### 3.673 $$\int \frac{1}{(-\coth (x)+\text{csch}(x))^5} \, dx$$

Optimal. Leaf size=30 $-\frac{4}{1-\cosh (x)}+\frac{2}{(1-\cosh (x))^2}-\log (1-\cosh (x))$

[Out]

2/(1 - Cosh[x])^2 - 4/(1 - Cosh[x]) - Log[1 - Cosh[x]]

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Rubi [A]  time = 0.0624783, antiderivative size = 30, 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} $-\frac{4}{1-\cosh (x)}+\frac{2}{(1-\cosh (x))^2}-\log (1-\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[(-Coth[x] + Csch[x])^(-5),x]

[Out]

2/(1 - Cosh[x])^2 - 4/(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))^5} \, dx &=i \int \frac{\sinh ^5(x)}{(i-i \cosh (x))^5} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(i-x)^2}{(i+x)^3} \, dx,x,-i \cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{4}{(i+x)^3}-\frac{4 i}{(i+x)^2}+\frac{1}{i+x}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac{2}{(i-i \cosh (x))^2}-\frac{4 i}{i-i \cosh (x)}-\log (1-\cosh (x))\\ \end{align*}

Mathematica [A]  time = 0.0238201, size = 32, normalized size = 1.07 $\frac{1}{2} \text{csch}^4\left (\frac{x}{2}\right )+2 \text{csch}^2\left (\frac{x}{2}\right )-2 \log \left (\sinh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-Coth[x] + Csch[x])^(-5),x]

[Out]

2*Csch[x/2]^2 + Csch[x/2]^4/2 - 2*Log[Sinh[x/2]]

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Maple [A]  time = 0.046, size = 37, normalized size = 1.2 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}-2\,\ln \left ( \tanh \left ( x/2 \right ) \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))^5,x)

[Out]

ln(tanh(1/2*x)+1)+1/2/tanh(1/2*x)^4+1/tanh(1/2*x)^2-2*ln(tanh(1/2*x))+ln(tanh(1/2*x)-1)

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Maxima [B]  time = 1.13417, size = 78, normalized size = 2.6 \begin{align*} -x - \frac{8 \,{\left (e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}}{4 \, e^{\left (-x\right )} - 6 \, e^{\left (-2 \, x\right )} + 4 \, e^{\left (-3 \, 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(1/(-coth(x)+csch(x))^5,x, algorithm="maxima")

[Out]

-x - 8*(e^(-x) - e^(-2*x) + e^(-3*x))/(4*e^(-x) - 6*e^(-2*x) + 4*e^(-3*x) - e^(-4*x) - 1) - 2*log(e^(-x) - 1)

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Fricas [B]  time = 1.93872, size = 923, normalized size = 30.77 \begin{align*} \frac{x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} - 4 \,{\left (x - 2\right )} \cosh \left (x\right )^{3} + 4 \,{\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right )^{3} + 2 \,{\left (3 \, x - 4\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, x \cosh \left (x\right )^{2} - 6 \,{\left (x - 2\right )} \cosh \left (x\right ) + 3 \, x - 4\right )} \sinh \left (x\right )^{2} - 4 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 4 \,{\left (x \cosh \left (x\right )^{3} - 3 \,{\left (x - 2\right )} \cosh \left (x\right )^{2} +{\left (3 \, x - 4\right )} \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x))^5,x, algorithm="fricas")

[Out]

(x*cosh(x)^4 + x*sinh(x)^4 - 4*(x - 2)*cosh(x)^3 + 4*(x*cosh(x) - x + 2)*sinh(x)^3 + 2*(3*x - 4)*cosh(x)^2 + 2
*(3*x*cosh(x)^2 - 6*(x - 2)*cosh(x) + 3*x - 4)*sinh(x)^2 - 4*(x - 2)*cosh(x) - 2*(cosh(x)^4 + 4*(cosh(x) - 1)*
sinh(x)^3 + sinh(x)^4 - 4*cosh(x)^3 + 6*(cosh(x)^2 - 2*cosh(x) + 1)*sinh(x)^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 - 3
*cosh(x)^2 + 3*cosh(x) - 1)*sinh(x) - 4*cosh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 4*(x*cosh(x)^3 - 3*(x - 2)*c
osh(x)^2 + (3*x - 4)*cosh(x) - x + 2)*sinh(x) + x)/(cosh(x)^4 + 4*(cosh(x) - 1)*sinh(x)^3 + sinh(x)^4 - 4*cosh
(x)^3 + 6*(cosh(x)^2 - 2*cosh(x) + 1)*sinh(x)^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x)^2 + 3*cosh(x) - 1)*si
nh(x) - 4*cosh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{\coth ^{5}{\left (x \right )} - 5 \coth ^{4}{\left (x \right )} \operatorname{csch}{\left (x \right )} + 10 \coth ^{3}{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )} - 10 \coth ^{2}{\left (x \right )} \operatorname{csch}^{3}{\left (x \right )} + 5 \coth{\left (x \right )} \operatorname{csch}^{4}{\left (x \right )} - \operatorname{csch}^{5}{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x))**5,x)

[Out]

-Integral(1/(coth(x)**5 - 5*coth(x)**4*csch(x) + 10*coth(x)**3*csch(x)**2 - 10*coth(x)**2*csch(x)**3 + 5*coth(
x)*csch(x)**4 - csch(x)**5), x)

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Giac [A]  time = 1.14177, size = 42, normalized size = 1.4 \begin{align*} x + \frac{8 \,{\left (e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} - 1\right )}^{4}} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x))^5,x, algorithm="giac")

[Out]

x + 8*(e^(3*x) - e^(2*x) + e^x)/(e^x - 1)^4 - 2*log(abs(e^x - 1))