3.177 \(\int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx\)
Optimal. Leaf size=159 \[ \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}+\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{15 \left (a^2-b^2\right )^2}+\frac {\text {csch}(x) \left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right )}{15 \left (a^2-b^2\right )^3}+\frac {2 b^6 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}} \]
[Out]
2*b^6*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b)^(7/2)/(a+b)^(7/2)+1/15*(15*b^5-a*(8*a^4-26*a^2*b^2+33
*b^4)*cosh(x))*csch(x)/(a^2-b^2)^3+1/15*(5*b^3+a*(4*a^2-9*b^2)*cosh(x))*csch(x)^3/(a^2-b^2)^2+1/5*(b-a*cosh(x)
)*csch(x)^5/(a^2-b^2)
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Rubi [A] time = 0.48, antiderivative size = 159, normalized size of antiderivative = 1.00,
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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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 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]
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*}
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Mathematica [A] time = 1.81, size = 201, normalized size = 1.26 \[ \frac {1}{480} \left (-\frac {2 \left (64 a^2-183 a b+149 b^2\right ) \tanh \left (\frac {x}{2}\right )}{(a-b)^3}-\frac {2 \left (64 a^2+183 a b+149 b^2\right ) \coth \left (\frac {x}{2}\right )}{(a+b)^3}+\frac {960 b^6 \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}-\frac {3 \sinh (x) \text {csch}^6\left (\frac {x}{2}\right )}{2 (a+b)}-\frac {96 \sinh ^6\left (\frac {x}{2}\right ) \text {csch}^5(x)}{a-b}+\frac {(19 a+29 b) \sinh (x) \text {csch}^4\left (\frac {x}{2}\right )}{2 (a+b)^2}-\frac {8 (19 a-29 b) \sinh ^4\left (\frac {x}{2}\right ) \text {csch}^3(x)}{(a-b)^2}\right ) \]
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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fricas [B] time = 0.65, size = 6381, normalized size = 40.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^6/(a+b*cosh(x)),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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giac [B] time = 0.19, size = 303, normalized size = 1.91 \[ \frac {2 \, b^{6} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, b^{5} e^{\left (9 \, x\right )} - 15 \, a b^{4} e^{\left (8 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (7 \, x\right )} - 80 \, b^{5} e^{\left (7 \, x\right )} - 30 \, a^{3} b^{2} e^{\left (6 \, x\right )} + 90 \, a b^{4} e^{\left (6 \, x\right )} + 48 \, a^{4} b e^{\left (5 \, x\right )} - 136 \, a^{2} b^{3} e^{\left (5 \, x\right )} + 178 \, b^{5} e^{\left (5 \, x\right )} - 80 \, a^{5} e^{\left (4 \, x\right )} + 230 \, a^{3} b^{2} e^{\left (4 \, x\right )} - 240 \, a b^{4} e^{\left (4 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (3 \, x\right )} - 80 \, b^{5} e^{\left (3 \, x\right )} + 40 \, a^{5} e^{\left (2 \, x\right )} - 130 \, a^{3} b^{2} e^{\left (2 \, x\right )} + 150 \, a b^{4} e^{\left (2 \, x\right )} + 15 \, b^{5} e^{x} - 8 \, a^{5} + 26 \, a^{3} b^{2} - 33 \, a b^{4}\right )}}{15 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \]
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)
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maple [A] time = 0.09, size = 213, normalized size = 1.34 \[ -\frac {\frac {a^{2} \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {2 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right ) a b}{5}+\frac {b^{2} \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {5 a^{2} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}+4 a \left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) b -\frac {7 b^{2} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}+10 a^{2} \tanh \left (\frac {x}{2}\right )-28 a b \tanh \left (\frac {x}{2}\right )+22 b^{2} \tanh \left (\frac {x}{2}\right )}{32 \left (a -b \right )^{3}}+\frac {2 b^{6} \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {1}{160 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{5}}-\frac {-5 a -7 b}{96 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {10 a^{2}+28 a b +22 b^{2}}{32 \left (a +b \right )^{3} \tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(x)^6/(a+b*cosh(x)),x)
[Out]
-1/32/(a-b)^3*(1/5*a^2*tanh(1/2*x)^5-2/5*tanh(1/2*x)^5*a*b+1/5*b^2*tanh(1/2*x)^5-5/3*a^2*tanh(1/2*x)^3+4*a*tan
h(1/2*x)^3*b-7/3*b^2*tanh(1/2*x)^3+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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?
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mupad [B] time = 2.61, size = 1031, normalized size = 6.48 \[ \frac {\frac {16\,\left (a\,b^2-a^3\right )}{{\left (a^2-b^2\right )}^2}+\frac {64\,{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{5\,{\left (a^2-b^2\right )}^2}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {\frac {2\,a\,b^4}{{\left (a^2-b^2\right )}^3}-\frac {2\,b^5\,{\mathrm {e}}^x}{{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {32\,a}{5\,\left (a^2-b^2\right )}-\frac {32\,b\,{\mathrm {e}}^x}{5\,\left (a^2-b^2\right )}}{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}+\frac {\frac {8\,\left (3\,a\,b^2-4\,a^3\right )}{3\,{\left (a^2-b^2\right )}^2}+\frac {8\,{\mathrm {e}}^x\,\left (12\,a^2\,b-7\,b^3\right )}{15\,{\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\,\left (a\,b^4-a^3\,b^2\right )}{{\left (a^2-b^2\right )}^3}-\frac {8\,{\mathrm {e}}^x\,\left (b^5-a^2\,b^3\right )}{3\,{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,b^4}{{\left (a^2-b^2\right )}^3\,\sqrt {b^{12}}\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {2\,a\,\left (a^7\,\sqrt {b^{12}}+3\,a^3\,b^4\,\sqrt {b^{12}}-3\,a^5\,b^2\,\sqrt {b^{12}}-a\,b^6\,\sqrt {b^{12}}\right )}{b^8\,\sqrt {-{\left (a^2-b^2\right )}^7}\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}\right )-\frac {2\,a\,\left (b^7\,\sqrt {b^{12}}-3\,a^2\,b^5\,\sqrt {b^{12}}+3\,a^4\,b^3\,\sqrt {b^{12}}-a^6\,b\,\sqrt {b^{12}}\right )}{b^8\,\sqrt {-{\left (a^2-b^2\right )}^7}\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}\right )\,\left (\frac {b^7\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}{2}-\frac {a^6\,b\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}{2}-\frac {3\,a^2\,b^5\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}{2}+\frac {3\,a^4\,b^3\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}{2}\right )\right )\,\sqrt {b^{12}}}{\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sinh(x)^6*(a + b*cosh(x))),x)
[Out]
((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))/(6*exp(4*x) - 4*exp(2*x) - 4*
exp(6*x) + exp(8*x) + 1) - ((2*a*b^4)/(a^2 - b^2)^3 - (2*b^5*exp(x))/(a^2 - b^2)^3)/(exp(2*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) + ((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))/(exp(4*x) - 2*exp(2*x) + 1) - (2*atan((exp(x)*((2*b^4)/((a^2 - b^2)^3*(b^12)^(1/2)*(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)*(b^14 - a^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*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)*(b^14 - a^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*(b^14 - a^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 -
(a^6*b*(b^14 - a^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*(b^14 - a^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*(b^14 - a^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))*(b^12)^(1/2))/(b^14 - a^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)
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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)),x)
[Out]
Timed out
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