### 3.124 $$\int \coth ^2(x) \text{csch}^4(x) \, dx$$

Optimal. Leaf size=17 $\frac{\coth ^3(x)}{3}-\frac{\coth ^5(x)}{5}$

[Out]

Coth[x]^3/3 - Coth[x]^5/5

________________________________________________________________________________________

Rubi [A]  time = 0.0263057, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {2607, 14} $\frac{\coth ^3(x)}{3}-\frac{\coth ^5(x)}{5}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[x]^2*Csch[x]^4,x]

[Out]

Coth[x]^3/3 - Coth[x]^5/5

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \coth ^2(x) \text{csch}^4(x) \, dx &=i \operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,i \coth (x)\right )\\ &=i \operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,i \coth (x)\right )\\ &=\frac{\coth ^3(x)}{3}-\frac{\coth ^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0225496, size = 27, normalized size = 1.59 $\frac{2 \coth (x)}{15}-\frac{1}{5} \coth (x) \text{csch}^4(x)-\frac{1}{15} \coth (x) \text{csch}^2(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[x]^2*Csch[x]^4,x]

[Out]

(2*Coth[x])/15 - (Coth[x]*Csch[x]^2)/15 - (Coth[x]*Csch[x]^4)/5

________________________________________________________________________________________

Maple [B]  time = 0.012, size = 28, normalized size = 1.7 \begin{align*} -{\frac{\cosh \left ( x \right ) }{4\, \left ( \sinh \left ( x \right ) \right ) ^{5}}}-{\frac{{\rm coth} \left (x\right )}{4} \left ( -{\frac{8}{15}}-{\frac{ \left ({\rm csch} \left (x\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm csch} \left (x\right ) \right ) ^{2}}{15}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2*csch(x)^4,x)

[Out]

-1/4/sinh(x)^5*cosh(x)-1/4*(-8/15-1/5*csch(x)^4+4/15*csch(x)^2)*coth(x)

________________________________________________________________________________________

Maxima [B]  time = 1.01819, size = 201, normalized size = 11.82 \begin{align*} \frac{4 \, e^{\left (-2 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac{4 \, e^{\left (-4 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac{4 \, e^{\left (-6 \, x\right )}}{5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1} - \frac{4}{15 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2*csch(x)^4,x, algorithm="maxima")

[Out]

4/3*e^(-2*x)/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1) + 4/3*e^(-4*x)/(5*e^(-2*x)
- 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1) + 4*e^(-6*x)/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x)
- 5*e^(-8*x) + e^(-10*x) - 1) - 4/15/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1)

________________________________________________________________________________________

Fricas [B]  time = 1.83057, size = 555, normalized size = 32.65 \begin{align*} -\frac{8 \,{\left (7 \, \cosh \left (x\right )^{3} + 24 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 21 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 8 \, \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )}}{15 \,{\left (\cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sinh \left (x\right )^{7} +{\left (21 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{5} - 5 \, \cosh \left (x\right )^{5} + 5 \,{\left (7 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} +{\left (35 \, \cosh \left (x\right )^{4} - 50 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{3} + 9 \, \cosh \left (x\right )^{3} +{\left (21 \, \cosh \left (x\right )^{5} - 50 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (7 \, \cosh \left (x\right )^{6} - 25 \, \cosh \left (x\right )^{4} + 33 \, \cosh \left (x\right )^{2} - 15\right )} \sinh \left (x\right ) - 5 \, \cosh \left (x\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2*csch(x)^4,x, algorithm="fricas")

[Out]

-8/15*(7*cosh(x)^3 + 24*cosh(x)^2*sinh(x) + 21*cosh(x)*sinh(x)^2 + 8*sinh(x)^3 + 5*cosh(x))/(cosh(x)^7 + 7*cos
h(x)*sinh(x)^6 + sinh(x)^7 + (21*cosh(x)^2 - 5)*sinh(x)^5 - 5*cosh(x)^5 + 5*(7*cosh(x)^3 - 5*cosh(x))*sinh(x)^
4 + (35*cosh(x)^4 - 50*cosh(x)^2 + 11)*sinh(x)^3 + 9*cosh(x)^3 + (21*cosh(x)^5 - 50*cosh(x)^3 + 27*cosh(x))*si
nh(x)^2 + (7*cosh(x)^6 - 25*cosh(x)^4 + 33*cosh(x)^2 - 15)*sinh(x) - 5*cosh(x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth ^{2}{\left (x \right )} \operatorname{csch}^{4}{\left (x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**2*csch(x)**4,x)

[Out]

Integral(coth(x)**2*csch(x)**4, x)

________________________________________________________________________________________

Giac [B]  time = 1.18252, size = 41, normalized size = 2.41 \begin{align*} -\frac{4 \,{\left (15 \, e^{\left (6 \, x\right )} + 5 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}}{15 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2*csch(x)^4,x, algorithm="giac")

[Out]

-4/15*(15*e^(6*x) + 5*e^(4*x) + 5*e^(2*x) - 1)/(e^(2*x) - 1)^5