### 3.655 $$\int (\coth (x)+\text{csch}(x))^4 \, dx$$

Optimal. Leaf size=30 $x+\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\frac{2 \sinh (x)}{1-\cosh (x)}$

[Out]

x + (2*Sinh[x])/(1 - Cosh[x]) + (2*Sinh[x]^3)/(3*(1 - Cosh[x])^3)

________________________________________________________________________________________

Rubi [A]  time = 0.113769, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.571, Rules used = {4392, 2670, 2680, 8} $x+\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\frac{2 \sinh (x)}{1-\cosh (x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x] + Csch[x])^4,x]

[Out]

x + (2*Sinh[x])/(1 - Cosh[x]) + (2*Sinh[x]^3)/(3*(1 - Cosh[x])^3)

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 2670

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

Rule 2680

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (\coth (x)+\text{csch}(x))^4 \, dx &=\int (i+i \cosh (x))^4 \text{csch}^4(x) \, dx\\ &=\int \frac{\sinh ^4(x)}{(i-i \cosh (x))^4} \, dx\\ &=\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}-\int \frac{\sinh ^2(x)}{(i-i \cosh (x))^2} \, dx\\ &=\frac{2 \sinh (x)}{1-\cosh (x)}+\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\int 1 \, dx\\ &=x+\frac{2 \sinh (x)}{1-\cosh (x)}+\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}\\ \end{align*}

Mathematica [A]  time = 0.0479179, size = 30, normalized size = 1. $x-\frac{8}{3} \coth \left (\frac{x}{2}\right )-\frac{2}{3} \coth \left (\frac{x}{2}\right ) \text{csch}^2\left (\frac{x}{2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[x] + Csch[x])^4,x]

[Out]

x - (8*Coth[x/2])/3 - (2*Coth[x/2]*Csch[x/2]^2)/3

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 57, normalized size = 1.9 \begin{align*} x-{\rm coth} \left (x\right )-{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{3}}{3}}-{\frac{8\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{3\, \left ( \sinh \left ( x \right ) \right ) ^{3}}}-{\frac{4\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{3\,\sinh \left ( x \right ) }}+{\frac{4\,\sinh \left ( x \right ) }{3}}-3\,{\frac{\cosh \left ( x \right ) }{ \left ( \sinh \left ( x \right ) \right ) ^{3}}}-2\, \left ( 2/3-1/3\, \left ({\rm csch} \left (x\right ) \right ) ^{2} \right ){\rm coth} \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((coth(x)+csch(x))^4,x)

[Out]

x-coth(x)-1/3*coth(x)^3-8/3/sinh(x)^3*cosh(x)^2-4/3*cosh(x)^2/sinh(x)+4/3*sinh(x)-3/sinh(x)^3*cosh(x)-2*(2/3-1
/3*csch(x)^2)*coth(x)

________________________________________________________________________________________

Maxima [B]  time = 1.18459, size = 247, normalized size = 8.23 \begin{align*} -2 \, \coth \left (x\right )^{3} + x - \frac{4 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{8 \, e^{\left (-x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac{4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac{16 \, e^{\left (-3 \, x\right )}}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{8 \, e^{\left (-5 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac{4}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{32}{3 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^4,x, algorithm="maxima")

[Out]

-2*coth(x)^3 + x - 4/3*(3*e^(-2*x) - 3*e^(-4*x) - 2)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) + 8*e^(-x)/(3*e^
(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) + 4*e^(-2*x)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) - 16/3*e^(-3*x)/(3*e
^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) + 8*e^(-5*x)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) - 4/3/(3*e^(-2*x) -
3*e^(-4*x) + e^(-6*x) - 1) + 32/3/(e^(-x) - e^x)^3

________________________________________________________________________________________

Fricas [B]  time = 1.99215, size = 234, normalized size = 7.8 \begin{align*} \frac{3 \, x \cosh \left (x\right )^{2} + 3 \, x \sinh \left (x\right )^{2} - 4 \,{\left (3 \, x + 10\right )} \cosh \left (x\right ) + 2 \,{\left (3 \, x \cosh \left (x\right ) - 3 \, x - 4\right )} \sinh \left (x\right ) + 9 \, x + 24}{3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) + 3\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^4,x, algorithm="fricas")

[Out]

1/3*(3*x*cosh(x)^2 + 3*x*sinh(x)^2 - 4*(3*x + 10)*cosh(x) + 2*(3*x*cosh(x) - 3*x - 4)*sinh(x) + 9*x + 24)/(cos
h(x)^2 + 2*(cosh(x) - 1)*sinh(x) + sinh(x)^2 - 4*cosh(x) + 3)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))**4,x)

[Out]

Integral((coth(x) + csch(x))**4, x)

________________________________________________________________________________________

Giac [A]  time = 1.11199, size = 30, normalized size = 1. \begin{align*} x - \frac{8 \,{\left (3 \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 2\right )}}{3 \,{\left (e^{x} - 1\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^4,x, algorithm="giac")

[Out]

x - 8/3*(3*e^(2*x) - 3*e^x + 2)/(e^x - 1)^3