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]

-1/4*csch(x)^4-1/6*csch(x)^6

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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 = {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 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 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])

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*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {1}{6} \text {csch}^6(x)-\frac {\text {csch}^4(x)}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*Csch[x]^4 - Csch[x]^6/6

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fricas [B]  time = 0.46, size = 222, normalized size = 13.06 \[ -\frac {4 \, {\left (3 \, \cosh \relax (x)^{4} + 12 \, \cosh \relax (x) \sinh \relax (x)^{3} + 3 \, \sinh \relax (x)^{4} + 2 \, {\left (9 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (3 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 3\right )}}{3 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 2 \, {\left (14 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x)^{6} - 6 \, \cosh \relax (x)^{6} + 4 \, {\left (14 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (35 \, \cosh \relax (x)^{4} - 45 \, \cosh \relax (x)^{2} + 8\right )} \sinh \relax (x)^{4} + 16 \, \cosh \relax (x)^{4} + 8 \, {\left (7 \, \cosh \relax (x)^{5} - 15 \, \cosh \relax (x)^{3} + 7 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (14 \, \cosh \relax (x)^{6} - 45 \, \cosh \relax (x)^{4} + 48 \, \cosh \relax (x)^{2} - 13\right )} \sinh \relax (x)^{2} - 26 \, \cosh \relax (x)^{2} + 4 \, {\left (2 \, \cosh \relax (x)^{7} - 9 \, \cosh \relax (x)^{5} + 14 \, \cosh \relax (x)^{3} - 7 \, \cosh \relax (x)\right )} \sinh \relax (x) + 15\right )}} \]

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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giac [B]  time = 0.14, size = 29, normalized size = 1.71 \[ -\frac {4 \, {\left (3 \, e^{\left (8 \, x\right )} + 2 \, e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )}\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{6}} \]

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

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maple [A]  time = 0.10, size = 18, normalized size = 1.06 \[ -\frac {\cosh ^{2}\relax (x )}{4 \sinh \relax (x )^{6}}+\frac {1}{12 \sinh \relax (x )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.31, size = 139, normalized size = 8.18 \[ \frac {4 \, e^{\left (-4 \, x\right )}}{6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1} + \frac {8 \, e^{\left (-6 \, x\right )}}{3 \, {\left (6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1\right )}} + \frac {4 \, e^{\left (-8 \, x\right )}}{6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1} \]

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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mupad [B]  time = 1.48, size = 210, normalized size = 12.35 \[ -\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{5}+\frac {12\,{\mathrm {e}}^{4\,x}}{5}+\frac {16\,{\mathrm {e}}^{6\,x}}{15}+\frac {4}{15}}{5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1}-\frac {\frac {4\,{\mathrm {e}}^{2\,x}}{3}+4\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+\frac {4\,{\mathrm {e}}^{8\,x}}{3}}{15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}-6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{15}+\frac {2}{5}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {4}{15\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\frac {6\,{\mathrm {e}}^{2\,x}}{5}+\frac {4\,{\mathrm {e}}^{4\,x}}{5}+\frac {2}{5}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/sinh(x)^4,x)

[Out]

- ((8*exp(2*x))/5 + (12*exp(4*x))/5 + (16*exp(6*x))/15 + 4/15)/(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*exp
(8*x) + exp(10*x) - 1) - ((4*exp(2*x))/3 + 4*exp(4*x) + 4*exp(6*x) + (4*exp(8*x))/3)/(15*exp(4*x) - 6*exp(2*x)
- 20*exp(6*x) + 15*exp(8*x) - 6*exp(10*x) + exp(12*x) + 1) - ((8*exp(2*x))/15 + 2/5)/(3*exp(2*x) - 3*exp(4*x)
+ exp(6*x) - 1) - 4/(15*(exp(4*x) - 2*exp(2*x) + 1)) - ((6*exp(2*x))/5 + (4*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 \[ \int \coth ^{3}{\relax (x )} \operatorname {csch}^{4}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(x)**3*csch(x)**4, x)

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