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

Optimal. Leaf size=130 \[ \frac {b \coth ^4(x)}{4 a^2}-\frac {b \left (a^2-b^2\right )^2 \log (\tanh (x))}{a^6}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{a^6}-\frac {\left (a^2-b^2\right )^2 \coth (x)}{a^5}-\frac {b \left (2 a^2-b^2\right ) \coth ^2(x)}{2 a^4}+\frac {\left (2 a^2-b^2\right ) \coth ^3(x)}{3 a^3}-\frac {\coth ^5(x)}{5 a} \]

[Out]

-(a^2-b^2)^2*coth(x)/a^5-1/2*b*(2*a^2-b^2)*coth(x)^2/a^4+1/3*(2*a^2-b^2)*coth(x)^3/a^3+1/4*b*coth(x)^4/a^2-1/5
*coth(x)^5/a-b*(a^2-b^2)^2*ln(tanh(x))/a^6+b*(a^2-b^2)^2*ln(a+b*tanh(x))/a^6

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Rubi [A]  time = 0.15, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3516, 894} \[ \frac {\left (2 a^2-b^2\right ) \coth ^3(x)}{3 a^3}-\frac {b \left (2 a^2-b^2\right ) \coth ^2(x)}{2 a^4}-\frac {\left (a^2-b^2\right )^2 \coth (x)}{a^5}-\frac {b \left (a^2-b^2\right )^2 \log (\tanh (x))}{a^6}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{a^6}+\frac {b \coth ^4(x)}{4 a^2}-\frac {\coth ^5(x)}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^6/(a + b*Tanh[x]),x]

[Out]

-(((a^2 - b^2)^2*Coth[x])/a^5) - (b*(2*a^2 - b^2)*Coth[x]^2)/(2*a^4) + ((2*a^2 - b^2)*Coth[x]^3)/(3*a^3) + (b*
Coth[x]^4)/(4*a^2) - Coth[x]^5/(5*a) - (b*(a^2 - b^2)^2*Log[Tanh[x]])/a^6 + (b*(a^2 - b^2)^2*Log[a + b*Tanh[x]
])/a^6

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 3516

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[(x^m*(a + x)^n)/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

Rubi steps

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

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Mathematica [A]  time = 0.69, size = 119, normalized size = 0.92 \[ \frac {15 b \left (a^4 \text {csch}^4(x)-2 a^2 \left (a^2-b^2\right ) \text {csch}^2(x)-4 \left (a^2-b^2\right )^2 (\log (\sinh (x))-\log (a \cosh (x)+b \sinh (x)))\right )-4 \coth (x) \left (3 a^5 \text {csch}^4(x)+8 a^5-25 a^3 b^2+\left (5 a^3 b^2-4 a^5\right ) \text {csch}^2(x)+15 a b^4\right )}{60 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Coth[x]*(8*a^5 - 25*a^3*b^2 + 15*a*b^4 + (-4*a^5 + 5*a^3*b^2)*Csch[x]^2 + 3*a^5*Csch[x]^4) + 15*b*(-2*a^2*
(a^2 - b^2)*Csch[x]^2 + a^4*Csch[x]^4 - 4*(a^2 - b^2)^2*(Log[Sinh[x]] - Log[a*Cosh[x] + b*Sinh[x]])))/(60*a^6)

