3.10.93 \(\int \frac {\text {sech}^2(x) \tanh ^2(x)}{(2+\tanh ^3(x))^2} \, dx\) [993]

Optimal. Leaf size=12 \[ -\frac {1}{3 \left (2+\tanh ^3(x)\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4427, 267} \begin {gather*} -\frac {1}{3 \left (\tanh ^3(x)+2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sech[x]^2*Tanh[x]^2)/(2 + Tanh[x]^3)^2,x]

[Out]

-1/3*1/(2 + Tanh[x]^3)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4427

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[d/
(b*c), Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{3 \left (2+\tanh ^3(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sech[x]^2*Tanh[x]^2)/(2 + Tanh[x]^3)^2,x]

[Out]

-1/3*1/(2 + Tanh[x]^3)

________________________________________________________________________________________

Maple [A]
time = 0.76, size = 11, normalized size = 0.92

method	result	size
derivativedivides	\(-\frac {1}{3 \left (2+\tanh ^{3}\left (x \right )\right )}\)	\(11\)
default	\(-\frac {1}{3 \left (2+\tanh ^{3}\left (x \right )\right )}\)	\(11\)
risch	\(-\frac {2 \left (3 \,{\mathrm e}^{4 x}+1\right )}{9 \left (3 \,{\mathrm e}^{6 x}+3 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{2 x}+1\right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^2*tanh(x)^2/(2+tanh(x)^3)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 10, normalized size = 0.83 \begin {gather*} -\frac {1}{3 \, {\left (\tanh \left (x\right )^{3} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2*tanh(x)^2/(2+tanh(x)^3)^2,x, algorithm="maxima")

[Out]

-1/3/(tanh(x)^3 + 2)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (10) = 20\).
time = 0.32, size = 73, normalized size = 6.08 \begin {gather*} -\frac {8 \, {\left (\cosh \left (x\right )^{2} + \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}}{9 \, {\left (3 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} + 2 \, {\left (9 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 4 \, {\left (3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2*tanh(x)^2/(2+tanh(x)^3)^2,x, algorithm="fricas")

[Out]

-8/9*(cosh(x)^2 + cosh(x)*sinh(x) + sinh(x)^2)/(3*cosh(x)^4 + 12*cosh(x)*sinh(x)^3 + 3*sinh(x)^4 + 2*(9*cosh(x
)^2 + 2)*sinh(x)^2 + 4*cosh(x)^2 + 4*(3*cosh(x)^3 + cosh(x))*sinh(x) + 9)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2*tanh(x)**2/(2+tanh(x)**3)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (10) = 20\).
time = 0.40, size = 32, normalized size = 2.67 \begin {gather*} -\frac {2 \, {\left (3 \, e^{\left (4 \, x\right )} + 1\right )}}{9 \, {\left (3 \, e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2*tanh(x)^2/(2+tanh(x)^3)^2,x, algorithm="giac")

[Out]

-2/9*(3*e^(4*x) + 1)/(3*e^(6*x) + 3*e^(4*x) + 9*e^(2*x) + 1)

________________________________________________________________________________________

Mupad [B]
time = 1.80, size = 32, normalized size = 2.67 \begin {gather*} -\frac {\frac {2\,{\mathrm {e}}^{4\,x}}{3}+\frac {2}{9}}{9\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+3\,{\mathrm {e}}^{6\,x}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^2/(cosh(x)^2*(tanh(x)^3 + 2)^2),x)

[Out]

-((2*exp(4*x))/3 + 2/9)/(9*exp(2*x) + 3*exp(4*x) + 3*exp(6*x) + 1)

________________________________________________________________________________________