3.100 \(\int \frac{\text{sech}^6(x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{4} (1-\tanh (x))^4-\frac{2}{3} (1-\tanh (x))^3 \]

[Out]

(-2*(1 - Tanh[x])^3)/3 + (1 - Tanh[x])^4/4

________________________________________________________________________________________

Rubi [A]  time = 0.0405961, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3487, 43} \[ \frac{1}{4} (1-\tanh (x))^4-\frac{2}{3} (1-\tanh (x))^3 \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^6/(1 + Tanh[x]),x]

[Out]

(-2*(1 - Tanh[x])^3)/3 + (1 - Tanh[x])^4/4

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

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}^6(x)}{1+\tanh (x)} \, dx &=\operatorname{Subst}\left (\int (1-x)^2 (1+x) \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (2 (1-x)^2-(1-x)^3\right ) \, dx,x,\tanh (x)\right )\\ &=-\frac{2}{3} (1-\tanh (x))^3+\frac{1}{4} (1-\tanh (x))^4\\ \end{align*}

Mathematica [A]  time = 0.0298958, size = 20, normalized size = 0.8 \[ \frac{1}{12} (4 \sinh (2 x)+\sinh (4 x)+3) \text{sech}^4(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^6/(1 + Tanh[x]),x]

[Out]

(Sech[x]^4*(3 + 4*Sinh[2*x] + Sinh[4*x]))/12

________________________________________________________________________________________

Maple [B]  time = 0.024, size = 56, normalized size = 2.2 \begin{align*} -2\,{\frac{- \left ( \tanh \left ( x/2 \right ) \right ) ^{7}+ \left ( \tanh \left ( x/2 \right ) \right ) ^{6}-5/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{5}-5/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}+ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-\tanh \left ( x/2 \right ) }{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^6/(1+tanh(x)),x)

[Out]

-2*(-tanh(1/2*x)^7+tanh(1/2*x)^6-5/3*tanh(1/2*x)^5-5/3*tanh(1/2*x)^3+tanh(1/2*x)^2-tanh(1/2*x))/(tanh(1/2*x)^2
+1)^4

________________________________________________________________________________________

Maxima [B]  time = 1.11806, size = 126, normalized size = 5.04 \begin{align*} \frac{16 \, e^{\left (-2 \, x\right )}}{3 \,{\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac{8 \, e^{\left (-4 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1} + \frac{4}{3 \,{\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^6/(1+tanh(x)),x, algorithm="maxima")

[Out]

16/3*e^(-2*x)/(4*e^(-2*x) + 6*e^(-4*x) + 4*e^(-6*x) + e^(-8*x) + 1) + 8*e^(-4*x)/(4*e^(-2*x) + 6*e^(-4*x) + 4*
e^(-6*x) + e^(-8*x) + 1) + 4/3/(4*e^(-2*x) + 6*e^(-4*x) + 4*e^(-6*x) + e^(-8*x) + 1)

________________________________________________________________________________________

Fricas [B]  time = 2.15348, size = 467, normalized size = 18.68 \begin{align*} -\frac{4 \,{\left (5 \, \cosh \left (x\right ) + 3 \, \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sinh \left (x\right )^{7} +{\left (21 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{5} + 5 \,{\left (7 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} +{\left (35 \, \cosh \left (x\right )^{4} + 40 \, \cosh \left (x\right )^{2} + 6\right )} \sinh \left (x\right )^{3} + 6 \, \cosh \left (x\right )^{3} +{\left (21 \, \cosh \left (x\right )^{5} + 40 \, \cosh \left (x\right )^{3} + 18 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (7 \, \cosh \left (x\right )^{6} + 20 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right ) + 5 \, \cosh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^6/(1+tanh(x)),x, algorithm="fricas")

[Out]

-4/3*(5*cosh(x) + 3*sinh(x))/(cosh(x)^7 + 7*cosh(x)*sinh(x)^6 + sinh(x)^7 + (21*cosh(x)^2 + 4)*sinh(x)^5 + 4*c
osh(x)^5 + 5*(7*cosh(x)^3 + 4*cosh(x))*sinh(x)^4 + (35*cosh(x)^4 + 40*cosh(x)^2 + 6)*sinh(x)^3 + 6*cosh(x)^3 +
(21*cosh(x)^5 + 40*cosh(x)^3 + 18*cosh(x))*sinh(x)^2 + (7*cosh(x)^6 + 20*cosh(x)^4 + 18*cosh(x)^2 + 3)*sinh(x
) + 5*cosh(x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{6}{\left (x \right )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**6/(1+tanh(x)),x)

[Out]

Integral(sech(x)**6/(tanh(x) + 1), x)

________________________________________________________________________________________

Giac [A]  time = 1.20393, size = 24, normalized size = 0.96 \begin{align*} -\frac{4 \,{\left (4 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^6/(1+tanh(x)),x, algorithm="giac")

[Out]

-4/3*(4*e^(2*x) + 1)/(e^(2*x) + 1)^4