3.176 \(\int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx\)

Optimal. Leaf size=151 \[ \frac {\left (3 a^2+9 a b+8 b^2\right ) \log (1-\cosh (x))}{16 (a+b)^3}-\frac {\left (3 a^2-9 a b+8 b^2\right ) \log (\cosh (x)+1)}{16 (a-b)^3}+\frac {\text {csch}^4(x) (b-a \cosh (x))}{4 \left (a^2-b^2\right )}+\frac {b^5 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3}+\frac {\text {csch}^2(x) \left (a \left (3 a^2-7 b^2\right ) \cosh (x)+4 b^3\right )}{8 \left (a^2-b^2\right )^2} \]

[Out]

1/8*(4*b^3+a*(3*a^2-7*b^2)*cosh(x))*csch(x)^2/(a^2-b^2)^2+1/4*(b-a*cosh(x))*csch(x)^4/(a^2-b^2)+1/16*(3*a^2+9*
a*b+8*b^2)*ln(1-cosh(x))/(a+b)^3-1/16*(3*a^2-9*a*b+8*b^2)*ln(1+cosh(x))/(a-b)^3+b^5*ln(a+b*cosh(x))/(a^2-b^2)^
3

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Rubi [A]  time = 0.25, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2668, 741, 823, 801} \[ \frac {b^5 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \log (1-\cosh (x))}{16 (a+b)^3}-\frac {\left (3 a^2-9 a b+8 b^2\right ) \log (\cosh (x)+1)}{16 (a-b)^3}+\frac {\text {csch}^4(x) (b-a \cosh (x))}{4 \left (a^2-b^2\right )}+\frac {\text {csch}^2(x) \left (a \left (3 a^2-7 b^2\right ) \cosh (x)+4 b^3\right )}{8 \left (a^2-b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^5/(a + b*Cosh[x]),x]

[Out]

((4*b^3 + a*(3*a^2 - 7*b^2)*Cosh[x])*Csch[x]^2)/(8*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^4)/(4*(a^2 - b^2)
) + ((3*a^2 + 9*a*b + 8*b^2)*Log[1 - Cosh[x]])/(16*(a + b)^3) - ((3*a^2 - 9*a*b + 8*b^2)*Log[1 + Cosh[x]])/(16
*(a - b)^3) + (b^5*Log[a + b*Cosh[x]])/(a^2 - b^2)^3

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx &=-\left (b^5 \operatorname {Subst}\left (\int \frac {1}{(a+x) \left (b^2-x^2\right )^3} \, dx,x,b \cosh (x)\right )\right )\\ &=\frac {(b-a \cosh (x)) \text {csch}^4(x)}{4 \left (a^2-b^2\right )}-\frac {b^3 \operatorname {Subst}\left (\int \frac {3 a^2-4 b^2+3 a x}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \cosh (x)\right )}{4 \left (a^2-b^2\right )}\\ &=\frac {\left (4 b^3+a \left (3 a^2-7 b^2\right ) \cosh (x)\right ) \text {csch}^2(x)}{8 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^4(x)}{4 \left (a^2-b^2\right )}+\frac {b \operatorname {Subst}\left (\int \frac {-3 a^4+7 a^2 b^2-8 b^4-a \left (3 a^2-7 b^2\right ) x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \cosh (x)\right )}{8 \left (a^2-b^2\right )^2}\\ &=\frac {\left (4 b^3+a \left (3 a^2-7 b^2\right ) \cosh (x)\right ) \text {csch}^2(x)}{8 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^4(x)}{4 \left (a^2-b^2\right )}+\frac {b \operatorname {Subst}\left (\int \left (-\frac {(a-b)^2 \left (3 a^2+9 a b+8 b^2\right )}{2 b (a+b) (b-x)}+\frac {8 b^4}{(a-b) (a+b) (a+x)}-\frac {(a+b)^2 \left (3 a^2-9 a b+8 b^2\right )}{2 (a-b) b (b+x)}\right ) \, dx,x,b \cosh (x)\right )}{8 \left (a^2-b^2\right )^2}\\ &=\frac {\left (4 b^3+a \left (3 a^2-7 b^2\right ) \cosh (x)\right ) \text {csch}^2(x)}{8 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^4(x)}{4 \left (a^2-b^2\right )}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \log (1-\cosh (x))}{16 (a+b)^3}-\frac {\left (3 a^2-9 a b+8 b^2\right ) \log (1+\cosh (x))}{16 (a-b)^3}+\frac {b^5 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3}\\ \end {align*}

