3.667 \(\int (-\coth (x)+\text {csch}(x))^2 \, dx\)

Optimal. Leaf size=12 \[ x-\frac {2 \sinh (x)}{\cosh (x)+1} \]

[Out]

x-2*sinh(x)/(1+cosh(x))

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Rubi [A]  time = 0.08, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4392, 2670, 2680, 8} \[ x-\frac {2 \sinh (x)}{\cosh (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(-Coth[x] + Csch[x])^2,x]

[Out]

x - (2*Sinh[x])/(1 + Cosh[x])

Rule 8

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

Rule 2670

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

Rule 2680

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

\begin {align*} \int (-\coth (x)+\text {csch}(x))^2 \, dx &=-\int (i-i \cosh (x))^2 \text {csch}^2(x) \, dx\\ &=-\int \frac {\sinh ^2(x)}{(i+i \cosh (x))^2} \, dx\\ &=-\frac {2 \sinh (x)}{1+\cosh (x)}+\int 1 \, dx\\ &=x-\frac {2 \sinh (x)}{1+\cosh (x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.50 \[ 2 \tanh ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Coth[x] + Csch[x])^2,x]

[Out]

2*ArcTanh[Tanh[x/2]] - 2*Tanh[x/2]

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fricas [A]  time = 0.39, size = 20, normalized size = 1.67 \[ \frac {x \cosh \relax (x) + x \sinh \relax (x) + x + 4}{\cosh \relax (x) + \sinh \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-coth(x)+csch(x))^2,x, algorithm="fricas")

[Out]

(x*cosh(x) + x*sinh(x) + x + 4)/(cosh(x) + sinh(x) + 1)

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giac [A]  time = 0.11, size = 10, normalized size = 0.83 \[ x + \frac {4}{e^{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-coth(x)+csch(x))^2,x, algorithm="giac")

[Out]

x + 4/(e^x + 1)

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maple [A]  time = 0.37, size = 13, normalized size = 1.08 \[ x -2 \coth \relax (x )+\frac {2}{\sinh \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-coth(x)+csch(x))^2,x)

[Out]

x-2*coth(x)+2/sinh(x)

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maxima [B]  time = 0.40, size = 25, normalized size = 2.08 \[ x - \frac {4}{e^{\left (-x\right )} - e^{x}} + \frac {4}{e^{\left (-2 \, x\right )} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-coth(x)+csch(x))^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 10, normalized size = 0.83 \[ x+\frac {4}{{\mathrm {e}}^x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(x) - 1/sinh(x))^2,x)

[Out]

x + 4/(exp(x) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- \coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-coth(x)+csch(x))**2,x)

[Out]

Integral((-coth(x) + csch(x))**2, x)

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