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

Optimal. Leaf size=23 \[ \frac{\text{csch}^3(x)}{3 a}-\frac{\coth ^3(x)}{3 a} \]

[Out]

-Coth[x]^3/(3*a) + Csch[x]^3/(3*a)

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Rubi [A]  time = 0.138909, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3872, 2839, 2606, 30, 2607} \[ \frac{\text{csch}^3(x)}{3 a}-\frac{\coth ^3(x)}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(a + a*Sech[x]),x]

[Out]

-Coth[x]^3/(3*a) + Csch[x]^3/(3*a)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rubi steps

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

Mathematica [A]  time = 0.0381574, size = 25, normalized size = 1.09 \[ -\frac{(2 \cosh (x)+\cosh (2 x)+3) \text{csch}(x)}{6 a (\cosh (x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*Cosh[x] + Cosh[2*x])*Csch[x])/(6*a*(1 + Cosh[x]))

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Maple [A]  time = 0.022, size = 23, normalized size = 1. \begin{align*}{\frac{1}{4\,a} \left ( -{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^2/(a+a*sech(x)),x)

[Out]

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

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Maxima [B]  time = 1.14342, size = 122, normalized size = 5.3 \begin{align*} -\frac{4 \, e^{\left (-x\right )}}{3 \,{\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} - \frac{2 \, e^{\left (-2 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a} - \frac{2}{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(csch(x)^2/(a+a*sech(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.29228, size = 230, normalized size = 10. \begin{align*} -\frac{4 \,{\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}}{3 \,{\left (a \cosh \left (x\right )^{3} + a \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} +{\left (3 \, a \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right )^{2} - a \cosh \left (x\right ) +{\left (3 \, a \cosh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - 2 \, a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.15226, size = 42, normalized size = 1.83 \begin{align*} -\frac{1}{2 \, a{\left (e^{x} - 1\right )}} + \frac{3 \, e^{\left (2 \, x\right )} + 1}{6 \, a{\left (e^{x} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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