3.652 \(\int \frac{1}{(a \coth (x)+b \text{csch}(x))^4} \, dx\)

Optimal. Leaf size=159 \[ -\frac{b \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \sinh ^3(x)}{2 a \left (a^2-b^2\right ) (a \cosh (x)+b)^2}-\frac{\sinh (x) \left (2 \left (a^2-b^2\right )-a b \cosh (x)\right )}{2 a^3 \left (a^2-b^2\right ) (a \cosh (x)+b)}+\frac{x}{a^4}-\frac{\sinh ^3(x)}{3 a (a \cosh (x)+b)^3} \]

[Out]

x/a^4 - (b*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)) - ((
2*(a^2 - b^2) - a*b*Cosh[x])*Sinh[x])/(2*a^3*(a^2 - b^2)*(b + a*Cosh[x])) - Sinh[x]^3/(3*a*(b + a*Cosh[x])^3)
- (b*Sinh[x]^3)/(2*a*(a^2 - b^2)*(b + a*Cosh[x])^2)

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Rubi [A]  time = 0.37267, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {4392, 2693, 2864, 2863, 2735, 2659, 205} \[ -\frac{b \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \sinh ^3(x)}{2 a \left (a^2-b^2\right ) (a \cosh (x)+b)^2}-\frac{\sinh (x) \left (2 \left (a^2-b^2\right )-a b \cosh (x)\right )}{2 a^3 \left (a^2-b^2\right ) (a \cosh (x)+b)}+\frac{x}{a^4}-\frac{\sinh ^3(x)}{3 a (a \cosh (x)+b)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*Coth[x] + b*Csch[x])^(-4),x]

[Out]

x/a^4 - (b*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)) - ((
2*(a^2 - b^2) - a*b*Cosh[x])*Sinh[x])/(2*a^3*(a^2 - b^2)*(b + a*Cosh[x])) - Sinh[x]^3/(3*a*(b + a*Cosh[x])^3)
- (b*Sinh[x]^3)/(2*a*(a^2 - b^2)*(b + a*Cosh[x])^2)

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2693

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(g^2*(p - 1))/(b*(m + 1)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]

Rule 2864

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a
^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*Sim
p[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 2863

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) -
a*d*p + b*d*(m + 1)*Sin[e + f*x]))/(b^2*f*(m + 1)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + 1)*(m + p +
1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Si
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && N
eQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

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 205

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

Rubi steps

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

Mathematica [A]  time = 0.433528, size = 150, normalized size = 0.94 \[ \frac{\sinh (x) \left (-\frac{a \left (8 a^2-11 b^2\right ) (a \cosh (x)+b)^2}{(a-b) (a+b)}-\frac{6 b \left (2 b^2-3 a^2\right ) \text{csch}(x) (a \cosh (x)+b)^3 \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+2 a \left (a^2-b^2\right )+7 a b (a \cosh (x)+b)+6 x \text{csch}(x) (a \cosh (x)+b)^3\right )}{6 a^4 (a \cosh (x)+b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Coth[x] + b*Csch[x])^(-4),x]

[Out]

((2*a*(a^2 - b^2) + 7*a*b*(b + a*Cosh[x]) - (a*(8*a^2 - 11*b^2)*(b + a*Cosh[x])^2)/((a - b)*(a + b)) + 6*x*(b
+ a*Cosh[x])^3*Csch[x] - (6*b*(-3*a^2 + 2*b^2)*ArcTan[((-a + b)*Tanh[x/2])/Sqrt[a^2 - b^2]]*(b + a*Cosh[x])^3*
Csch[x])/(a^2 - b^2)^(3/2))*Sinh[x])/(6*a^4*(b + a*Cosh[x])^3)

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Maple [B]  time = 0.066, size = 507, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*coth(x)+b*csch(x))^4,x)

[Out]

1/a^4*ln(tanh(1/2*x)+1)-2/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3/(a+b)*tanh(1/2*x)^5+1/a/(a*tanh(1/2*x)^2-tan
h(1/2*x)^2*b+a+b)^3/(a+b)*tanh(1/2*x)^5*b+3/a^2/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3/(a+b)*tanh(1/2*x)^5*b^
2-2/a^3/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3/(a+b)*tanh(1/2*x)^5*b^3-20/3/a/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*
b+a+b)^3*tanh(1/2*x)^3+4/a^3/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3*tanh(1/2*x)^3*b^2-2/(a*tanh(1/2*x)^2-tanh
(1/2*x)^2*b+a+b)^3/(a-b)*tanh(1/2*x)-1/a/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3/(a-b)*tanh(1/2*x)*b+3/a^2/(a*
tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3/(a-b)*tanh(1/2*x)*b^2+2/a^3/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)^3/(a-b)
*tanh(1/2*x)*b^3-3/a^2*b/(a^2-b^2)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))+2/a^4*b^3
/(a^2-b^2)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))-1/a^4*ln(tanh(1/2*x)-1)