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fricas [B]  time = 0.60, size = 2972, normalized size = 22.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(30*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)^8 + 240*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)*sinh
(x)^7 + 30*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*sinh(x)^8 - 30*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh
(x)^6 - 30*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4 - 28*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)^2)*sinh
(x)^6 + 60*(28*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)^3 - 3*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*c
osh(x))*sinh(x)^5 + 16*a^5 - 50*a^3*b^2 + 30*a*b^4 + 10*(16*a^5 + 15*a^4*b - 32*a^3*b^2 - 9*a^2*b^3 + 18*a*b^4
)*cosh(x)^4 + 10*(16*a^5 + 15*a^4*b - 32*a^3*b^2 - 9*a^2*b^3 + 18*a*b^4 + 210*(a^4*b - a^3*b^2 - a^2*b^3 + a*b
^4)*cosh(x)^4 - 45*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh(x)^2)*sinh(x)^4 + 40*(42*(a^4*b - a^3*b^2
- a^2*b^3 + a*b^4)*cosh(x)^5 - 15*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh(x)^3 + (16*a^5 + 15*a^4*b -
32*a^3*b^2 - 9*a^2*b^3 + 18*a*b^4)*cosh(x))*sinh(x)^3 - 10*(8*a^5 + 3*a^4*b - 22*a^3*b^2 - 3*a^2*b^3 + 12*a*b
^4)*cosh(x)^2 + 10*(84*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)^6 - 8*a^5 - 3*a^4*b + 22*a^3*b^2 + 3*a^2*b^
3 - 12*a*b^4 - 45*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh(x)^4 + 6*(16*a^5 + 15*a^4*b - 32*a^3*b^2 -
9*a^2*b^3 + 18*a*b^4)*cosh(x)^2)*sinh(x)^2 - 15*((a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^10 + 10*(a^4*b - 2*a^2*b^3
+ b^5)*cosh(x)*sinh(x)^9 + (a^4*b - 2*a^2*b^3 + b^5)*sinh(x)^10 - 5*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^8 - 5*(a
^4*b - 2*a^2*b^3 + b^5 - 9*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^4*b - 2*a^2*b^3 + b^5)*co
sh(x)^3 - (a^4*b - 2*a^2*b^3 + b^5)*cosh(x))*sinh(x)^7 + 10*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^6 + 10*(a^4*b -
2*a^2*b^3 + b^5 + 21*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^4 - 14*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^2)*sinh(x)^6 +
4*(63*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^5 - 70*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^3 + 15*(a^4*b - 2*a^2*b^3 +
b^5)*cosh(x))*sinh(x)^5 - a^4*b + 2*a^2*b^3 - b^5 - 10*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^4 + 10*(21*(a^4*b - 2
*a^2*b^3 + b^5)*cosh(x)^6 - a^4*b + 2*a^2*b^3 - b^5 - 35*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^4 + 15*(a^4*b - 2*a
^2*b^3 + b^5)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^7 - 7*(a^4*b - 2*a^2*b^3 + b^5)*c
osh(x)^5 + 5*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^3 - (a^4*b - 2*a^2*b^3 + b^5)*cosh(x))*sinh(x)^3 + 5*(a^4*b - 2
*a^2*b^3 + b^5)*cosh(x)^2 + 5*(9*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^8 - 28*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^6
+ a^4*b - 2*a^2*b^3 + b^5 + 30*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^4 - 12*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^2)*s
inh(x)^2 + 10*((a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^9 - 4*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^7 + 6*(a^4*b - 2*a^2*
b^3 + b^5)*cosh(x)^5 - 4*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^3 + (a^4*b - 2*a^2*b^3 + b^5)*cosh(x))*sinh(x))*log
(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) + 15*((a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^10 + 10*(a^4*b - 2*a^2
*b^3 + b^5)*cosh(x)*sinh(x)^9 + (a^4*b - 2*a^2*b^3 + b^5)*sinh(x)^10 - 5*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^8 -
5*(a^4*b - 2*a^2*b^3 + b^5 - 9*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^4*b - 2*a^2*b^3 + b^
5)*cosh(x)^3 - (a^4*b - 2*a^2*b^3 + b^5)*cosh(x))*sinh(x)^7 + 10*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^6 + 10*(a^4
*b - 2*a^2*b^3 + b^5 + 21*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^4 - 14*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^2)*sinh(x
)^6 + 4*(63*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^5 - 70*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^3 + 15*(a^4*b - 2*a^2*b
^3 + b^5)*cosh(x))*sinh(x)^5 - a^4*b + 2*a^2*b^3 - b^5 - 10*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^4 + 10*(21*(a^4*
b - 2*a^2*b^3 + b^5)*cosh(x)^6 - a^4*b + 2*a^2*b^3 - b^5 - 35*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^4 + 15*(a^4*b
- 2*a^2*b^3 + b^5)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^7 - 7*(a^4*b - 2*a^2*b^3 + b
^5)*cosh(x)^5 + 5*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^3 - (a^4*b - 2*a^2*b^3 + b^5)*cosh(x))*sinh(x)^3 + 5*(a^4*
b - 2*a^2*b^3 + b^5)*cosh(x)^2 + 5*(9*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^8 - 28*(a^4*b - 2*a^2*b^3 + b^5)*cosh(
x)^6 + a^4*b - 2*a^2*b^3 + b^5 + 30*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^4 - 12*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)
^2)*sinh(x)^2 + 10*((a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^9 - 4*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^7 + 6*(a^4*b - 2
*a^2*b^3 + b^5)*cosh(x)^5 - 4*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^3 + (a^4*b - 2*a^2*b^3 + b^5)*cosh(x))*sinh(x)
)*log(2*sinh(x)/(cosh(x) - sinh(x))) + 20*(12*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)^7 - 9*(5*a^4*b - 6*a
^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh(x)^5 + 2*(16*a^5 + 15*a^4*b - 32*a^3*b^2 - 9*a^2*b^3 + 18*a*b^4)*cosh(x)^3
- (8*a^5 + 3*a^4*b - 22*a^3*b^2 - 3*a^2*b^3 + 12*a*b^4)*cosh(x))*sinh(x))/(a^6*cosh(x)^10 + 10*a^6*cosh(x)*sin
h(x)^9 + a^6*sinh(x)^10 - 5*a^6*cosh(x)^8 + 10*a^6*cosh(x)^6 - 10*a^6*cosh(x)^4 + 5*(9*a^6*cosh(x)^2 - a^6)*si
nh(x)^8 + 5*a^6*cosh(x)^2 + 40*(3*a^6*cosh(x)^3 - a^6*cosh(x))*sinh(x)^7 + 10*(21*a^6*cosh(x)^4 - 14*a^6*cosh(
x)^2 + a^6)*sinh(x)^6 - a^6 + 4*(63*a^6*cosh(x)^5 - 70*a^6*cosh(x)^3 + 15*a^6*cosh(x))*sinh(x)^5 + 10*(21*a^6*
cosh(x)^6 - 35*a^6*cosh(x)^4 + 15*a^6*cosh(x)^2 - a^6)*sinh(x)^4 + 40*(3*a^6*cosh(x)^7 - 7*a^6*cosh(x)^5 + 5*a
^6*cosh(x)^3 - a^6*cosh(x))*sinh(x)^3 + 5*(9*a^6*cosh(x)^8 - 28*a^6*cosh(x)^6 + 30*a^6*cosh(x)^4 - 12*a^6*cosh
(x)^2 + a^6)*sinh(x)^2 + 10*(a^6*cosh(x)^9 - 4*a^6*cosh(x)^7 + 6*a^6*cosh(x)^5 - 4*a^6*cosh(x)^3 + a^6*cosh(x)
)*sinh(x))

