3.7.61 \(\int \frac {1}{(\coth (x)+\text {csch}(x))^3} \, dx\) [661]

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

[Out]

2/(1+cosh(x))+ln(1+cosh(x))

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Rubi [A]
time = 0.05, antiderivative size = 14, 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 = {4477, 2746, 45} \begin {gather*} \frac {2}{\cosh (x)+1}+\log (\cosh (x)+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Coth[x] + Csch[x])^(-3),x]

[Out]

2/(1 + Cosh[x]) + Log[1 + Cosh[x]]

Rule 45

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 2746

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 4477

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))^3} \, dx &=-\left (i \int \frac {\sinh ^3(x)}{(i+i \cosh (x))^3} \, dx\right )\\ &=-\text {Subst}\left (\int \frac {i-x}{(i+x)^2} \, dx,x,i \cosh (x)\right )\\ &=-\text {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*}

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.29 \begin {gather*} 2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+\text {sech}^2\left (\frac {x}{2}\right ) \end {gather*}

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 = 1.06, size = 28, normalized size = 2.00

method	result	size
risch	\(-x +\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}+1\right )^{2}}+2 \ln \left ({\mathrm e}^{x}+1\right )\)	\(22\)
default	\(-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
time = 0.26, size = 31, normalized size = 2.21 \begin {gather*} 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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (14) = 28\).
time = 0.38, size = 89, normalized size = 6.36 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{3}}\, dx \end {gather*}

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 = 0.41, size = 21, normalized size = 1.50 \begin {gather*} -x + \frac {4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) \end {gather*}

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)

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Mupad [B]
time = 1.55, size = 33, normalized size = 2.36 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^x+1\right )-x-\frac {4}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}+\frac {4}{{\mathrm {e}}^x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x) + 1/sinh(x))^3,x)

[Out]

2*log(exp(x) + 1) - x - 4/(exp(2*x) + 2*exp(x) + 1) + 4/(exp(x) + 1)

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