### 3.127 $$\int \coth ^4(x) \text{csch}^3(x) \, dx$$

Optimal. Leaf size=38 $\frac{1}{16} \tanh ^{-1}(\cosh (x))-\frac{1}{6} \coth ^3(x) \text{csch}^3(x)-\frac{1}{8} \coth (x) \text{csch}^3(x)-\frac{1}{16} \coth (x) \text{csch}(x)$

[Out]

ArcTanh[Cosh[x]]/16 - (Coth[x]*Csch[x])/16 - (Coth[x]*Csch[x]^3)/8 - (Coth[x]^3*Csch[x]^3)/6

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Rubi [A]  time = 0.0645452, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {2611, 3768, 3770} $\frac{1}{16} \tanh ^{-1}(\cosh (x))-\frac{1}{6} \coth ^3(x) \text{csch}^3(x)-\frac{1}{8} \coth (x) \text{csch}^3(x)-\frac{1}{16} \coth (x) \text{csch}(x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

ArcTanh[Cosh[x]]/16 - (Coth[x]*Csch[x])/16 - (Coth[x]*Csch[x]^3)/8 - (Coth[x]^3*Csch[x]^3)/6

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \coth ^4(x) \text{csch}^3(x) \, dx &=-\frac{1}{6} \coth ^3(x) \text{csch}^3(x)+\frac{1}{2} \int \coth ^2(x) \text{csch}^3(x) \, dx\\ &=-\frac{1}{8} \coth (x) \text{csch}^3(x)-\frac{1}{6} \coth ^3(x) \text{csch}^3(x)+\frac{1}{8} \int \text{csch}^3(x) \, dx\\ &=-\frac{1}{16} \coth (x) \text{csch}(x)-\frac{1}{8} \coth (x) \text{csch}^3(x)-\frac{1}{6} \coth ^3(x) \text{csch}^3(x)-\frac{1}{16} \int \text{csch}(x) \, dx\\ &=\frac{1}{16} \tanh ^{-1}(\cosh (x))-\frac{1}{16} \coth (x) \text{csch}(x)-\frac{1}{8} \coth (x) \text{csch}^3(x)-\frac{1}{6} \coth ^3(x) \text{csch}^3(x)\\ \end{align*}

Mathematica [B]  time = 0.0197441, size = 84, normalized size = 2.21 $-\frac{1}{384} \text{csch}^6\left (\frac{x}{2}\right )-\frac{1}{64} \text{csch}^4\left (\frac{x}{2}\right )-\frac{1}{64} \text{csch}^2\left (\frac{x}{2}\right )-\frac{1}{384} \text{sech}^6\left (\frac{x}{2}\right )+\frac{1}{64} \text{sech}^4\left (\frac{x}{2}\right )-\frac{1}{64} \text{sech}^2\left (\frac{x}{2}\right )-\frac{1}{16} \log \left (\tanh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Csch[x/2]^2/64 - Csch[x/2]^4/64 - Csch[x/2]^6/384 - Log[Tanh[x/2]]/16 - Sech[x/2]^2/64 + Sech[x/2]^4/64 - Sec
h[x/2]^6/384

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Maple [A]  time = 0.016, size = 46, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{3}}{3\, \left ( \sinh \left ( x \right ) \right ) ^{6}}}+{\frac{\cosh \left ( x \right ) }{5\, \left ( \sinh \left ( x \right ) \right ) ^{6}}}+{\frac{{\rm coth} \left (x\right )}{5} \left ( -{\frac{ \left ({\rm csch} \left (x\right ) \right ) ^{5}}{6}}+{\frac{5\, \left ({\rm csch} \left (x\right ) \right ) ^{3}}{24}}-{\frac{5\,{\rm csch} \left (x\right )}{16}} \right ) }+{\frac{{\it Artanh} \left ({{\rm e}^{x}} \right ) }{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/sinh(x)^6*cosh(x)^3+1/5/sinh(x)^6*cosh(x)+1/5*(-1/6*csch(x)^5+5/24*csch(x)^3-5/16*csch(x))*coth(x)+1/8*ar
ctanh(exp(x))

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Maxima [B]  time = 1.04158, size = 132, normalized size = 3.47 \begin{align*} \frac{3 \, e^{\left (-x\right )} + 47 \, e^{\left (-3 \, x\right )} + 78 \, e^{\left (-5 \, x\right )} + 78 \, e^{\left (-7 \, x\right )} + 47 \, e^{\left (-9 \, x\right )} + 3 \, e^{\left (-11 \, x\right )}}{24 \,{\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{1}{16} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac{1}{16} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*e^(-x) + 47*e^(-3*x) + 78*e^(-5*x) + 78*e^(-7*x) + 47*e^(-9*x) + 3*e^(-11*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) + 1/16*log(e^(-x) + 1) - 1/16*log(e^(-x) - 1)

