3.663 \(\int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx\)

Optimal. Leaf size=22 \[ \frac {4}{\cosh (x)+1}-\frac {2}{(\cosh (x)+1)^2}+\log (\cosh (x)+1) \]

[Out]

-2/(1+cosh(x))^2+4/(1+cosh(x))+ln(1+cosh(x))

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Rubi [A]  time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \[ \frac {4}{\cosh (x)+1}-\frac {2}{(\cosh (x)+1)^2}+\log (\cosh (x)+1) \]

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.45 \[ -\frac {1}{2} \text {sech}^4\left (\frac {x}{2}\right )+2 \text {sech}^2\left (\frac {x}{2}\right )+2 \log \left (\cosh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Coth[x] + Csch[x])^(-5),x]

[Out]

2*Log[Cosh[x/2]] + 2*Sech[x/2]^2 - Sech[x/2]^4/2

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fricas [B]  time = 0.41, size = 266, normalized size = 12.09 \[ -\frac {x \cosh \relax (x)^{4} + x \sinh \relax (x)^{4} + 4 \, {\left (x - 2\right )} \cosh \relax (x)^{3} + 4 \, {\left (x \cosh \relax (x) + x - 2\right )} \sinh \relax (x)^{3} + 2 \, {\left (3 \, x - 4\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, x \cosh \relax (x)^{2} + 6 \, {\left (x - 2\right )} \cosh \relax (x) + 3 \, x - 4\right )} \sinh \relax (x)^{2} + 4 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{4} + 4 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} + 6 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{2} + 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x)^{2} + 3 \, \cosh \relax (x) + 1\right )} \sinh \relax (x) + 4 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 4 \, {\left (x \cosh \relax (x)^{3} + 3 \, {\left (x - 2\right )} \cosh \relax (x)^{2} + {\left (3 \, x - 4\right )} \cosh \relax (x) + x - 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{4} + 4 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} + 6 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{2} + 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x)^{2} + 3 \, \cosh \relax (x) + 1\right )} \sinh \relax (x) + 4 \, \cosh \relax (x) + 1} \]

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)*
cosh(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*cos
h(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)*s
inh(x) + 4*cosh(x) + 1)

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giac [A]  time = 0.12, size = 30, normalized size = 1.36 \[ -x + \frac {8 \, {\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} + 1\right )}^{4}} + 2 \, \log \left (e^{x} + 1\right ) \]

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(e^x + 1)

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maple [A]  time = 0.20, size = 36, normalized size = 1.64 \[ -\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{2}-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)+csch(x))^5,x)

[Out]

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

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maxima [B]  time = 0.68, size = 52, normalized size = 2.36 \[ x + \frac {8 \, {\left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}}{4 \, e^{\left (-x\right )} + 6 \, e^{\left (-2 \, x\right )} + 4 \, e^{\left (-3 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \]

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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mupad [B]  time = 1.61, size = 79, normalized size = 3.59 \[ 2\,\ln \left ({\mathrm {e}}^x+1\right )-x-\frac {16}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {8}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}+\frac {8}{{\mathrm {e}}^x+1}+\frac {16}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x) + 1/sinh(x))^5,x)

[Out]

2*log(exp(x) + 1) - x - 16/(exp(2*x) + 2*exp(x) + 1) - 8/(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1) +
8/(exp(x) + 1) + 16/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((coth(x) + csch(x))**(-5), x)

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