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giac [B]  time = 0.13, size = 412, normalized size = 3.17 \[ \frac {{\left (a^{5} b + a^{4} b^{2} - 2 \, a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5} + b^{6}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{7} + a^{6} b} - \frac {{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{a^{6}} + \frac {137 \, a^{4} b e^{\left (10 \, x\right )} - 274 \, a^{2} b^{3} e^{\left (10 \, x\right )} + 137 \, b^{5} e^{\left (10 \, x\right )} - 805 \, a^{4} b e^{\left (8 \, x\right )} + 120 \, a^{3} b^{2} e^{\left (8 \, x\right )} + 1490 \, a^{2} b^{3} e^{\left (8 \, x\right )} - 120 \, a b^{4} e^{\left (8 \, x\right )} - 685 \, b^{5} e^{\left (8 \, x\right )} + 1970 \, a^{4} b e^{\left (6 \, x\right )} - 720 \, a^{3} b^{2} e^{\left (6 \, x\right )} - 3100 \, a^{2} b^{3} e^{\left (6 \, x\right )} + 480 \, a b^{4} e^{\left (6 \, x\right )} + 1370 \, b^{5} e^{\left (6 \, x\right )} - 640 \, a^{5} e^{\left (4 \, x\right )} - 1970 \, a^{4} b e^{\left (4 \, x\right )} + 1280 \, a^{3} b^{2} e^{\left (4 \, x\right )} + 3100 \, a^{2} b^{3} e^{\left (4 \, x\right )} - 720 \, a b^{4} e^{\left (4 \, x\right )} - 1370 \, b^{5} e^{\left (4 \, x\right )} + 320 \, a^{5} e^{\left (2 \, x\right )} + 805 \, a^{4} b e^{\left (2 \, x\right )} - 880 \, a^{3} b^{2} e^{\left (2 \, x\right )} - 1490 \, a^{2} b^{3} e^{\left (2 \, x\right )} + 480 \, a b^{4} e^{\left (2 \, x\right )} + 685 \, b^{5} e^{\left (2 \, x\right )} - 64 \, a^{5} - 137 \, a^{4} b + 200 \, a^{3} b^{2} + 274 \, a^{2} b^{3} - 120 \, a b^{4} - 137 \, b^{5}}{60 \, a^{6} {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^5*b + a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + a*b^5 + b^6)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^7 + a^6*b)
- (a^4*b - 2*a^2*b^3 + b^5)*log(abs(e^(2*x) - 1))/a^6 + 1/60*(137*a^4*b*e^(10*x) - 274*a^2*b^3*e^(10*x) + 137*
b^5*e^(10*x) - 805*a^4*b*e^(8*x) + 120*a^3*b^2*e^(8*x) + 1490*a^2*b^3*e^(8*x) - 120*a*b^4*e^(8*x) - 685*b^5*e^
(8*x) + 1970*a^4*b*e^(6*x) - 720*a^3*b^2*e^(6*x) - 3100*a^2*b^3*e^(6*x) + 480*a*b^4*e^(6*x) + 1370*b^5*e^(6*x)
- 640*a^5*e^(4*x) - 1970*a^4*b*e^(4*x) + 1280*a^3*b^2*e^(4*x) + 3100*a^2*b^3*e^(4*x) - 720*a*b^4*e^(4*x) - 13
70*b^5*e^(4*x) + 320*a^5*e^(2*x) + 805*a^4*b*e^(2*x) - 880*a^3*b^2*e^(2*x) - 1490*a^2*b^3*e^(2*x) + 480*a*b^4*
e^(2*x) + 685*b^5*e^(2*x) - 64*a^5 - 137*a^4*b + 200*a^3*b^2 + 274*a^2*b^3 - 120*a*b^4 - 137*b^5)/(a^6*(e^(2*x
) - 1)^5)