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Fricas [B]  time = 1.873, size = 4327, normalized size = 113.87 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(6*cosh(x)^11 + 66*cosh(x)*sinh(x)^10 + 6*sinh(x)^11 + 2*(165*cosh(x)^2 + 47)*sinh(x)^9 + 94*cosh(x)^9 +
18*(55*cosh(x)^3 + 47*cosh(x))*sinh(x)^8 + 12*(165*cosh(x)^4 + 282*cosh(x)^2 + 13)*sinh(x)^7 + 156*cosh(x)^7
+ 84*(33*cosh(x)^5 + 94*cosh(x)^3 + 13*cosh(x))*sinh(x)^6 + 12*(231*cosh(x)^6 + 987*cosh(x)^4 + 273*cosh(x)^2
+ 13)*sinh(x)^5 + 156*cosh(x)^5 + 12*(165*cosh(x)^7 + 987*cosh(x)^5 + 455*cosh(x)^3 + 65*cosh(x))*sinh(x)^4 +
2*(495*cosh(x)^8 + 3948*cosh(x)^6 + 2730*cosh(x)^4 + 780*cosh(x)^2 + 47)*sinh(x)^3 + 94*cosh(x)^3 + 6*(55*cosh
(x)^9 + 564*cosh(x)^7 + 546*cosh(x)^5 + 260*cosh(x)^3 + 47*cosh(x))*sinh(x)^2 - 3*(cosh(x)^12 + 12*cosh(x)*sin
h(x)^11 + sinh(x)^12 + 6*(11*cosh(x)^2 - 1)*sinh(x)^10 - 6*cosh(x)^10 + 20*(11*cosh(x)^3 - 3*cosh(x))*sinh(x)^
9 + 15*(33*cosh(x)^4 - 18*cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 - 30*cosh(x)^3 + 5*cosh(x
))*sinh(x)^7 + 4*(231*cosh(x)^6 - 315*cosh(x)^4 + 105*cosh(x)^2 - 5)*sinh(x)^6 - 20*cosh(x)^6 + 24*(33*cosh(x)
^7 - 63*cosh(x)^5 + 35*cosh(x)^3 - 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 - 84*cosh(x)^6 + 70*cosh(x)^4 - 20*
cosh(x)^2 + 1)*sinh(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 - 36*cosh(x)^7 + 42*cosh(x)^5 - 20*cosh(x)^3 + 3*co
sh(x))*sinh(x)^3 + 6*(11*cosh(x)^10 - 45*cosh(x)^8 + 70*cosh(x)^6 - 50*cosh(x)^4 + 15*cosh(x)^2 - 1)*sinh(x)^2
- 6*cosh(x)^2 + 12*(cosh(x)^11 - 5*cosh(x)^9 + 10*cosh(x)^7 - 10*cosh(x)^5 + 5*cosh(x)^3 - cosh(x))*sinh(x) +
1)*log(cosh(x) + sinh(x) + 1) + 3*(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 6*(11*cosh(x)^2 - 1)*sin
h(x)^10 - 6*cosh(x)^10 + 20*(11*cosh(x)^3 - 3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^4 - 18*cosh(x)^2 + 1)*sinh(x
)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 - 30*cosh(x)^3 + 5*cosh(x))*sinh(x)^7 + 4*(231*cosh(x)^6 - 315*cosh(x)^4
+ 105*cosh(x)^2 - 5)*sinh(x)^6 - 20*cosh(x)^6 + 24*(33*cosh(x)^7 - 63*cosh(x)^5 + 35*cosh(x)^3 - 5*cosh(x))*s
inh(x)^5 + 15*(33*cosh(x)^8 - 84*cosh(x)^6 + 70*cosh(x)^4 - 20*cosh(x)^2 + 1)*sinh(x)^4 + 15*cosh(x)^4 + 20*(1
1*cosh(x)^9 - 36*cosh(x)^7 + 42*cosh(x)^5 - 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)^10 - 45*cosh(x
)^8 + 70*cosh(x)^6 - 50*cosh(x)^4 + 15*cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 12*(cosh(x)^11 - 5*cosh(x)^9 +
10*cosh(x)^7 - 10*cosh(x)^5 + 5*cosh(x)^3 - cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 6*(11*cosh(x)^
10 + 141*cosh(x)^8 + 182*cosh(x)^6 + 130*cosh(x)^4 + 47*cosh(x)^2 + 1)*sinh(x) + 6*cosh(x))/(cosh(x)^12 + 12*c
osh(x)*sinh(x)^11 + sinh(x)^12 + 6*(11*cosh(x)^2 - 1)*sinh(x)^10 - 6*cosh(x)^10 + 20*(11*cosh(x)^3 - 3*cosh(x)
)*sinh(x)^9 + 15*(33*cosh(x)^4 - 18*cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 - 30*cosh(x)^3
+ 5*cosh(x))*sinh(x)^7 + 4*(231*cosh(x)^6 - 315*cosh(x)^4 + 105*cosh(x)^2 - 5)*sinh(x)^6 - 20*cosh(x)^6 + 24*(
33*cosh(x)^7 - 63*cosh(x)^5 + 35*cosh(x)^3 - 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 - 84*cosh(x)^6 + 70*cosh(
x)^4 - 20*cosh(x)^2 + 1)*sinh(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 - 36*cosh(x)^7 + 42*cosh(x)^5 - 20*cosh(x
)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)^10 - 45*cosh(x)^8 + 70*cosh(x)^6 - 50*cosh(x)^4 + 15*cosh(x)^2 - 1)
*sinh(x)^2 - 6*cosh(x)^2 + 12*(cosh(x)^11 - 5*cosh(x)^9 + 10*cosh(x)^7 - 10*cosh(x)^5 + 5*cosh(x)^3 - cosh(x))
*sinh(x) + 1)

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

[Out]

Timed out

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Giac [B]  time = 1.15026, size = 96, normalized size = 2.53 \begin{align*} -\frac{3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{5} + 32 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 48 \, e^{\left (-x\right )} - 48 \, e^{x}}{24 \,{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{3}} + \frac{1}{32} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac{1}{32} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(3*(e^(-x) + e^x)^5 + 32*(e^(-x) + e^x)^3 - 48*e^(-x) - 48*e^x)/((e^(-x) + e^x)^2 - 4)^3 + 1/32*log(e^(-
x) + e^x + 2) - 1/32*log(e^(-x) + e^x - 2)