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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 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*}

________________________________________________________________________________________

Mathematica [B]  time = 0.05, 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

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 88, normalized size = 5.50 \[ \frac {x \cosh \relax (x)^{2} + x \sinh \relax (x)^{2} + 2 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (x \cosh \relax (x) + x - 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.11, size = 19, normalized size = 1.19 \[ x - \frac {4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} - 2 \, \log \left (e^{x} + 1\right ) \]

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(e^x + 1)

________________________________________________________________________________________

maple [B]  time = 0.37, size = 37, normalized size = 2.31 \[ -\ln \left (\sinh \relax (x )\right )+\frac {\left (\coth ^{2}\relax (x )\right )}{2}-\frac {3 \cosh \relax (x )}{\sinh \relax (x )^{2}}+\coth \relax (x ) \mathrm {csch}\relax (x )-2 \arctanh \left ({\mathrm e}^{x}\right )+\frac {3}{2 \sinh \relax (x )^{2}} \]

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

________________________________________________________________________________________

maxima [B]  time = 0.51, size = 68, normalized size = 4.25 \[ \frac {3}{2} \, \coth \relax (x)^{2} - x + \frac {4 \, {\left (e^{\left (-x\right )} + e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.54, size = 31, normalized size = 1.94 \[ x-2\,\ln \left ({\mathrm {e}}^x+1\right )+\frac {4}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {4}{{\mathrm {e}}^x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(coth(x) - 1/sinh(x))^3,x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int 3 \coth {\relax (x )} \operatorname {csch}^{2}{\relax (x )}\, dx - \int \left (- 3 \coth ^{2}{\relax (x )} \operatorname {csch}{\relax (x )}\right )\, dx - \int \coth ^{3}{\relax (x )}\, dx - \int \left (- \operatorname {csch}^{3}{\relax (x )}\right )\, dx \]

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)

________________________________________________________________________________________