### 3.435 $$\int \coth ^2(a+b x) \, dx$$

Optimal. Leaf size=13 $x-\frac{\coth (a+b x)}{b}$

[Out]

x - Coth[a + b*x]/b

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Rubi [A]  time = 0.00946, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {3473, 8} $x-\frac{\coth (a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[a + b*x]^2,x]

[Out]

x - Coth[a + b*x]/b

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

Rubi steps

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

Mathematica [C]  time = 0.0131388, size = 27, normalized size = 2.08 $-\frac{\coth (a+b x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(a+b x)\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[a + b*x]^2,x]

[Out]

-((Coth[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[a + b*x]^2])/b)

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Maple [A]  time = 0.013, size = 18, normalized size = 1.4 \begin{align*}{\frac{bx+a-{\rm coth} \left (bx+a\right )}{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^2*csch(b*x+a)^2,x)

[Out]

1/b*(b*x+a-coth(b*x+a))

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Maxima [A]  time = 1.09518, size = 34, normalized size = 2.62 \begin{align*} x + \frac{a}{b} + \frac{2}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*csch(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.20794, size = 82, normalized size = 6.31 \begin{align*} \frac{{\left (b x + 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \sinh \left (b x + a\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*csch(b*x+a)^2,x, algorithm="fricas")

[Out]

((b*x + 1)*sinh(b*x + a) - cosh(b*x + a))/(b*sinh(b*x + a))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**2*csch(b*x+a)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.15989, size = 38, normalized size = 2.92 \begin{align*} \frac{b x + a}{b} - \frac{2}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*csch(b*x+a)^2,x, algorithm="giac")

[Out]

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