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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(1/(a*coth(x)+b*csch(x))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.95758, size = 13226, normalized size = 83.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x))^4,x, algorithm="fricas")

[Out]

[1/6*(6*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x)^6 + 6*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*sinh(x)^6 + 16*a^7 - 38*a^5*
b^2 + 22*a^3*b^4 + 6*(5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh(x)^5 + 6*(5*a
^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x) + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*s
inh(x)^5 + 6*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4 + 18*a*b^6 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x)
^4 + 6*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4 + 18*a*b^6 + 15*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x)^2 + 3*(a^7 + 2*a^
5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x + 5*(5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh
(x))*sinh(x)^4 + 4*(24*a^6*b - 41*a^4*b^3 - 5*a^2*b^5 + 22*b^7 + 6*(3*a^6*b - 4*a^4*b^3 - a^2*b^5 + 2*b^7)*x)*
cosh(x)^3 + 4*(24*a^6*b - 41*a^4*b^3 - 5*a^2*b^5 + 22*b^7 + 30*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x)^3 + 15*(5
*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh(x)^2 + 6*(3*a^6*b - 4*a^4*b^3 - a^2*
b^5 + 2*b^7)*x + 6*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4 + 18*a*b^6 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*c
osh(x))*sinh(x)^3 + 6*(4*a^7 + 8*a^5*b^2 - 38*a^3*b^4 + 26*a*b^6 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x
)*cosh(x)^2 + 6*(4*a^7 + 8*a^5*b^2 - 38*a^3*b^4 + 26*a*b^6 + 15*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x)^4 + 10*(
5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh(x)^3 + 6*(4*a^7 + 5*a^5*b^2 - 27*a^
3*b^4 + 18*a*b^6 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x)^2 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4
*a*b^6)*x + 2*(24*a^6*b - 41*a^4*b^3 - 5*a^2*b^5 + 22*b^7 + 6*(3*a^6*b - 4*a^4*b^3 - a^2*b^5 + 2*b^7)*x)*cosh(
x))*sinh(x)^2 - 3*((3*a^5*b - 2*a^3*b^3)*cosh(x)^6 + (3*a^5*b - 2*a^3*b^3)*sinh(x)^6 + 3*a^5*b - 2*a^3*b^3 + 6
*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x)^5 + 6*(3*a^4*b^2 - 2*a^2*b^4 + (3*a^5*b - 2*a^3*b^3)*cosh(x))*sinh(x)^5 + 3*(
3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*cosh(x)^4 + 3*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5 + 5*(3*a^5*b - 2*a^3*b^3)*cosh(x
)^2 + 10*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x))*sinh(x)^4 + 4*(9*a^4*b^2 - 4*b^6)*cosh(x)^3 + 4*(9*a^4*b^2 - 4*b^6 +
5*(3*a^5*b - 2*a^3*b^3)*cosh(x)^3 + 15*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x)^2 + 3*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5)
*cosh(x))*sinh(x)^3 + 3*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*cosh(x)^2 + 3*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5 + 5*(3*
a^5*b - 2*a^3*b^3)*cosh(x)^4 + 20*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x)^3 + 6*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*cosh(
x)^2 + 4*(9*a^4*b^2 - 4*b^6)*cosh(x))*sinh(x)^2 + 6*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x) + 6*(3*a^4*b^2 - 2*a^2*b^4
+ (3*a^5*b - 2*a^3*b^3)*cosh(x)^5 + 5*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x)^4 + 2*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*
cosh(x)^3 + 2*(9*a^4*b^2 - 4*b^6)*cosh(x)^2 + (3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*cosh(x))*sinh(x))*sqrt(-a^2 + b
^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) - a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(
-a^2 + b^2)*(a*cosh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x)
+ a)) + 6*(a^7 - 2*a^5*b^2 + a^3*b^4)*x + 6*(11*a^6*b - 27*a^4*b^3 + 16*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b
