3.66 \(\int \frac{1}{4-6 \tanh (c+d x)} \, dx\)

Optimal. Leaf size=31 \[ -\frac{3 \log (2 \cosh (c+d x)-3 \sinh (c+d x))}{10 d}-\frac{x}{5} \]

[Out]

-x/5 - (3*Log[2*Cosh[c + d*x] - 3*Sinh[c + d*x]])/(10*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0422853, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3484, 3530} \[ -\frac{3 \log (2 \cosh (c+d x)-3 \sinh (c+d x))}{10 d}-\frac{x}{5} \]

Antiderivative was successfully verified.

[In]

Int[(4 - 6*Tanh[c + d*x])^(-1),x]

[Out]

-x/5 - (3*Log[2*Cosh[c + d*x] - 3*Sinh[c + d*x]])/(10*d)

Rule 3484

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[(a*x)/(a^2 + b^2), x] + Dist[b/(a^2 + b^2),
Int[(b - a*Tan[c + d*x])/(a + b*Tan[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

\begin{align*} \int \frac{1}{4-6 \tanh (c+d x)} \, dx &=-\frac{x}{5}-\frac{3}{10} i \int \frac{6 i-4 i \tanh (c+d x)}{4-6 \tanh (c+d x)} \, dx\\ &=-\frac{x}{5}-\frac{3 \log (2 \cosh (c+d x)-3 \sinh (c+d x))}{10 d}\\ \end{align*}

Mathematica [A]  time = 0.0316149, size = 53, normalized size = 1.71 \[ -\frac{3 \log (2-3 \tanh (c+d x))}{10 d}+\frac{\log (1-\tanh (c+d x))}{4 d}+\frac{\log (\tanh (c+d x)+1)}{20 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 - 6*Tanh[c + d*x])^(-1),x]

[Out]

(-3*Log[2 - 3*Tanh[c + d*x]])/(10*d) + Log[1 - Tanh[c + d*x]]/(4*d) + Log[1 + Tanh[c + d*x]]/(20*d)

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 46, normalized size = 1.5 \begin{align*} -{\frac{3\,\ln \left ( -2+3\,\tanh \left ( dx+c \right ) \right ) }{10\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{20\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4-6*tanh(d*x+c)),x)

[Out]

-3/10/d*ln(-2+3*tanh(d*x+c))+1/20/d*ln(tanh(d*x+c)+1)+1/4/d*ln(tanh(d*x+c)-1)

________________________________________________________________________________________

Maxima [A]  time = 1.11217, size = 39, normalized size = 1.26 \begin{align*} -\frac{1}{2} \, x - \frac{c}{2 \, d} - \frac{3 \, \log \left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}{10 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-6*tanh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*x - 1/2*c/d - 3/10*log(5*e^(-2*d*x - 2*c) - 1)/d

________________________________________________________________________________________

Fricas [A]  time = 2.26511, size = 127, normalized size = 4.1 \begin{align*} \frac{d x - 3 \, \log \left (-\frac{2 \,{\left (2 \, \cosh \left (d x + c\right ) - 3 \, \sinh \left (d x + c\right )\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{10 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-6*tanh(d*x+c)),x, algorithm="fricas")

[Out]

1/10*(d*x - 3*log(-2*(2*cosh(d*x + c) - 3*sinh(d*x + c))/(cosh(d*x + c) - sinh(d*x + c))))/d

________________________________________________________________________________________

Sympy [A]  time = 0.571761, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} - \frac{x}{2} - \frac{3 \log{\left (\tanh{\left (c + d x \right )} - \frac{2}{3} \right )}}{10 d} + \frac{3 \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{10 d} & \text{for}\: d \neq 0 \\\frac{x}{4 - 6 \tanh{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-6*tanh(d*x+c)),x)

[Out]

Piecewise((-x/2 - 3*log(tanh(c + d*x) - 2/3)/(10*d) + 3*log(tanh(c + d*x) + 1)/(10*d), Ne(d, 0)), (x/(4 - 6*ta
nh(c)), True))

________________________________________________________________________________________

Giac [A]  time = 1.18794, size = 39, normalized size = 1.26 \begin{align*} \frac{d x + c}{10 \, d} - \frac{3 \, \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 5 \right |}\right )}{10 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-6*tanh(d*x+c)),x, algorithm="giac")

[Out]

1/10*(d*x + c)/d - 3/10*log(abs(e^(2*d*x + 2*c) - 5))/d