### 3.666 $$\int (-\coth (x)+\text{csch}(x))^3 \, dx$$

Optimal. Leaf size=16 $-\frac{2}{\cosh (x)+1}-\log (\cosh (x)+1)$

[Out]

-2/(1 + Cosh[x]) - Log[1 + Cosh[x]]

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Rubi [A]  time = 0.0581354, antiderivative size = 16, 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 = {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 (-\coth (x)+\text{csch}(x))^3 \, dx &=i \int (i-i \cosh (x))^3 \text{csch}^3(x) \, dx\\ &=-\operatorname{Subst}\left (\int \frac{i+x}{(i-x)^2} \, dx,x,-i \cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{2 i}{(-i+x)^2}+\frac{1}{-i+x}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac{2 i}{i+i \cosh (x)}-\log (1+\cosh (x))\\ \end{align*}

Mathematica [B]  time = 0.0519006, size = 43, normalized size = 2.69 $-\text{sech}^2\left (\frac{x}{2}\right )-2 \log \left (\sinh \left (\frac{x}{2}\right )\right )-\log (\sinh (x))+3 \log \left (\tanh \left (\frac{x}{2}\right )\right )+2 \log \left (\cosh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-Coth[x] + Csch[x])^3,x]

[Out]

2*Log[Cosh[x/2]] - 2*Log[Sinh[x/2]] - Log[Sinh[x]] + 3*Log[Tanh[x/2]] - Sech[x/2]^2

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Maple [B]  time = 0.02, size = 41, normalized size = 2.6 \begin{align*} -\ln \left ( \sinh \left ( x \right ) \right ) +{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{2}}{2}}-3\,{\frac{\cosh \left ( x \right ) }{ \left ( \sinh \left ( x \right ) \right ) ^{2}}}+{\rm csch} \left (x\right ){\rm coth} \left (x\right )-2\,{\it Artanh} \left ({{\rm e}^{x}} \right ) +{\frac{3\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{2\, \left ( \sinh \left ( x \right ) \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-coth(x)+csch(x))^3,x)

[Out]

-ln(sinh(x))+1/2*coth(x)^2-3/sinh(x)^2*cosh(x)+csch(x)*coth(x)-2*arctanh(exp(x))+3/2*cosh(x)^2/sinh(x)^2

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Maxima [B]  time = 1.19223, size = 92, normalized size = 5.75 \begin{align*} \frac{3}{2} \, \coth \left (x\right )^{2} - x + \frac{4 \,{\left (e^{\left (-x\right )} + e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac{2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, 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((-coth(x)+csch(x))^3,x, algorithm="maxima")

[Out]

3/2*coth(x)^2 - x + 4*(e^(-x) + e^(-3*x))/(2*e^(-2*x) - e^(-4*x) - 1) - 2*e^(-2*x)/(2*e^(-2*x) - e^(-4*x) - 1)
- 2*log(e^(-x) + 1)

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Fricas [B]  time = 2.07745, size = 335, normalized size = 20.94 \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((-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 3 \coth{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )}\, dx - \int - 3 \coth ^{2}{\left (x \right )} \operatorname{csch}{\left (x \right )}\, dx - \int \coth ^{3}{\left (x \right )}\, dx - \int - \operatorname{csch}^{3}{\left (x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(3*coth(x)*csch(x)**2, x) - Integral(-3*coth(x)**2*csch(x), x) - Integral(coth(x)**3, x) - Integral(-
csch(x)**3, x)

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Giac [A]  time = 1.11969, size = 26, normalized size = 1.62 \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((-coth(x)+csch(x))^3,x, algorithm="giac")

[Out]

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