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maple [B]  time = 0.14, size = 333, normalized size = 2.56 \[ -\frac {\tanh ^{5}\left (\frac {x}{2}\right )}{160 a}+\frac {b \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{64 a^{2}}+\frac {5 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{96 a}-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) b^{2}}{24 a^{3}}-\frac {3 b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{16 a^{2}}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b^{3}}{8 a^{4}}-\frac {5 \tanh \left (\frac {x}{2}\right )}{16 a}+\frac {7 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{3}}-\frac {\tanh \left (\frac {x}{2}\right ) b^{4}}{2 a^{5}}+\frac {b \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right )}{a^{2}}-\frac {2 b^{3} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right )}{a^{4}}+\frac {b^{5} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right )}{a^{6}}-\frac {1}{160 a \tanh \left (\frac {x}{2}\right )^{5}}+\frac {5}{96 a \tanh \left (\frac {x}{2}\right )^{3}}-\frac {b^{2}}{24 a^{3} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {5}{16 a \tanh \left (\frac {x}{2}\right )}+\frac {7 b^{2}}{8 a^{3} \tanh \left (\frac {x}{2}\right )}-\frac {b^{4}}{2 a^{5} \tanh \left (\frac {x}{2}\right )}+\frac {b}{64 a^{2} \tanh \left (\frac {x}{2}\right )^{4}}-\frac {3 b}{16 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {b^{3}}{8 a^{4} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{2}}+\frac {2 b^{3} \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{4}}-\frac {b^{5} \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/160/a*tanh(1/2*x)^5+1/64/a^2*b*tanh(1/2*x)^4+5/96/a*tanh(1/2*x)^3-1/24/a^3*tanh(1/2*x)^3*b^2-3/16/a^2*b*tan
h(1/2*x)^2+1/8/a^4*tanh(1/2*x)^2*b^3-5/16/a*tanh(1/2*x)+7/8/a^3*b^2*tanh(1/2*x)-1/2/a^5*tanh(1/2*x)*b^4+1/a^2*
b*ln(a*tanh(1/2*x)^2+2*tanh(1/2*x)*b+a)-2/a^4*b^3*ln(a*tanh(1/2*x)^2+2*tanh(1/2*x)*b+a)+1/a^6*b^5*ln(a*tanh(1/
2*x)^2+2*tanh(1/2*x)*b+a)-1/160/a/tanh(1/2*x)^5+5/96/a/tanh(1/2*x)^3-1/24/a^3/tanh(1/2*x)^3*b^2-5/16/a/tanh(1/
2*x)+7/8/a^3/tanh(1/2*x)*b^2-1/2/a^5/tanh(1/2*x)*b^4+1/64/a^2*b/tanh(1/2*x)^4-3/16/a^2*b/tanh(1/2*x)^2+1/8/a^4
*b^3/tanh(1/2*x)^2-1/a^2*b*ln(tanh(1/2*x))+2/a^4*b^3*ln(tanh(1/2*x))-1/a^6*b^5*ln(tanh(1/2*x))