^5)*x)*cosh(x) + 6*(11*a^6*b - 27*a^4*b^3 + 16*a^2*b^5 + 6*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x)^5 + 5*(5*a^6*
b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh(x)^4 + 4*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4
+ 18*a*b^6 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x)^3 + 2*(24*a^6*b - 41*a^4*b^3 - 5*a^2*b^5 + 2
2*b^7 + 6*(3*a^6*b - 4*a^4*b^3 - a^2*b^5 + 2*b^7)*x)*cosh(x)^2 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x + 2*(4*a^7
+ 8*a^5*b^2 - 38*a^3*b^4 + 26*a*b^6 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x))*sinh(x))/(a^11 - 2
*a^9*b^2 + a^7*b^4 + (a^11 - 2*a^9*b^2 + a^7*b^4)*cosh(x)^6 + (a^11 - 2*a^9*b^2 + a^7*b^4)*sinh(x)^6 + 6*(a^10
*b - 2*a^8*b^3 + a^6*b^5)*cosh(x)^5 + 6*(a^10*b - 2*a^8*b^3 + a^6*b^5 + (a^11 - 2*a^9*b^2 + a^7*b^4)*cosh(x))*
sinh(x)^5 + 3*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6)*cosh(x)^4 + 3*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b
^6 + 5*(a^11 - 2*a^9*b^2 + a^7*b^4)*cosh(x)^2 + 10*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(x))*sinh(x)^4 + 4*(3*a^
10*b - 4*a^8*b^3 - a^6*b^5 + 2*a^4*b^7)*cosh(x)^3 + 4*(3*a^10*b - 4*a^8*b^3 - a^6*b^5 + 2*a^4*b^7 + 5*(a^11 -
2*a^9*b^2 + a^7*b^4)*cosh(x)^3 + 15*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(x)^2 + 3*(a^11 + 2*a^9*b^2 - 7*a^7*b^4
+ 4*a^5*b^6)*cosh(x))*sinh(x)^3 + 3*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6)*cosh(x)^2 + 3*(a^11 + 2*a^9*b^
2 - 7*a^7*b^4 + 4*a^5*b^6 + 5*(a^11 - 2*a^9*b^2 + a^7*b^4)*cosh(x)^4 + 20*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(
x)^3 + 6*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6)*cosh(x)^2 + 4*(3*a^10*b - 4*a^8*b^3 - a^6*b^5 + 2*a^4*b^7)
*cosh(x))*sinh(x)^2 + 6*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(x) + 6*(a^10*b - 2*a^8*b^3 + a^6*b^5 + (a^11 - 2*a
^9*b^2 + a^7*b^4)*cosh(x)^5 + 5*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(x)^4 + 2*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4
*a^5*b^6)*cosh(x)^3 + 2*(3*a^10*b - 4*a^8*b^3 - a^6*b^5 + 2*a^4*b^7)*cosh(x)^2 + (a^11 + 2*a^9*b^2 - 7*a^7*b^4
+ 4*a^5*b^6)*cosh(x))*sinh(x)), 1/3*(3*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x)^6 + 3*(a^7 - 2*a^5*b^2 + a^3*b^4
)*x*sinh(x)^6 + 8*a^7 - 19*a^5*b^2 + 11*a^3*b^4 + 3*(5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 +
a^2*b^5)*x)*cosh(x)^5 + 3*(5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x) + 6*(a^
6*b - 2*a^4*b^3 + a^2*b^5)*x)*sinh(x)^5 + 3*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4 + 18*a*b^6 + 3*(a^7 + 2*a^5*b^2 -
7*a^3*b^4 + 4*a*b^6)*x)*cosh(x)^4 + 3*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4 + 18*a*b^6 + 15*(a^7 - 2*a^5*b^2 + a^3*b
^4)*x*cosh(x)^2 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x + 5*(5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b
- 2*a^4*b^3 + a^2*b^5)*x)*cosh(x))*sinh(x)^4 + 2*(24*a^6*b - 41*a^4*b^3 - 5*a^2*b^5 + 22*b^7 + 6*(3*a^6*b - 4
*a^4*b^3 - a^2*b^5 + 2*b^7)*x)*cosh(x)^3 + 2*(24*a^6*b - 41*a^4*b^3 - 5*a^2*b^5 + 22*b^7 + 30*(a^7 - 2*a^5*b^2
+ a^3*b^4)*x*cosh(x)^3 + 15*(5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh(x)^2
+ 6*(3*a^6*b - 4*a^4*b^3 - a^2*b^5 + 2*b^7)*x + 6*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4 + 18*a*b^6 + 3*(a^7 + 2*a^5*
b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x))*sinh(x)^3 + 3*(4*a^7 + 8*a^5*b^2 - 38*a^3*b^4 + 26*a*b^6 + 3*(a^7 + 2*a
^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x)^2 + 3*(4*a^7 + 8*a^5*b^2 - 38*a^3*b^4 + 26*a*b^6 + 15*(a^7 - 2*a^5*b^
2 + a^3*b^4)*x*cosh(x)^4 + 10*(5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh(x)^3
+ 6*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4 + 18*a*b^6 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x)^2 + 3*(
a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x + 2*(24*a^6*b - 41*a^4*b^3 - 5*a^2*b^5 + 22*b^7 + 6*(3*a^6*b - 4*a^4*
