3.129 \(\int \coth ^5(6 x) \text {csch}(6 x) \, dx\)

Optimal. Leaf size=29 \[ -\frac {1}{30} \text {csch}^5(6 x)-\frac {1}{9} \text {csch}^3(6 x)-\frac {1}{6} \text {csch}(6 x) \]

[Out]

-1/6*csch(6*x)-1/9*csch(6*x)^3-1/30*csch(6*x)^5

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2606, 194} \[ -\frac {1}{30} \text {csch}^5(6 x)-\frac {1}{9} \text {csch}^3(6 x)-\frac {1}{6} \text {csch}(6 x) \]

Antiderivative was successfully verified.

[In]

Int[Coth[6*x]^5*Csch[6*x],x]

[Out]

-Csch[6*x]/6 - Csch[6*x]^3/9 - Csch[6*x]^5/30

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 ^5(6 x) \text {csch}(6 x) \, dx &=-\left (\frac {1}{6} i \operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,-i \text {csch}(6 x)\right )\right )\\ &=-\left (\frac {1}{6} i \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-i \text {csch}(6 x)\right )\right )\\ &=-\frac {1}{6} \text {csch}(6 x)-\frac {1}{9} \text {csch}^3(6 x)-\frac {1}{30} \text {csch}^5(6 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 29, normalized size = 1.00 \[ -\frac {1}{30} \text {csch}^5(6 x)-\frac {1}{9} \text {csch}^3(6 x)-\frac {1}{6} \text {csch}(6 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[6*x]^5*Csch[6*x],x]

[Out]

-1/6*Csch[6*x] - Csch[6*x]^3/9 - Csch[6*x]^5/30

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fricas [B]  time = 0.42, size = 250, normalized size = 8.62 \[ -\frac {15 \, \cosh \left (6 \, x\right )^{5} + 75 \, \cosh \left (6 \, x\right ) \sinh \left (6 \, x\right )^{4} + 15 \, \sinh \left (6 \, x\right )^{5} + 5 \, {\left (30 \, \cosh \left (6 \, x\right )^{2} - 7\right )} \sinh \left (6 \, x\right )^{3} - 5 \, \cosh \left (6 \, x\right )^{3} + 15 \, {\left (10 \, \cosh \left (6 \, x\right )^{3} - \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right )^{2} + 3 \, {\left (25 \, \cosh \left (6 \, x\right )^{4} - 35 \, \cosh \left (6 \, x\right )^{2} + 26\right )} \sinh \left (6 \, x\right ) + 38 \, \cosh \left (6 \, x\right )}{45 \, {\left (\cosh \left (6 \, x\right )^{6} + 6 \, \cosh \left (6 \, x\right ) \sinh \left (6 \, x\right )^{5} + \sinh \left (6 \, x\right )^{6} + 3 \, {\left (5 \, \cosh \left (6 \, x\right )^{2} - 2\right )} \sinh \left (6 \, x\right )^{4} - 6 \, \cosh \left (6 \, x\right )^{4} + 4 \, {\left (5 \, \cosh \left (6 \, x\right )^{3} - 4 \, \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right )^{3} + 3 \, {\left (5 \, \cosh \left (6 \, x\right )^{4} - 12 \, \cosh \left (6 \, x\right )^{2} + 5\right )} \sinh \left (6 \, x\right )^{2} + 15 \, \cosh \left (6 \, x\right )^{2} + 2 \, {\left (3 \, \cosh \left (6 \, x\right )^{5} - 8 \, \cosh \left (6 \, x\right )^{3} + 5 \, \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right ) - 10\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(6*x)^5*csch(6*x),x, algorithm="fricas")

[Out]

