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 = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.24, 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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fricas [B]  time = 0.47, size = 70, normalized size = 3.50 \[ -\frac {{\left (b \cosh \relax (x) + a \sinh \relax (x)\right )} \cosh \left (n \log \left (\frac {b \cosh \relax (x) + a \sinh \relax (x)}{\sinh \relax (x)}\right )\right ) + {\left (b \cosh \relax (x) + a \sinh \relax (x)\right )} \sinh \left (n \log \left (\frac {b \cosh \relax (x) + a \sinh \relax (x)}{\sinh \relax (x)}\right )\right )}{{\left (b n + b\right )} \sinh \relax (x)} \]

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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giac [A]  time = 0.14, size = 40, normalized size = 2.00 \[ -\frac {\left (\frac {a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b}{e^{\left (2 \, x\right )} - 1}\right )^{n + 1}}{b {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.16, size = 21, normalized size = 1.05 \[ -\frac {\left (a +b \coth \relax (x )\right )^{n +1}}{b \left (n +1\right )} \]

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 [A]  time = 0.31, size = 20, normalized size = 1.00 \[ -\frac {{\left (b \coth \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*coth(x) + a)^(n + 1)/(b*(n + 1))

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mupad [B]  time = 1.99, size = 55, normalized size = 2.75 \[ -\frac {{\left (a+\frac {b\,\left ({\mathrm {e}}^{2\,x}+1\right )}{{\mathrm {e}}^{2\,x}-1}\right )}^n\,\left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{b\,\left ({\mathrm {e}}^{2\,x}-1\right )\,\left (n+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*coth(x))^n/sinh(x)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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