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maxima [B]  time = 0.33, size = 308, normalized size = 2.37 \[ \frac {2 \, {\left (8 \, a^{4} - 25 \, a^{2} b^{2} + 15 \, b^{4} - 5 \, {\left (8 \, a^{4} - 3 \, a^{3} b - 22 \, a^{2} b^{2} + 3 \, a b^{3} + 12 \, b^{4}\right )} e^{\left (-2 \, x\right )} + 5 \, {\left (16 \, a^{4} - 15 \, a^{3} b - 32 \, a^{2} b^{2} + 9 \, a b^{3} + 18 \, b^{4}\right )} e^{\left (-4 \, x\right )} + 15 \, {\left (5 \, a^{3} b + 6 \, a^{2} b^{2} - 3 \, a b^{3} - 4 \, b^{4}\right )} e^{\left (-6 \, x\right )} - 15 \, {\left (a^{3} b + a^{2} b^{2} - a b^{3} - b^{4}\right )} e^{\left (-8 \, x\right )}\right )}}{15 \, {\left (5 \, a^{5} e^{\left (-2 \, x\right )} - 10 \, a^{5} e^{\left (-4 \, x\right )} + 10 \, a^{5} e^{\left (-6 \, x\right )} - 5 \, a^{5} e^{\left (-8 \, x\right )} + a^{5} e^{\left (-10 \, x\right )} - a^{5}\right )}} + \frac {{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6}} - \frac {{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{6}} - \frac {{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(8*a^4 - 25*a^2*b^2 + 15*b^4 - 5*(8*a^4 - 3*a^3*b - 22*a^2*b^2 + 3*a*b^3 + 12*b^4)*e^(-2*x) + 5*(16*a^4 -
15*a^3*b - 32*a^2*b^2 + 9*a*b^3 + 18*b^4)*e^(-4*x) + 15*(5*a^3*b + 6*a^2*b^2 - 3*a*b^3 - 4*b^4)*e^(-6*x) - 15
*(a^3*b + a^2*b^2 - a*b^3 - b^4)*e^(-8*x))/(5*a^5*e^(-2*x) - 10*a^5*e^(-4*x) + 10*a^5*e^(-6*x) - 5*a^5*e^(-8*x
) + a^5*e^(-10*x) - a^5) + (a^4*b - 2*a^2*b^3 + b^5)*log(-(a - b)*e^(-2*x) - a - b)/a^6 - (a^4*b - 2*a^2*b^3 +
b^5)*log(e^(-x) + 1)/a^6 - (a^4*b - 2*a^2*b^3 + b^5)*log(e^(-x) - 1)/a^6

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mupad [B]  time = 1.33, size = 237, normalized size = 1.82 \[ \frac {2\,\left (a-b\right )\,\left (a\,b-b^2\right )}{a^4\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {8\,\left (4\,a^2-3\,a\,b+b^2\right )}{3\,a^3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {4\,\left (4\,a-b\right )}{a^2\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {32}{5\,a\,\left (5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1\right )}-\frac {2\,\left (a+b\right )\,\left (a-b\right )\,\left (a\,b-b^2\right )}{a^5\,\left ({\mathrm {e}}^{2\,x}-1\right )}+\frac {b\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{a^6}-\frac {b\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{a^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^6*(a + b*tanh(x))),x)

[Out]

(2*(a - b)*(a*b - b^2))/(a^4*(exp(4*x) - 2*exp(2*x) + 1)) - (8*(4*a^2 - 3*a*b + b^2))/(3*a^3*(3*exp(2*x) - 3*e
xp(4*x) + exp(6*x) - 1)) - (4*(4*a - b))/(a^2*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) - 32/(5*a
*(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*exp(8*x) + exp(10*x) - 1)) - (2*(a + b)*(a - b)*(a*b - b^2))/(a^5
*(exp(2*x) - 1)) + (b*log(a - b + a*exp(2*x) + b*exp(2*x))*(a + b)^2*(a - b)^2)/a^6 - (b*log(exp(2*x) - 1)*(a
+ b)^2*(a - b)^2)/a^6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{6}{\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(x)**6/(a + b*tanh(x)), x)

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