### 3.125 $$\int \coth ^3(x) \text{csch}^4(x) \, dx$$

Optimal. Leaf size=17 $-\frac{1}{6} \text{csch}^6(x)-\frac{\text{csch}^4(x)}{4}$

[Out]

-Csch[x]^4/4 - Csch[x]^6/6

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Rubi [A]  time = 0.0280708, 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 = {2606, 14} $-\frac{1}{6} \text{csch}^6(x)-\frac{\text{csch}^4(x)}{4}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[x]^3*Csch[x]^4,x]

[Out]

-Csch[x]^4/4 - Csch[x]^6/6

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 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 ^3(x) \text{csch}^4(x) \, dx &=\operatorname{Subst}\left (\int x^3 \left (-1+x^2\right ) \, dx,x,-i \text{csch}(x)\right )\\ &=\operatorname{Subst}\left (\int \left (-x^3+x^5\right ) \, dx,x,-i \text{csch}(x)\right )\\ &=-\frac{1}{4} \text{csch}^4(x)-\frac{\text{csch}^6(x)}{6}\\ \end{align*}

Mathematica [A]  time = 0.0090641, size = 17, normalized size = 1. $-\frac{1}{6} \text{csch}^6(x)-\frac{\text{csch}^4(x)}{4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[x]^3*Csch[x]^4,x]

[Out]

-Csch[x]^4/4 - Csch[x]^6/6

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Maple [B]  time = 0.013, size = 32, normalized size = 1.9 \begin{align*} -{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{6\, \left ( \sinh \left ( x \right ) \right ) ^{6}}}-{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{12\, \left ( \sinh \left ( x \right ) \right ) ^{4}}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{12\, \left ( \sinh \left ( x \right ) \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3*csch(x)^4,x)

[Out]

-1/6/sinh(x)^6*cosh(x)^2-1/12/sinh(x)^4*cosh(x)^2+1/12*cosh(x)^2/sinh(x)^2

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Maxima [B]  time = 1.02277, size = 188, normalized size = 11.06 \begin{align*} \frac{4 \, e^{\left (-4 \, x\right )}}{6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1} + \frac{8 \, e^{\left (-6 \, x\right )}}{3 \,{\left (6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1\right )}} + \frac{4 \, e^{\left (-8 \, x\right )}}{6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3*csch(x)^4,x, algorithm="maxima")

[Out]

4*e^(-4*x)/(6*e^(-2*x) - 15*e^(-4*x) + 20*e^(-6*x) - 15*e^(-8*x) + 6*e^(-10*x) - e^(-12*x) - 1) + 8/3*e^(-6*x)
/(6*e^(-2*x) - 15*e^(-4*x) + 20*e^(-6*x) - 15*e^(-8*x) + 6*e^(-10*x) - e^(-12*x) - 1) + 4*e^(-8*x)/(6*e^(-2*x)
- 15*e^(-4*x) + 20*e^(-6*x) - 15*e^(-8*x) + 6*e^(-10*x) - e^(-12*x) - 1)

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Fricas [B]  time = 1.70895, size = 737, normalized size = 43.35 \begin{align*} -\frac{4 \,{\left (3 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} + 2 \,{\left (9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}}{3 \,{\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \,{\left (14 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{6} - 6 \, \cosh \left (x\right )^{6} + 4 \,{\left (14 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \,{\left (35 \, \cosh \left (x\right )^{4} - 45 \, \cosh \left (x\right )^{2} + 8\right )} \sinh \left (x\right )^{4} + 16 \, \cosh \left (x\right )^{4} + 8 \,{\left (7 \, \cosh \left (x\right )^{5} - 15 \, \cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \,{\left (14 \, \cosh \left (x\right )^{6} - 45 \, \cosh \left (x\right )^{4} + 48 \, \cosh \left (x\right )^{2} - 13\right )} \sinh \left (x\right )^{2} - 26 \, \cosh \left (x\right )^{2} + 4 \,{\left (2 \, \cosh \left (x\right )^{7} - 9 \, \cosh \left (x\right )^{5} + 14 \, \cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 15\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3*csch(x)^4,x, algorithm="fricas")

[Out]

-4/3*(3*cosh(x)^4 + 12*cosh(x)*sinh(x)^3 + 3*sinh(x)^4 + 2*(9*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(3*co
sh(x)^3 + cosh(x))*sinh(x) + 3)/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 2*(14*cosh(x)^2 - 3)*sinh(x)^6
- 6*cosh(x)^6 + 4*(14*cosh(x)^3 - 9*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 45*cosh(x)^2 + 8)*sinh(x)^4 + 16*co
sh(x)^4 + 8*(7*cosh(x)^5 - 15*cosh(x)^3 + 7*cosh(x))*sinh(x)^3 + 2*(14*cosh(x)^6 - 45*cosh(x)^4 + 48*cosh(x)^2
- 13)*sinh(x)^2 - 26*cosh(x)^2 + 4*(2*cosh(x)^7 - 9*cosh(x)^5 + 14*cosh(x)^3 - 7*cosh(x))*sinh(x) + 15)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**3*csch(x)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.18309, size = 39, normalized size = 2.29 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (8 \, x\right )} + 2 \, e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )}\right )}}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3*csch(x)^4,x, algorithm="giac")

[Out]

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