3.85 \(\int \frac{1}{4+6 \coth (c+d x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0438151, 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} \[ \frac{3 \log (2 \sinh (c+d x)+3 \cosh (c+d x))}{10 d}-\frac{x}{5} \]

Antiderivative was successfully verified.

[In]

Int[(4 + 6*Coth[c + d*x])^(-1),x]

[Out]

-x/5 + (3*Log[3*Cosh[c + d*x] + 2*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 \coth (c+d x)} \, dx &=-\frac{x}{5}+\frac{3}{10} i \int \frac{-6 i-4 i \coth (c+d x)}{4+6 \coth (c+d x)} \, dx\\ &=-\frac{x}{5}+\frac{3 \log (3 \cosh (c+d x)+2 \sinh (c+d x))}{10 d}\\ \end{align*}

Mathematica [A]  time = 0.0387123, size = 53, normalized size = 1.71 \[ -\frac{\log (1-\tanh (c+d x))}{20 d}-\frac{\log (\tanh (c+d x)+1)}{4 d}+\frac{3 \log (2 \tanh (c+d x)+3)}{10 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10128, size = 38, normalized size = 1.23 \begin{align*} \frac{d x + c}{10 \, d} + \frac{3 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 5\right )}{10 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.55375, size = 130, normalized size = 4.19 \begin{align*} -\frac{5 \, d x - 3 \, \log \left (\frac{2 \,{\left (3 \, \cosh \left (d x + c\right ) + 2 \, \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*coth(d*x+c)),x, algorithm="fricas")

[Out]

-1/10*(5*d*x - 3*log(2*(3*cosh(d*x + c) + 2*sinh(d*x + c))/(cosh(d*x + c) - sinh(d*x + c))))/d

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Sympy [A]  time = 1.36237, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} \frac{x}{10} - \frac{3 \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{10 d} + \frac{3 \log{\left (\tanh{\left (c + d x \right )} + \frac{3}{2} \right )}}{10 d} & \text{for}\: d \neq 0 \\\frac{x}{6 \coth{\left (c \right )} + 4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17926, size = 41, normalized size = 1.32 \begin{align*} -\frac{d x + 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*coth(d*x+c)),x, algorithm="giac")

[Out]

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