-1/45*(15*cosh(6*x)^5 + 75*cosh(6*x)*sinh(6*x)^4 + 15*sinh(6*x)^5 + 5*(30*cosh(6*x)^2 - 7)*sinh(6*x)^3 - 5*cos
h(6*x)^3 + 15*(10*cosh(6*x)^3 - cosh(6*x))*sinh(6*x)^2 + 3*(25*cosh(6*x)^4 - 35*cosh(6*x)^2 + 26)*sinh(6*x) +
38*cosh(6*x))/(cosh(6*x)^6 + 6*cosh(6*x)*sinh(6*x)^5 + sinh(6*x)^6 + 3*(5*cosh(6*x)^2 - 2)*sinh(6*x)^4 - 6*cos
h(6*x)^4 + 4*(5*cosh(6*x)^3 - 4*cosh(6*x))*sinh(6*x)^3 + 3*(5*cosh(6*x)^4 - 12*cosh(6*x)^2 + 5)*sinh(6*x)^2 +
15*cosh(6*x)^2 + 2*(3*cosh(6*x)^5 - 8*cosh(6*x)^3 + 5*cosh(6*x))*sinh(6*x) - 10)

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giac [B]  time = 0.13, size = 47, normalized size = 1.62 \[ -\frac {15 \, {\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{4} + 40 \, {\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{2} + 48}{45 \, {\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(6*x)^5*csch(6*x),x, algorithm="giac")

[Out]

-1/45*(15*(e^(6*x) - e^(-6*x))^4 + 40*(e^(6*x) - e^(-6*x))^2 + 48)/(e^(6*x) - e^(-6*x))^5

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maple [A]  time = 0.13, size = 38, normalized size = 1.31 \[ -\frac {\cosh ^{4}\left (6 x \right )}{6 \sinh \left (6 x \right )^{5}}+\frac {2 \left (\cosh ^{2}\left (6 x \right )\right )}{9 \sinh \left (6 x \right )^{5}}-\frac {4}{45 \sinh \left (6 x \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(6*x)^5*csch(6*x),x)

[Out]

-1/6/sinh(6*x)^5*cosh(6*x)^4+2/9/sinh(6*x)^5*cosh(6*x)^2-4/45/sinh(6*x)^5

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maxima [B]  time = 0.54, size = 191, normalized size = 6.59 \[ \frac {e^{\left (-6 \, x\right )}}{3 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} - \frac {4 \, e^{\left (-18 \, x\right )}}{9 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} + \frac {58 \, e^{\left (-30 \, x\right )}}{45 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} - \frac {4 \, e^{\left (-42 \, x\right )}}{9 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} + \frac {e^{\left (-54 \, x\right )}}{3 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(6*x)^5*csch(6*x),x, algorithm="maxima")

[Out]

1/3*e^(-6*x)/(5*e^(-12*x) - 10*e^(-24*x) + 10*e^(-36*x) - 5*e^(-48*x) + e^(-60*x) - 1) - 4/9*e^(-18*x)/(5*e^(-
12*x) - 10*e^(-24*x) + 10*e^(-36*x) - 5*e^(-48*x) + e^(-60*x) - 1) + 58/45*e^(-30*x)/(5*e^(-12*x) - 10*e^(-24*
x) + 10*e^(-36*x) - 5*e^(-48*x) + e^(-60*x) - 1) - 4/9*e^(-42*x)/(5*e^(-12*x) - 10*e^(-24*x) + 10*e^(-36*x) -
5*e^(-48*x) + e^(-60*x) - 1) + 1/3*e^(-54*x)/(5*e^(-12*x) - 10*e^(-24*x) + 10*e^(-36*x) - 5*e^(-48*x) + e^(-60
*x) - 1)

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mupad [B]  time = 1.45, size = 40, normalized size = 1.38 \[ -\frac {{\mathrm {e}}^{6\,x}\,\left (58\,{\mathrm {e}}^{24\,x}-20\,{\mathrm {e}}^{12\,x}-20\,{\mathrm {e}}^{36\,x}+15\,{\mathrm {e}}^{48\,x}+15\right )}{45\,{\left ({\mathrm {e}}^{12\,x}-1\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(6*x)^5/sinh(6*x),x)

[Out]

-(exp(6*x)*(58*exp(24*x) - 20*exp(12*x) - 20*exp(36*x) + 15*exp(48*x) + 15))/(45*(exp(12*x) - 1)^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{5}{\left (6 x \right )} \operatorname {csch}{\left (6 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(6*x)**5*csch(6*x),x)

[Out]

Integral(coth(6*x)**5*csch(6*x), x)

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