3.110 \(\int \frac{\coth ^2(x)}{a+a \text{sech}(x)} \, dx\)

Optimal. Leaf size=38 \[ \frac{x}{a}-\frac{\coth ^3(x) (1-\text{sech}(x))}{3 a}-\frac{\coth (x) (3-2 \text{sech}(x))}{3 a} \]

[Out]

x/a - (Coth[x]*(3 - 2*Sech[x]))/(3*a) - (Coth[x]^3*(1 - Sech[x]))/(3*a)

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Rubi [A]  time = 0.0893435, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3882, 8} \[ \frac{x}{a}-\frac{\coth ^3(x) (1-\text{sech}(x))}{3 a}-\frac{\coth (x) (3-2 \text{sech}(x))}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2/(a + a*Sech[x]),x]

[Out]

x/a - (Coth[x]*(3 - 2*Sech[x]))/(3*a) - (Coth[x]^3*(1 - Sech[x]))/(3*a)

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rule 3882

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[((e*Cot[c
+ d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(
a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0718144, size = 33, normalized size = 0.87 \[ \frac{6 x-4 \tanh (x)-4 \coth (x)-2 \text{csch}(x)+6 x \text{sech}(x)}{6 a \text{sech}(x)+6 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^2/(a + a*Sech[x]),x]

[Out]

(6*x - 4*Coth[x] - 2*Csch[x] + 6*x*Sech[x] - 4*Tanh[x])/(6*a + 6*a*Sech[x])

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Maple [A]  time = 0.03, size = 56, normalized size = 1.5 \begin{align*} -{\frac{1}{12\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\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)^2/(a+a*sech(x)),x)

[Out]

-1/12/a*tanh(1/2*x)^3-1/a*tanh(1/2*x)+1/a*ln(tanh(1/2*x)+1)-1/4/a/tanh(1/2*x)-1/a*ln(tanh(1/2*x)-1)

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Maxima [A]  time = 1.09619, size = 63, normalized size = 1.66 \begin{align*} \frac{x}{a} - \frac{2 \,{\left (5 \, e^{\left (-x\right )} - 3 \, e^{\left (-3 \, x\right )} + 4\right )}}{3 \,{\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2/(a+a*sech(x)),x, algorithm="maxima")

[Out]

x/a - 2/3*(5*e^(-x) - 3*e^(-3*x) + 4)/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x) + a)

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Fricas [A]  time = 2.23765, size = 149, normalized size = 3.92 \begin{align*} -\frac{2 \, \cosh \left (x\right )^{2} -{\left ({\left (3 \, x + 4\right )} \cosh \left (x\right ) + 3 \, x + 4\right )} \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + \cosh \left (x\right )}{3 \,{\left (a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2/(a+a*sech(x)),x, algorithm="fricas")

[Out]

-1/3*(2*cosh(x)^2 - ((3*x + 4)*cosh(x) + 3*x + 4)*sinh(x) + 2*sinh(x)^2 + cosh(x))/((a*cosh(x) + a)*sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\coth ^{2}{\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)**2/(a+a*sech(x)),x)

[Out]

Integral(coth(x)**2/(sech(x) + 1), x)/a

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Giac [A]  time = 1.17326, size = 54, normalized size = 1.42 \begin{align*} \frac{x}{a} - \frac{1}{2 \, a{\left (e^{x} - 1\right )}} + \frac{15 \, e^{\left (2 \, x\right )} + 24 \, e^{x} + 13}{6 \, a{\left (e^{x} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2/(a+a*sech(x)),x, algorithm="giac")

[Out]

x/a - 1/2/(a*(e^x - 1)) + 1/6*(15*e^(2*x) + 24*e^x + 13)/(a*(e^x + 1)^3)