b^3 - a^2*b^5 + 2*b^7)*x)*cosh(x))*sinh(x)^2 + 3*((3*a^5*b - 2*a^3*b^3)*cosh(x)^6 + (3*a^5*b - 2*a^3*b^3)*sinh
(x)^6 + 3*a^5*b - 2*a^3*b^3 + 6*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x)^5 + 6*(3*a^4*b^2 - 2*a^2*b^4 + (3*a^5*b - 2*a^
3*b^3)*cosh(x))*sinh(x)^5 + 3*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*cosh(x)^4 + 3*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5 +
5*(3*a^5*b - 2*a^3*b^3)*cosh(x)^2 + 10*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x))*sinh(x)^4 + 4*(9*a^4*b^2 - 4*b^6)*cos
h(x)^3 + 4*(9*a^4*b^2 - 4*b^6 + 5*(3*a^5*b - 2*a^3*b^3)*cosh(x)^3 + 15*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x)^2 + 3*(
3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*cosh(x))*sinh(x)^3 + 3*(3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*cosh(x)^2 + 3*(3*a^5*b
+ 10*a^3*b^3 - 8*a*b^5 + 5*(3*a^5*b - 2*a^3*b^3)*cosh(x)^4 + 20*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x)^3 + 6*(3*a^5*
b + 10*a^3*b^3 - 8*a*b^5)*cosh(x)^2 + 4*(9*a^4*b^2 - 4*b^6)*cosh(x))*sinh(x)^2 + 6*(3*a^4*b^2 - 2*a^2*b^4)*cos
h(x) + 6*(3*a^4*b^2 - 2*a^2*b^4 + (3*a^5*b - 2*a^3*b^3)*cosh(x)^5 + 5*(3*a^4*b^2 - 2*a^2*b^4)*cosh(x)^4 + 2*(3
*a^5*b + 10*a^3*b^3 - 8*a*b^5)*cosh(x)^3 + 2*(9*a^4*b^2 - 4*b^6)*cosh(x)^2 + (3*a^5*b + 10*a^3*b^3 - 8*a*b^5)*
cosh(x))*sinh(x))*sqrt(a^2 - b^2)*arctan(-(a*cosh(x) + a*sinh(x) + b)/sqrt(a^2 - b^2)) + 3*(a^7 - 2*a^5*b^2 +
a^3*b^4)*x + 3*(11*a^6*b - 27*a^4*b^3 + 16*a^2*b^5 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh(x) + 3*(11*a^6*b
- 27*a^4*b^3 + 16*a^2*b^5 + 6*(a^7 - 2*a^5*b^2 + a^3*b^4)*x*cosh(x)^5 + 5*(5*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 +
6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x)*cosh(x)^4 + 4*(4*a^7 + 5*a^5*b^2 - 27*a^3*b^4 + 18*a*b^6 + 3*(a^7 + 2*a^5*b
^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x)^3 + 2*(24*a^6*b - 41*a^4*b^3 - 5*a^2*b^5 + 22*b^7 + 6*(3*a^6*b - 4*a^4*b^
3 - a^2*b^5 + 2*b^7)*x)*cosh(x)^2 + 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*x + 2*(4*a^7 + 8*a^5*b^2 - 38*a^3*b^4 + 26
*a*b^6 + 3*(a^7 + 2*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*x)*cosh(x))*sinh(x))/(a^11 - 2*a^9*b^2 + a^7*b^4 + (a^11 -
2*a^9*b^2 + a^7*b^4)*cosh(x)^6 + (a^11 - 2*a^9*b^2 + a^7*b^4)*sinh(x)^6 + 6*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cos
h(x)^5 + 6*(a^10*b - 2*a^8*b^3 + a^6*b^5 + (a^11 - 2*a^9*b^2 + a^7*b^4)*cosh(x))*sinh(x)^5 + 3*(a^11 + 2*a^9*b
^2 - 7*a^7*b^4 + 4*a^5*b^6)*cosh(x)^4 + 3*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6 + 5*(a^11 - 2*a^9*b^2 + a^
7*b^4)*cosh(x)^2 + 10*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(x))*sinh(x)^4 + 4*(3*a^10*b - 4*a^8*b^3 - a^6*b^5 +
2*a^4*b^7)*cosh(x)^3 + 4*(3*a^10*b - 4*a^8*b^3 - a^6*b^5 + 2*a^4*b^7 + 5*(a^11 - 2*a^9*b^2 + a^7*b^4)*cosh(x)^
3 + 15*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(x)^2 + 3*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6)*cosh(x))*sinh(x
)^3 + 3*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6)*cosh(x)^2 + 3*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6 + 5
*(a^11 - 2*a^9*b^2 + a^7*b^4)*cosh(x)^4 + 20*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(x)^3 + 6*(a^11 + 2*a^9*b^2 -
7*a^7*b^4 + 4*a^5*b^6)*cosh(x)^2 + 4*(3*a^10*b - 4*a^8*b^3 - a^6*b^5 + 2*a^4*b^7)*cosh(x))*sinh(x)^2 + 6*(a^10
*b - 2*a^8*b^3 + a^6*b^5)*cosh(x) + 6*(a^10*b - 2*a^8*b^3 + a^6*b^5 + (a^11 - 2*a^9*b^2 + a^7*b^4)*cosh(x)^5 +
5*(a^10*b - 2*a^8*b^3 + a^6*b^5)*cosh(x)^4 + 2*(a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6)*cosh(x)^3 + 2*(3*a^
10*b - 4*a^8*b^3 - a^6*b^5 + 2*a^4*b^7)*cosh(x)^2 + (a^11 + 2*a^9*b^2 - 7*a^7*b^4 + 4*a^5*b^6)*cosh(x))*sinh(x
))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \coth{\left (x \right )} + b \operatorname{csch}{\left (x \right )}\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x))**4,x)

