*x)^5*a*b-1/160/(a+b)^3*b^2*tanh(1/2*x)^5+5/96/(a+b)^3*a
^2*tanh(1/2*x)^3+7/48/(a+b)^3*a*tanh(1/2*x)^3*b+3/32/(a+b)^3*b^2*tanh(1/2*x)^3-5/16/(a+b)^3*a^2*tanh(1/2*x)-1/
(a+b)^3*a*b*tanh(1/2*x)-19/16/(a+b)^3*b^2*tanh(1/2*x)+1/2*b^3/(a+b)^(7/2)/a^(1/2)*ln((a+b)^(1/2)*tanh(1/2*x)^2
-2*a^(1/2)*tanh(1/2*x)+(a+b)^(1/2))-1/2*b^3/(a+b)^(7/2)/a^(1/2)*ln((a+b)^(1/2)*tanh(1/2*x)^2+2*a^(1/2)*tanh(1/
2*x)+(a+b)^(1/2))-1/160/(a+b)/tanh(1/2*x)^5+5/96/(a+b)^2/tanh(1/2*x)^3*a+3/32/(a+b)^2/tanh(1/2*x)^3*b-5/16/(a+
b)^3/tanh(1/2*x)*a^2-1/(a+b)^3/tanh(1/2*x)*a*b-19/16/(a+b)^3/tanh(1/2*x)*b^2

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maxima [B]  time = 0.48, size = 307, normalized size = 3.45 \[ \frac {b^{3} \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {2 \, {\left (15 \, b^{2} e^{\left (-8 \, x\right )} + 8 \, a^{2} + 26 \, a b + 33 \, b^{2} - 10 \, {\left (4 \, a^{2} + 13 \, a b + 15 \, b^{2}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (8 \, a^{2} + 23 \, a b + 24 \, b^{2}\right )} e^{\left (-4 \, x\right )} - 30 \, {\left (a b + 3 \, b^{2}\right )} e^{\left (-6 \, x\right )}\right )}}{15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, x\right )} - 10 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-6 \, x\right )} + 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-8 \, x\right )} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-10 \, x\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^6/(a+b*cosh(x)^2),x, algorithm="maxima")

[Out]

1/2*b^3*log((b*e^(-2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(-2*x) + 2*a + b + 2*sqrt((a + b)*a)))/((a^3 + 3*a
^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*a)) - 2/15*(15*b^2*e^(-8*x) + 8*a^2 + 26*a*b + 33*b^2 - 10*(4*a^2 + 13*a*b
+ 15*b^2)*e^(-2*x) + 10*(8*a^2 + 23*a*b + 24*b^2)*e^(-4*x) - 30*(a*b + 3*b^2)*e^(-6*x))/(a^3 + 3*a^2*b + 3*a*b
^2 + b^3 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-2*x) + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-4*x) - 10*(a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*e^(-6*x) + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-8*x) - (a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*e^(-10*x))

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mupad [B]  time = 1.55, size = 333, normalized size = 3.74 \[ \frac {4\,\left (b^2+a\,b\right )}{{\left (a+b\right )}^3\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {16}{\left (a+b\right )\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {2\,b^2}{{\left (a+b\right )}^3\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {32}{5\,\left (a+b\right )\,\left (5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1\right )}-\frac {8\,\left (4\,a+3\,b\right )}{3\,{\left (a+b\right )}^2\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}+\frac {b^3\,\ln \left (\frac {4\,b^4\,\left (2\,a\,b+8\,a^2\,{\mathrm {e}}^{2\,x}+b^2\,{\mathrm {e}}^{2\,x}+b^2+8\,a\,b\,{\mathrm {e}}^{2\,x}\right )}{a\,{\left (a+b\right )}^7}-\frac {8\,b^4\,\left (b+4\,a\,{\mathrm {e}}^{2\,x}+2\,b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,{\left (a+b\right )}^{13/2}}\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{7/2}}-\frac {b^3\,\ln \left (\frac {8\,b^4\,\left (b+4\,a\,{\mathrm {e}}^{2\,x}+2\,b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,{\left (a+b\right )}^{13/2}}+\frac {4\,b^4\,\left (2\,a\,b+8\,a^2\,{\mathrm {e}}^{2\,x}+b^2\,{\mathrm {e}}^{2\,x}+b^2+8\,a\,b\,{\mathrm {e}}^{2\,x}\right )}{a\,{\left (a+b\right )}^7}\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^6*(a + b*cosh(x)^2)),x)

[Out]

(4*(a*b + b^2))/((a + b)^3*(exp(4*x) - 2*exp(2*x) + 1)) - 16/((a + b)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) +
exp(8*x) + 1)) - (2*b^2)/((a + b)^3*(exp(2*x) - 1)) - 32/(5*(a + b)*(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) -
5*exp(8*x) + exp(10*x) - 1)) - (8*(4*a + 3*b))/(3*(a + b)^2*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) + (b^3*l
og((4*b^4*(2*a*b + 8*a^2*exp(2*x) + b^2*exp(2*x) + b^2 + 8*a*b*exp(2*x)))/(a*(a + b)^7) - (8*b^4*(b + 4*a*exp(
2*x) + 2*b*exp(2*x)))/(a^(1/2)*(a + b)^(13/2))))/(2*a^(1/2)*(a + b)^(7/2)) - (b^3*log((8*b^4*(b + 4*a*exp(2*x)
+ 2*b*exp(2*x)))/(a^(1/2)*(a + b)^(13/2)) + (4*b^4*(2*a*b + 8*a^2*exp(2*x) + b^2*exp(2*x) + b^2 + 8*a*b*exp(2
*x)))/(a*(a + b)^7)))/(2*a^(1/2)*(a + b)^(7/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**6/(a+b*cosh(x)**2),x)

[Out]

Timed out

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