3.661 $$\int \frac{1}{(\coth (x)+\text{csch}(x))^3} \, dx$$

Optimal. Leaf size=14 $\frac{2}{\cosh (x)+1}+\log (\cosh (x)+1)$

[Out]

2/(1 + Cosh[x]) + Log[1 + Cosh[x]]

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Rubi [A]  time = 0.0574001, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.429, Rules used = {4392, 2667, 43} $\frac{2}{\cosh (x)+1}+\log (\cosh (x)+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x] + Csch[x])^(-3),x]

[Out]

2/(1 + Cosh[x]) + Log[1 + Cosh[x]]

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(\coth (x)+\text{csch}(x))^3} \, dx &=-\left (i \int \frac{\sinh ^3(x)}{(i+i \cosh (x))^3} \, dx\right )\\ &=-\operatorname{Subst}\left (\int \frac{i-x}{(i+x)^2} \, dx,x,i \cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{-i-x}+\frac{2 i}{(i+x)^2}\right ) \, dx,x,i \cosh (x)\right )\\ &=\frac{2 i}{i+i \cosh (x)}+\log (1+\cosh (x))\\ \end{align*}

Mathematica [A]  time = 0.0200618, size = 18, normalized size = 1.29 $\text{sech}^2\left (\frac{x}{2}\right )+2 \log \left (\cosh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[x] + Csch[x])^(-3),x]

[Out]

2*Log[Cosh[x/2]] + Sech[x/2]^2

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Maple [A]  time = 0.034, size = 28, normalized size = 2. \begin{align*} - \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)+csch(x))^3,x)

[Out]

-tanh(1/2*x)^2-ln(tanh(1/2*x)-1)-ln(tanh(1/2*x)+1)

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Maxima [B]  time = 1.17757, size = 42, normalized size = 3. \begin{align*} x + \frac{4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^3,x, algorithm="maxima")

[Out]

x + 4*e^(-x)/(2*e^(-x) + e^(-2*x) + 1) + 2*log(e^(-x) + 1)

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Fricas [B]  time = 2.10252, size = 336, normalized size = 24. \begin{align*} -\frac{x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} + 2 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \,{\left (x \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^3,x, algorithm="fricas")

[Out]

-(x*cosh(x)^2 + x*sinh(x)^2 + 2*(x - 2)*cosh(x) - 2*(cosh(x)^2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^2 + 2*cosh(
x) + 1)*log(cosh(x) + sinh(x) + 1) + 2*(x*cosh(x) + x - 2)*sinh(x) + x)/(cosh(x)^2 + 2*(cosh(x) + 1)*sinh(x) +
sinh(x)^2 + 2*cosh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))**3,x)

[Out]

Integral((coth(x) + csch(x))**(-3), x)

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Giac [A]  time = 1.12657, size = 28, normalized size = 2. \begin{align*} -x + \frac{4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^3,x, algorithm="giac")

[Out]

-x + 4*e^x/(e^x + 1)^2 + 2*log(e^x + 1)