3.112 \(\int \frac{\text{sech}^4(x)}{1+\coth (x)} \, dx\)

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

[Out]

Tanh[x]^2/2 - Tanh[x]^3/3

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Rubi [A]  time = 0.0454175, 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} \[ \frac{\tanh ^2(x)}{2}-\frac{\tanh ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^4/(1 + Coth[x]),x]

[Out]

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

Mathematica [A]  time = 0.0498663, size = 17, normalized size = 1. \[ \frac{1}{6} \left (-2 \tanh ^3(x)-3 \text{sech}^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^4/(1 + Coth[x]),x]

[Out]

(-3*Sech[x]^2 - 2*Tanh[x]^3)/6

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Maple [B]  time = 0.029, size = 38, normalized size = 2.2 \begin{align*} -4\,{\frac{-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+2/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}}{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^4/(1+coth(x)),x)

[Out]

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

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Maxima [B]  time = 1.03484, 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(sech(x)^4/(1+coth(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 = 2.52306, size = 286, normalized size = 16.82 \begin{align*} -\frac{4 \,{\left (\cosh \left (x\right ) + 2 \, \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} + 2\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^4/(1+coth(x)),x, algorithm="fricas")

[Out]

-4/3*(cosh(x) + 2*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 + 2)*sinh(x) + 4*cosh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**4/(1+coth(x)),x)

[Out]

Integral(sech(x)**4/(coth(x) + 1), x)

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Giac [A]  time = 1.14445, 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(sech(x)^4/(1+coth(x)),x, algorithm="giac")

[Out]

-2/3*(3*e^(2*x) - 1)/(e^(2*x) + 1)^3