3.76 \(\int \frac{\text{csch}^4(x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=17 \[ \frac{\coth ^2(x)}{2}-\frac{\coth ^3(x)}{3} \]

[Out]

Coth[x]^2/2 - Coth[x]^3/3

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Rubi [A]  time = 0.044374, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3516, 848, 43} \[ \frac{\coth ^2(x)}{2}-\frac{\coth ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(1 + Tanh[x]),x]

[Out]

Coth[x]^2/2 - Coth[x]^3/3

Rule 3516

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[(x^m*(a + x)^n)/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

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{\text{csch}^4(x)}{1+\tanh (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{-1+x^2}{x^4 (1+x)} \, dx,x,\tanh (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{-1+x}{x^4} \, dx,x,\tanh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{1}{x^4}+\frac{1}{x^3}\right ) \, dx,x,\tanh (x)\right )\\ &=\frac{\coth ^2(x)}{2}-\frac{\coth ^3(x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0477782, size = 20, normalized size = 1.18 \[ -\frac{1}{6} \text{csch}(x) (2 \cosh (x)+(2 \coth (x)-3) \text{csch}(x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(1 + Tanh[x]),x]

[Out]

-(Csch[x]*(2*Cosh[x] + (-3 + 2*Coth[x])*Csch[x]))/6

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Maple [B]  time = 0.03, size = 48, normalized size = 2.8 \begin{align*} -{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{8}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(1+tanh(x)),x)

[Out]

-1/24*tanh(1/2*x)^3+1/8*tanh(1/2*x)^2-1/8*tanh(1/2*x)+1/8/tanh(1/2*x)^2-1/8/tanh(1/2*x)-1/24/tanh(1/2*x)^3

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Maxima [B]  time = 1.16533, size = 101, normalized size = 5.94 \begin{align*} -\frac{2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac{4 \, e^{\left (-4 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac{2}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(1+tanh(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.28356, size = 286, normalized size = 16.82 \begin{align*} -\frac{4 \,{\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} +{\left (10 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{3} +{\left (10 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right ) + 2 \, \cosh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(1+tanh(x)),x, algorithm="fricas")

[Out]

-4/3*(2*cosh(x) + sinh(x))/(cosh(x)^5 + 5*cosh(x)*sinh(x)^4 + sinh(x)^5 + (10*cosh(x)^2 - 3)*sinh(x)^3 - 3*cos
h(x)^3 + (10*cosh(x)^3 - 9*cosh(x))*sinh(x)^2 + (5*cosh(x)^4 - 9*cosh(x)^2 + 4)*sinh(x) + 2*cosh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**4/(1+tanh(x)),x)

[Out]

Integral(csch(x)**4/(tanh(x) + 1), x)

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Giac [A]  time = 1.22717, size = 24, normalized size = 1.41 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(1+tanh(x)),x, algorithm="giac")

[Out]

-2/3*(3*e^(2*x) + 1)/(e^(2*x) - 1)^3