### 3.117 $$\int \coth ^n(a+b x) \text{csch}^2(a+b x) \, dx$$

Optimal. Leaf size=20 $-\frac{\coth ^{n+1}(a+b x)}{b (n+1)}$

[Out]

-(Coth[a + b*x]^(1 + n)/(b*(1 + n)))

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Rubi [A]  time = 0.033743, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {2607, 32} $-\frac{\coth ^{n+1}(a+b x)}{b (n+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[a + b*x]^n*Csch[a + b*x]^2,x]

[Out]

-(Coth[a + b*x]^(1 + n)/(b*(1 + n)))

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])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \coth ^n(a+b x) \text{csch}^2(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int (-i x)^n \, dx,x,i \coth (a+b x)\right )}{b}\\ &=-\frac{\coth ^{1+n}(a+b x)}{b (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.022899, size = 20, normalized size = 1. $-\frac{\coth ^{n+1}(a+b x)}{b (n+1)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[a + b*x]^n*Csch[a + b*x]^2,x]

[Out]

-(Coth[a + b*x]^(1 + n)/(b*(1 + n)))

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Maple [A]  time = 0.013, size = 21, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\rm coth} \left (bx+a\right ) \right ) ^{n+1}}{b \left ( n+1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.01245, size = 192, normalized size = 9.6 \begin{align*} -\frac{\cosh \left (b x + a\right ) \cosh \left (n \log \left (\frac{\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right )\right ) + \cosh \left (b x + a\right ) \sinh \left (n \log \left (\frac{\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right )\right )}{{\left (b n + b\right )} \sinh \left (b x + a\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cosh(b*x + a)*cosh(n*log(cosh(b*x + a)/sinh(b*x + a))) + cosh(b*x + a)*sinh(n*log(cosh(b*x + a)/sinh(b*x + a
))))/((b*n + 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(coth(b*x+a)**n*csch(b*x+a)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth \left (b x + a\right )^{n} \operatorname{csch}\left (b x + a\right )^{2}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth(b*x + a)^n*csch(b*x + a)^2, x)