3.75 \(\int \frac{1}{5+3 \cosh (c+d x)} \, dx\)

Optimal. Leaf size=31 \[ \frac{x}{4}-\frac{\tanh ^{-1}\left (\frac{\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{2 d} \]

[Out]

x/4 - ArcTanh[Sinh[c + d*x]/(3 + Cosh[c + d*x])]/(2*d)

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Rubi [A]  time = 0.0128323, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2657} \[ \frac{x}{4}-\frac{\tanh ^{-1}\left (\frac{\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(5 + 3*Cosh[c + d*x])^(-1),x]

[Out]

x/4 - ArcTanh[Sinh[c + d*x]/(3 + Cosh[c + d*x])]/(2*d)

Rule 2657

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2*ArcTan[(b*Cos[c + d*x])/(a + q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]

Rubi steps

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

Mathematica [B]  time = 0.0291241, size = 77, normalized size = 2.48 \[ \frac{\log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )+2 \cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{4 d}-\frac{\log \left (2 \cosh \left (\frac{c}{2}+\frac{d x}{2}\right )-\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + 3*Cosh[c + d*x])^(-1),x]

[Out]

-Log[2*Cosh[c/2 + (d*x)/2] - Sinh[c/2 + (d*x)/2]]/(4*d) + Log[2*Cosh[c/2 + (d*x)/2] + Sinh[c/2 + (d*x)/2]]/(4*
d)

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Maple [A]  time = 0.014, size = 36, normalized size = 1.2 \begin{align*}{\frac{1}{4\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }-{\frac{1}{4\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5+3*cosh(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.03287, size = 50, normalized size = 1.61 \begin{align*} -\frac{\log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{4 \, d} + \frac{\log \left (e^{\left (-d x - c\right )} + 3\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*cosh(d*x+c)),x, algorithm="maxima")

[Out]

-1/4*log(3*e^(-d*x - c) + 1)/d + 1/4*log(e^(-d*x - c) + 3)/d

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Fricas [A]  time = 2.32233, size = 126, normalized size = 4.06 \begin{align*} \frac{\log \left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) + 1\right ) - \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 3\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*cosh(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(log(3*cosh(d*x + c) + 3*sinh(d*x + c) + 1) - log(cosh(d*x + c) + sinh(d*x + c) + 3))/d

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Sympy [A]  time = 0.726658, size = 41, normalized size = 1.32 \begin{align*} \begin{cases} - \frac{\log{\left (\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 2 \right )}}{4 d} + \frac{\log{\left (\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 \right )}}{4 d} & \text{for}\: d \neq 0 \\\frac{x}{3 \cosh{\left (c \right )} + 5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*cosh(d*x+c)),x)

[Out]

Piecewise((-log(tanh(c/2 + d*x/2) - 2)/(4*d) + log(tanh(c/2 + d*x/2) + 2)/(4*d), Ne(d, 0)), (x/(3*cosh(c) + 5)
, True))

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Giac [A]  time = 1.16046, size = 42, normalized size = 1.35 \begin{align*} \frac{\log \left (3 \, e^{\left (d x + c\right )} + 1\right )}{4 \, d} - \frac{\log \left (e^{\left (d x + c\right )} + 3\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*cosh(d*x+c)),x, algorithm="giac")

[Out]

1/4*log(3*e^(d*x + c) + 1)/d - 1/4*log(e^(d*x + c) + 3)/d