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Mathematica [A]  time = 0.91, size = 148, normalized size = 0.98 \[ \frac {1}{64} \left (\frac {2 \left (3 a^2-8 a b+5 b^2\right ) \text {sech}^2\left (\frac {x}{2}\right )+\frac {8 \left (a \left (3 a^4-10 a^2 b^2+15 b^4\right ) \log \left (\tanh \left (\frac {x}{2}\right )\right )+8 b^5 \log (a+b \cosh (x))-8 b^5 \log (\sinh (x))\right )}{(a+b)^3}+(a-b)^2 \text {sech}^4\left (\frac {x}{2}\right )}{(a-b)^3}-\frac {\text {csch}^4\left (\frac {x}{2}\right )}{a+b}+\frac {2 (3 a+5 b) \text {csch}^2\left (\frac {x}{2}\right )}{(a+b)^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^5/(a + b*Cosh[x]),x]

[Out]

((2*(3*a + 5*b)*Csch[x/2]^2)/(a + b)^2 - Csch[x/2]^4/(a + b) + ((8*(8*b^5*Log[a + b*Cosh[x]] - 8*b^5*Log[Sinh[
x]] + a*(3*a^4 - 10*a^2*b^2 + 15*b^4)*Log[Tanh[x/2]]))/(a + b)^3 + 2*(3*a^2 - 8*a*b + 5*b^2)*Sech[x/2]^2 + (a
- b)^2*Sech[x/2]^4)/(a - b)^3)/64

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fricas [B]  time = 0.59, size = 3450, normalized size = 22.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="fricas")

[Out]

