### 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]

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

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Rubi [A]  time = 0.0191935, antiderivative size = 29, normalized size of antiderivative = 1., 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 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 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]

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*}

Mathematica [A]  time = 0.0171361, size = 29, normalized size = 1. $-\frac{1}{30} \text{csch}^5(6 x)-\frac{1}{9} \text{csch}^3(6 x)-\frac{1}{6} \text{csch}(6 x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.017, size = 64, normalized size = 2.2 \begin{align*} -{\frac{ \left ( \cosh \left ( 6\,x \right ) \right ) ^{4}}{6\, \left ( \sinh \left ( 6\,x \right ) \right ) ^{5}}}+{\frac{2\, \left ( \cosh \left ( 6\,x \right ) \right ) ^{2}}{15\, \left ( \sinh \left ( 6\,x \right ) \right ) ^{5}}}+{\frac{4\, \left ( \cosh \left ( 6\,x \right ) \right ) ^{2}}{45\, \left ( \sinh \left ( 6\,x \right ) \right ) ^{3}}}-{\frac{4\, \left ( \cosh \left ( 6\,x \right ) \right ) ^{2}}{45\,\sinh \left ( 6\,x \right ) }}+{\frac{4\,\sinh \left ( 6\,x \right ) }{45}} \end{align*}

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+2/15/sinh(6*x)^5*cosh(6*x)^2+4/45/sinh(6*x)^3*cosh(6*x)^2-4/45/sinh(6*x)*cosh(6*x
)^2+4/45*sinh(6*x)

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Maxima [B]  time = 1.09573, size = 258, normalized size = 8.9 \begin{align*} \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 )}} \end{align*}

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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Fricas [B]  time = 1.68757, size = 705, normalized size = 24.31 \begin{align*} -\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 )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth ^{5}{\left (6 x \right )} \operatorname{csch}{\left (6 x \right )}\, dx \end{align*}

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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Giac [B]  time = 1.14253, size = 63, normalized size = 2.17 \begin{align*} -\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}} \end{align*}

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