3.656 \(\int (\coth (x)+\text {csch}(x))^3 \, dx\)

Optimal. Leaf size=18 \[ \frac {2}{1-\cosh (x)}+\log (1-\cosh (x)) \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 18, 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} \[ \frac {2}{1-\cosh (x)}+\log (1-\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Coth[x] + Csch[x])^3,x]

[Out]

2/(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 (\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*}

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Mathematica [B]  time = 0.05, size = 41, normalized size = 2.28 \[ -\text {csch}^2\left (\frac {x}{2}\right )-2 \log \left (\sinh \left (\frac {x}{2}\right )\right )+\log (\sinh (x))+3 \log \left (\tanh \left (\frac {x}{2}\right )\right )+2 \log \left (\cosh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Coth[x] + Csch[x])^3,x]

[Out]

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

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fricas [B]  time = 0.43, size = 91, normalized size = 5.06 \[ -\frac {x \cosh \relax (x)^{2} + x \sinh \relax (x)^{2} - 2 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, {\left (x \cosh \relax (x) - x + 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1} \]

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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giac [A]  time = 0.13, size = 22, normalized size = 1.22 \[ -x - \frac {4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} + 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

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(abs(e^x - 1))

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maple [A]  time = 0.37, size = 35, normalized size = 1.94 \[ \ln \left (\sinh \relax (x )\right )-\frac {\left (\coth ^{2}\relax (x )\right )}{2}-\frac {3 \cosh \relax (x )}{\sinh \relax (x )^{2}}+\coth \relax (x ) \mathrm {csch}\relax (x )-2 \arctanh \left ({\mathrm e}^{x}\right )-\frac {3}{2 \sinh \relax (x )^{2}} \]

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

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maxima [B]  time = 0.36, size = 66, normalized size = 3.67 \[ -\frac {3}{2} \, \coth \relax (x)^{2} + x + \frac {4 \, {\left (e^{\left (-x\right )} + e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]

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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mupad [B]  time = 0.05, size = 33, normalized size = 1.83 \[ 2\,\ln \left ({\mathrm {e}}^x-1\right )-x-\frac {4}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1}-\frac {4}{{\mathrm {e}}^x-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((coth(x) + csch(x))**3, x)

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