1/8*(2*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^7 + 2*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*sinh(x)^7 + 16*(a^2*b^3 - b
^5)*cosh(x)^6 + 2*(8*a^2*b^3 - 8*b^5 + 7*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x))*sinh(x)^6 - 2*(11*a^5 - 26*a^
3*b^2 + 15*a*b^4)*cosh(x)^5 - 2*(11*a^5 - 26*a^3*b^2 + 15*a*b^4 - 21*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^2
- 48*(a^2*b^3 - b^5)*cosh(x))*sinh(x)^5 + 32*(a^4*b - 3*a^2*b^3 + 2*b^5)*cosh(x)^4 + 2*(16*a^4*b - 48*a^2*b^3
+ 32*b^5 + 35*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^3 + 120*(a^2*b^3 - b^5)*cosh(x)^2 - 5*(11*a^5 - 26*a^3*b^
2 + 15*a*b^4)*cosh(x))*sinh(x)^4 - 2*(11*a^5 - 26*a^3*b^2 + 15*a*b^4)*cosh(x)^3 - 2*(11*a^5 - 26*a^3*b^2 + 15*
a*b^4 - 35*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^4 - 160*(a^2*b^3 - b^5)*cosh(x)^3 + 10*(11*a^5 - 26*a^3*b^2
+ 15*a*b^4)*cosh(x)^2 - 64*(a^4*b - 3*a^2*b^3 + 2*b^5)*cosh(x))*sinh(x)^3 + 16*(a^2*b^3 - b^5)*cosh(x)^2 + 2*(
21*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^5 + 8*a^2*b^3 - 8*b^5 + 120*(a^2*b^3 - b^5)*cosh(x)^4 - 10*(11*a^5 -
26*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 96*(a^4*b - 3*a^2*b^3 + 2*b^5)*cosh(x)^2 - 3*(11*a^5 - 26*a^3*b^2 + 15*a*b
^4)*cosh(x))*sinh(x)^2 + 2*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x) + 8*(b^5*cosh(x)^8 + 8*b^5*cosh(x)*sinh(x)^7
+ b^5*sinh(x)^8 - 4*b^5*cosh(x)^6 + 6*b^5*cosh(x)^4 - 4*b^5*cosh(x)^2 + 4*(7*b^5*cosh(x)^2 - b^5)*sinh(x)^6 +
8*(7*b^5*cosh(x)^3 - 3*b^5*cosh(x))*sinh(x)^5 + b^5 + 2*(35*b^5*cosh(x)^4 - 30*b^5*cosh(x)^2 + 3*b^5)*sinh(x)
^4 + 8*(7*b^5*cosh(x)^5 - 10*b^5*cosh(x)^3 + 3*b^5*cosh(x))*sinh(x)^3 + 4*(7*b^5*cosh(x)^6 - 15*b^5*cosh(x)^4
+ 9*b^5*cosh(x)^2 - b^5)*sinh(x)^2 + 8*(b^5*cosh(x)^7 - 3*b^5*cosh(x)^5 + 3*b^5*cosh(x)^3 - b^5*cosh(x))*sinh(
x))*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))) - ((3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^8 + 8*(3*a^5
- 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)*sinh(x)^7 + (3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*sinh(x)^8 - 4*(3
*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^6 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5 - 7*(3*a^5 - 10*a^3
*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^3 - 3*(3*
a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^5 + 3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5 + 6*(3*a^5 - 1
0*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^4 + 2*(9*a^5 - 30*a^3*b^2 + 45*a*b^4 + 24*b^5 + 35*(3*a^5 - 10*a^3*b^2 +
15*a*b^4 + 8*b^5)*cosh(x)^4 - 30*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(3*a^5 -
10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^5 - 10*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^3 + 3*(3*a^5 -
10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^3 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^2 + 4*(7
*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^6 - 3*a^5 + 10*a^3*b^2 - 15*a*b^4 - 8*b^5 - 15*(3*a^5 - 10*a^
3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^4 + 9*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((3*a
^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^7 - 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^5 + 3*(3*a^5
- 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^3 - (3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x))*sinh(x))*log(co
sh(x) + sinh(x) + 1) + ((3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^8 + 8*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 -
8*b^5)*cosh(x)*sinh(x)^7 + (3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*sinh(x)^8 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b
^4 - 8*b^5)*cosh(x)^6 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5 - 7*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*c
osh(x)^2)*sinh(x)^6 + 8*(7*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^3 - 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^
4 - 8*b^5)*cosh(x))*sinh(x)^5 + 3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5 + 6*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b
^5)*cosh(x)^4 + 2*(9*a^5 - 30*a^3*b^2 + 45*a*b^4 - 24*b^5 + 35*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)
^4 - 30*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8
*b^5)*cosh(x)^5 - 10*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^3 + 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*
b^5)*cosh(x))*sinh(x)^3 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^2 + 4*(7*(3*a^5 - 10*a^3*b^2 + 15*
a*b^4 - 8*b^5)*cosh(x)^6 - 3*a^5 + 10*a^3*b^2 - 15*a*b^4 + 8*b^5 - 15*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*
cosh(x)^4 + 9*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^5 - 10*a^3*b^2 + 15*a*b^4
- 8*b^5)*cosh(x)^7 - 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^5 + 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 -
8*b^5)*cosh(x)^3 - (3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(
7*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^6 + 48*(a^2*b^3 - b^5)*cosh(x)^5 + 3*a^5 - 10*a^3*b^2 + 7*a*b^4 - 5*(
11*a^5 - 26*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 64*(a^4*b - 3*a^2*b^3 + 2*b^5)*cosh(x)^3 - 3*(11*a^5 - 26*a^3*b^2
+ 15*a*b^4)*cosh(x)^2 + 16*(a^2*b^3 - b^5)*cosh(x))*sinh(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^8 +
8*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)*sinh(x)^7 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)^8 - 4*(a
^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 - 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 7*(a^6 - 3*a^4*b^2 + 3*a^
2*b^4 - b^6)*cosh(x)^2)*sinh(x)^6 + a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 8*(7*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^
6)*cosh(x)^3 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^5 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6
)*cosh(x)^4 + 2*(3*a^6 - 9*a^4*b^2 + 9*a^2*b^4 - 3*b^6 + 35*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 - 30
*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5
- 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^3
- 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2 + 4*(7*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 - a^6
+ 3*a^4*b^2 - 3*a^2*b^4 + b^6 - 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 9*(a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*cosh(x)^2)*sinh(x)^2 + 8*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 - 3*(a^6 - 3*a^4*b^2 + 3*a^
2*b^4 - b^6)*cosh(x)^5 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)
*cosh(x))*sinh(x))

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giac [B]  time = 0.16, size = 338, normalized size = 2.24 \[ \frac {b^{6} \log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}} - \frac {{\left (3 \, a^{2} - 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (3 \, a^{2} + 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {3 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 3 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 10 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 7 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 8 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 32 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 20 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )} + 56 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )} - 36 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )} + 16 \, a^{4} b - 64 \, a^{2} b^{3} + 96 \, b^{5}}{4 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="giac")

