### 3.1020 $$\int (a+b \coth (x))^n \text{csch}^2(x) \, dx$$

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

[Out]

-((a + b*Coth[x])^(1 + n)/(b*(1 + n)))

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Rubi [A]  time = 0.0483491, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {3506, 32} $-\frac{(a+b \coth (x))^{n+1}}{b (n+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*Coth[x])^n*Csch[x]^2,x]

[Out]

-((a + b*Coth[x])^(1 + n)/(b*(1 + n)))

Rule 3506

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

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

Mathematica [A]  time = 0.196076, size = 19, normalized size = 0.95 $-\frac{(a+b \coth (x))^{n+1}}{b n+b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((a + b*Coth[x])^(1 + n)/(b + b*n))

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Maple [A]  time = 0.021, size = 21, normalized size = 1.1 \begin{align*} -{\frac{ \left ( a+b{\rm coth} \left (x\right ) \right ) ^{n+1}}{b \left ( n+1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.15537, size = 221, normalized size = 11.05 \begin{align*} -\frac{{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )} \cosh \left (n \log \left (\frac{b \cosh \left (x\right ) + a \sinh \left (x\right )}{\sinh \left (x\right )}\right )\right ) +{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )} \sinh \left (n \log \left (\frac{b \cosh \left (x\right ) + a \sinh \left (x\right )}{\sinh \left (x\right )}\right )\right )}{{\left (b n + b\right )} \sinh \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b*cosh(x) + a*sinh(x))*cosh(n*log((b*cosh(x) + a*sinh(x))/sinh(x))) + (b*cosh(x) + a*sinh(x))*sinh(n*log((b
*cosh(x) + a*sinh(x))/sinh(x))))/((b*n + b)*sinh(x))

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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