[Out]

Integral((a*coth(x) + b*csch(x))**(-4), x)

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Giac [A]  time = 1.15381, size = 327, normalized size = 2.06 \begin{align*} -\frac{{\left (3 \, a^{2} b - 2 \, b^{3}\right )} \arctan \left (\frac{a e^{x} + b}{\sqrt{a^{2} - b^{2}}}\right )}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt{a^{2} - b^{2}}} + \frac{15 \, a^{4} b e^{\left (5 \, x\right )} - 18 \, a^{2} b^{3} e^{\left (5 \, x\right )} + 12 \, a^{5} e^{\left (4 \, x\right )} + 27 \, a^{3} b^{2} e^{\left (4 \, x\right )} - 54 \, a b^{4} e^{\left (4 \, x\right )} + 48 \, a^{4} b e^{\left (3 \, x\right )} - 34 \, a^{2} b^{3} e^{\left (3 \, x\right )} - 44 \, b^{5} e^{\left (3 \, x\right )} + 12 \, a^{5} e^{\left (2 \, x\right )} + 36 \, a^{3} b^{2} e^{\left (2 \, x\right )} - 78 \, a b^{4} e^{\left (2 \, x\right )} + 33 \, a^{4} b e^{x} - 48 \, a^{2} b^{3} e^{x} + 8 \, a^{5} - 11 \, a^{3} b^{2}}{3 \,{\left (a^{6} - a^{4} b^{2}\right )}{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )}^{3}} + \frac{x}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x))^4,x, algorithm="giac")

[Out]

-(3*a^2*b - 2*b^3)*arctan((a*e^x + b)/sqrt(a^2 - b^2))/((a^6 - a^4*b^2)*sqrt(a^2 - b^2)) + 1/3*(15*a^4*b*e^(5*
x) - 18*a^2*b^3*e^(5*x) + 12*a^5*e^(4*x) + 27*a^3*b^2*e^(4*x) - 54*a*b^4*e^(4*x) + 48*a^4*b*e^(3*x) - 34*a^2*b
^3*e^(3*x) - 44*b^5*e^(3*x) + 12*a^5*e^(2*x) + 36*a^3*b^2*e^(2*x) - 78*a*b^4*e^(2*x) + 33*a^4*b*e^x - 48*a^2*b
^3*e^x + 8*a^5 - 11*a^3*b^2)/((a^6 - a^4*b^2)*(a*e^(2*x) + 2*b*e^x + a)^3) + x/a^4