[Out]

b^6*log(abs(b*(e^(-x) + e^x) + 2*a))/(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7) - 1/16*(3*a^2 - 9*a*b + 8*b^2)*log(
e^(-x) + e^x + 2)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/16*(3*a^2 + 9*a*b + 8*b^2)*log(e^(-x) + e^x - 2)/(a^3 +
3*a^2*b + 3*a*b^2 + b^3) + 1/4*(3*b^5*(e^(-x) + e^x)^4 + 3*a^5*(e^(-x) + e^x)^3 - 10*a^3*b^2*(e^(-x) + e^x)^3
+ 7*a*b^4*(e^(-x) + e^x)^3 + 8*a^2*b^3*(e^(-x) + e^x)^2 - 32*b^5*(e^(-x) + e^x)^2 - 20*a^5*(e^(-x) + e^x) + 56
*a^3*b^2*(e^(-x) + e^x) - 36*a*b^4*(e^(-x) + e^x) + 16*a^4*b - 64*a^2*b^3 + 96*b^5)/((a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*((e^(-x) + e^x)^2 - 4)^2)

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maple [A]  time = 0.09, size = 191, normalized size = 1.26 \[ \frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right ) a}{64 \left (a -b \right )^{2}}-\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right ) b}{64 \left (a -b \right )^{2}}-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) a}{8 \left (a -b \right )^{2}}+\frac {3 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b}{16 \left (a -b \right )^{2}}+\frac {b^{5} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {1}{64 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{4}}+\frac {a}{8 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 b}{16 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) a^{2}}{8 \left (a +b \right )^{3}}+\frac {9 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) a b}{8 \left (a +b \right )^{3}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) b^{2}}{\left (a +b \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^5/(a+b*cosh(x)),x)

[Out]

1/64/(a-b)^2*tanh(1/2*x)^4*a-1/64/(a-b)^2*tanh(1/2*x)^4*b-1/8/(a-b)^2*tanh(1/2*x)^2*a+3/16/(a-b)^2*tanh(1/2*x)
^2*b+1/(a-b)^3*b^5/(a+b)^3*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-a-b)-1/64/(a+b)/tanh(1/2*x)^4+1/8/(a+b)^2/tanh(1
/2*x)^2*a+3/16/(a+b)^2/tanh(1/2*x)^2*b+3/8/(a+b)^3*ln(tanh(1/2*x))*a^2+9/8/(a+b)^3*ln(tanh(1/2*x))*a*b+1/(a+b)
^3*ln(tanh(1/2*x))*b^2

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maxima [B]  time = 0.40, size = 348, normalized size = 2.30 \[ \frac {b^{5} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {{\left (3 \, a^{2} - 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (3 \, a^{2} + 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {8 \, b^{3} e^{\left (-2 \, x\right )} + 8 \, b^{3} e^{\left (-6 \, x\right )} + {\left (3 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-x\right )} - {\left (11 \, a^{3} - 15 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-4 \, x\right )} - {\left (11 \, a^{3} - 15 \, a b^{2}\right )} e^{\left (-5 \, x\right )} + {\left (3 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="maxima")

[Out]

b^5*log(2*a*e^(-x) + b*e^(-2*x) + b)/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - 1/8*(3*a^2 - 9*a*b + 8*b^2)*log(e^(
-x) + 1)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/8*(3*a^2 + 9*a*b + 8*b^2)*log(e^(-x) - 1)/(a^3 + 3*a^2*b + 3*a*b^
2 + b^3) + 1/4*(8*b^3*e^(-2*x) + 8*b^3*e^(-6*x) + (3*a^3 - 7*a*b^2)*e^(-x) - (11*a^3 - 15*a*b^2)*e^(-3*x) + 16
*(a^2*b - 2*b^3)*e^(-4*x) - (11*a^3 - 15*a*b^2)*e^(-5*x) + (3*a^3 - 7*a*b^2)*e^(-7*x))/(a^4 - 2*a^2*b^2 + b^4
- 4*(a^4 - 2*a^2*b^2 + b^4)*e^(-2*x) + 6*(a^4 - 2*a^2*b^2 + b^4)*e^(-4*x) - 4*(a^4 - 2*a^2*b^2 + b^4)*e^(-6*x)
+ (a^4 - 2*a^2*b^2 + b^4)*e^(-8*x))

