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]

1/3*coth(x)^3-1/5*coth(x)^5

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Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 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]]

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])

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*}

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Mathematica [A]  time = 0.02, 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

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fricas [B]  time = 0.48, size = 164, normalized size = 9.65 \[ -\frac {8 \, {\left (7 \, \cosh \relax (x)^{3} + 24 \, \cosh \relax (x)^{2} \sinh \relax (x) + 21 \, \cosh \relax (x) \sinh \relax (x)^{2} + 8 \, \sinh \relax (x)^{3} + 5 \, \cosh \relax (x)\right )}}{15 \, {\left (\cosh \relax (x)^{7} + 7 \, \cosh \relax (x) \sinh \relax (x)^{6} + \sinh \relax (x)^{7} + {\left (21 \, \cosh \relax (x)^{2} - 5\right )} \sinh \relax (x)^{5} - 5 \, \cosh \relax (x)^{5} + 5 \, {\left (7 \, \cosh \relax (x)^{3} - 5 \, \cosh \relax (x)\right )} \sinh \relax (x)^{4} + {\left (35 \, \cosh \relax (x)^{4} - 50 \, \cosh \relax (x)^{2} + 11\right )} \sinh \relax (x)^{3} + 9 \, \cosh \relax (x)^{3} + {\left (21 \, \cosh \relax (x)^{5} - 50 \, \cosh \relax (x)^{3} + 27 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (7 \, \cosh \relax (x)^{6} - 25 \, \cosh \relax (x)^{4} + 33 \, \cosh \relax (x)^{2} - 15\right )} \sinh \relax (x) - 5 \, \cosh \relax (x)\right )}} \]

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))

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giac [B]  time = 0.13, size = 30, normalized size = 1.76 \[ -\frac {4 \, {\left (15 \, e^{\left (6 \, x\right )} + 5 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}}{15 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \]

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

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maple [B]  time = 0.34, size = 28, normalized size = 1.65 \[ -\frac {\cosh \relax (x )}{4 \sinh \relax (x )^{5}}-\frac {\left (-\frac {8}{15}-\frac {\mathrm {csch}\relax (x )^{4}}{5}+\frac {4 \mathrm {csch}\relax (x )^{2}}{15}\right ) \coth \relax (x )}{4} \]

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)

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maxima [B]  time = 0.32, size = 149, normalized size = 8.76 \[ \frac {4 \, e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac {4 \, e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac {4 \, e^{\left (-6 \, x\right )}}{5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1} - \frac {4}{15 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} \]

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)

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mupad [B]  time = 1.46, size = 144, normalized size = 8.47 \[ -\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{5}+\frac {16\,{\mathrm {e}}^{4\,x}}{5}+\frac {8\,{\mathrm {e}}^{6\,x}}{5}}{5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1}-\frac {\frac {4\,{\mathrm {e}}^{2\,x}}{5}+\frac {8}{15}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {2}{5\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{5}+\frac {6\,{\mathrm {e}}^{4\,x}}{5}+\frac {2}{5}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2/sinh(x)^4,x)

[Out]

- ((8*exp(2*x))/5 + (16*exp(4*x))/5 + (8*exp(6*x))/5)/(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*exp(8*x) + e
xp(10*x) - 1) - ((4*exp(2*x))/5 + 8/15)/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1) - 2/(5*(exp(4*x) - 2*exp(2*x)
+ 1)) - ((8*exp(2*x))/5 + (6*exp(4*x))/5 + 2/5)/(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{2}{\relax (x )} \operatorname {csch}^{4}{\relax (x )}\, dx \]

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)

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