3.109 \(\int \frac{\coth (x)}{a+a \text{sech}(x)} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{2 a (\cosh (x)+1)}+\frac{\log (1-\cosh (x))}{4 a}+\frac{3 \log (\cosh (x)+1)}{4 a} \]

[Out]

1/(2*a*(1 + Cosh[x])) + Log[1 - Cosh[x]]/(4*a) + (3*Log[1 + Cosh[x]])/(4*a)

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Rubi [A]  time = 0.0581539, antiderivative size = 40, 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 = {3879, 88} \[ \frac{1}{2 a (\cosh (x)+1)}+\frac{\log (1-\cosh (x))}{4 a}+\frac{3 \log (\cosh (x)+1)}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(a + a*Sech[x]),x]

[Out]

1/(2*a*(1 + Cosh[x])) + Log[1 - Cosh[x]]/(4*a) + (3*Log[1 + Cosh[x]])/(4*a)

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{\coth (x)}{a+a \text{sech}(x)} \, dx &=-\left (a^2 \operatorname{Subst}\left (\int \frac{x^2}{(a-a x) (a+a x)^2} \, dx,x,\cosh (x)\right )\right )\\ &=-\left (a^2 \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^3 (-1+x)}+\frac{1}{2 a^3 (1+x)^2}-\frac{3}{4 a^3 (1+x)}\right ) \, dx,x,\cosh (x)\right )\right )\\ &=\frac{1}{2 a (1+\cosh (x))}+\frac{\log (1-\cosh (x))}{4 a}+\frac{3 \log (1+\cosh (x))}{4 a}\\ \end{align*}

Mathematica [A]  time = 0.048076, size = 44, normalized size = 1.1 \[ \frac{\text{sech}(x) \left (2 \cosh ^2\left (\frac{x}{2}\right ) \left (\log \left (\sinh \left (\frac{x}{2}\right )\right )+3 \log \left (\cosh \left (\frac{x}{2}\right )\right )\right )+1\right )}{2 a (\text{sech}(x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(a + a*Sech[x]),x]

[Out]

((1 + 2*Cosh[x/2]^2*(3*Log[Cosh[x/2]] + Log[Sinh[x/2]]))*Sech[x])/(2*a*(1 + Sech[x]))

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Maple [A]  time = 0.029, size = 47, normalized size = 1.2 \begin{align*} -{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(a+a*sech(x)),x)

[Out]

-1/4/a*tanh(1/2*x)^2-1/a*ln(tanh(1/2*x)+1)+1/2/a*ln(tanh(1/2*x))-1/a*ln(tanh(1/2*x)-1)

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Maxima [A]  time = 1.15523, size = 70, normalized size = 1.75 \begin{align*} \frac{x}{a} + \frac{e^{\left (-x\right )}}{2 \, a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a} + \frac{3 \, \log \left (e^{\left (-x\right )} + 1\right )}{2 \, a} + \frac{\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+a*sech(x)),x, algorithm="maxima")

[Out]

x/a + e^(-x)/(2*a*e^(-x) + a*e^(-2*x) + a) + 3/2*log(e^(-x) + 1)/a + 1/2*log(e^(-x) - 1)/a

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Fricas [B]  time = 2.46441, size = 497, normalized size = 12.42 \begin{align*} -\frac{2 \, x \cosh \left (x\right )^{2} + 2 \, x \sinh \left (x\right )^{2} + 2 \,{\left (2 \, x - 1\right )} \cosh \left (x\right ) - 3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \,{\left (2 \, x \cosh \left (x\right ) + 2 \, x - 1\right )} \sinh \left (x\right ) + 2 \, x}{2 \,{\left (a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+a*sech(x)),x, algorithm="fricas")

[Out]

-1/2*(2*x*cosh(x)^2 + 2*x*sinh(x)^2 + 2*(2*x - 1)*cosh(x) - 3*(cosh(x)^2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^2
+ 2*cosh(x) + 1)*log(cosh(x) + sinh(x) + 1) - (cosh(x)^2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^2 + 2*cosh(x) +
1)*log(cosh(x) + sinh(x) - 1) + 2*(2*x*cosh(x) + 2*x - 1)*sinh(x) + 2*x)/(a*cosh(x)^2 + a*sinh(x)^2 + 2*a*cosh
(x) + 2*(a*cosh(x) + a)*sinh(x) + a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+a*sech(x)),x)

[Out]

Integral(coth(x)/(sech(x) + 1), x)/a

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Giac [A]  time = 1.16699, size = 76, normalized size = 1.9 \begin{align*} \frac{3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, a} + \frac{\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, a} - \frac{3 \, e^{\left (-x\right )} + 3 \, e^{x} + 2}{4 \, a{\left (e^{\left (-x\right )} + e^{x} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+a*sech(x)),x, algorithm="giac")

[Out]

3/4*log(e^(-x) + e^x + 2)/a + 1/4*log(e^(-x) + e^x - 2)/a - 1/4*(3*e^(-x) + 3*e^x + 2)/(a*(e^(-x) + e^x + 2))