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mupad [B]  time = 1.83, size = 559, normalized size = 3.70 \[ \frac {\frac {4\,b}{a^2-b^2}-\frac {4\,a\,{\mathrm {e}}^x}{a^2-b^2}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {\frac {2\,\left (b^5-a^2\,b^3\right )}{{\left (a^2-b^2\right )}^3}-\frac {{\mathrm {e}}^x\,\left (3\,a^5-10\,a^3\,b^2+7\,a\,b^4\right )}{4\,{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{2\,x}-1}+\frac {\frac {8\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}+\frac {6\,{\mathrm {e}}^x\,\left (a\,b^2-a^3\right )}{{\left (a^2-b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}+\frac {\frac {2\,\left (2\,a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (a^3+3\,a\,b^2\right )}{2\,{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {b^5\,\ln \left (256\,b^{11}\,{\mathrm {e}}^{2\,x}-9\,a^{10}\,b+256\,b^{11}-225\,a^2\,b^9+300\,a^4\,b^7-190\,a^6\,b^5+60\,a^8\,b^3-18\,a^{11}\,{\mathrm {e}}^x-225\,a^2\,b^9\,{\mathrm {e}}^{2\,x}+300\,a^4\,b^7\,{\mathrm {e}}^{2\,x}-190\,a^6\,b^5\,{\mathrm {e}}^{2\,x}+60\,a^8\,b^3\,{\mathrm {e}}^{2\,x}+512\,a\,b^{10}\,{\mathrm {e}}^x-9\,a^{10}\,b\,{\mathrm {e}}^{2\,x}-450\,a^3\,b^8\,{\mathrm {e}}^x+600\,a^5\,b^6\,{\mathrm {e}}^x-380\,a^7\,b^4\,{\mathrm {e}}^x+120\,a^9\,b^2\,{\mathrm {e}}^x\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (3\,a^2+9\,a\,b+8\,b^2\right )}{8\,a^3+24\,a^2\,b+24\,a\,b^2+8\,b^3}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (3\,a^2-9\,a\,b+8\,b^2\right )}{8\,a^3-24\,a^2\,b+24\,a\,b^2-8\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^5*(a + b*cosh(x))),x)

[Out]

((4*b)/(a^2 - b^2) - (4*a*exp(x))/(a^2 - b^2))/(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1) - ((2*(b^
5 - a^2*b^3))/(a^2 - b^2)^3 - (exp(x)*(7*a*b^4 + 3*a^5 - 10*a^3*b^2))/(4*(a^2 - b^2)^3))/(exp(2*x) - 1) + ((8*
(a^2*b - b^3))/(a^2 - b^2)^2 + (6*exp(x)*(a*b^2 - a^3))/(a^2 - b^2)^2)/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1
) + ((2*(2*a^2*b - b^3))/(a^2 - b^2)^2 - (exp(x)*(3*a*b^2 + a^3))/(2*(a^2 - b^2)^2))/(exp(4*x) - 2*exp(2*x) +
1) + (b^5*log(256*b^11*exp(2*x) - 9*a^10*b + 256*b^11 - 225*a^2*b^9 + 300*a^4*b^7 - 190*a^6*b^5 + 60*a^8*b^3 -
18*a^11*exp(x) - 225*a^2*b^9*exp(2*x) + 300*a^4*b^7*exp(2*x) - 190*a^6*b^5*exp(2*x) + 60*a^8*b^3*exp(2*x) + 5
12*a*b^10*exp(x) - 9*a^10*b*exp(2*x) - 450*a^3*b^8*exp(x) + 600*a^5*b^6*exp(x) - 380*a^7*b^4*exp(x) + 120*a^9*
b^2*exp(x)))/(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2) + (log(exp(x) - 1)*(9*a*b + 3*a^2 + 8*b^2))/(24*a*b^2 + 24*a^
2*b + 8*a^3 + 8*b^3) - (log(exp(x) + 1)*(3*a^2 - 9*a*b + 8*b^2))/(24*a*b^2 - 24*a^2*b + 8*a^3 - 8*b^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{5}{\relax (x )}}{a + b \cosh {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**5/(a+b*cosh(x)),x)

[Out]

Integral(csch(x)**5/(a + b*cosh(x)), x)

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