3.239 \(\int \cosh (x) \coth (n x) \, dx\)

Optimal. Leaf size=76 \[ e^{-x} \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );e^{2 n x}\right )-\frac{e^{-x}}{2}+\frac{e^x}{2} \]

[Out]

-1/(2*E^x) + E^x/2 + Hypergeometric2F1[1, -1/(2*n), 1 - 1/(2*n), E^(2*n*x)]/E^x - E^x*Hypergeometric2F1[1, 1/(
2*n), (2 + n^(-1))/2, E^(2*n*x)]

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Rubi [A]  time = 0.0722335, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5602, 2194, 2251} \[ e^{-x} \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );e^{2 n x}\right )-\frac{e^{-x}}{2}+\frac{e^x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Coth[n*x],x]

[Out]

-1/(2*E^x) + E^x/2 + Hypergeometric2F1[1, -1/(2*n), 1 - 1/(2*n), E^(2*n*x)]/E^x - E^x*Hypergeometric2F1[1, 1/(
2*n), (2 + n^(-1))/2, E^(2*n*x)]

Rule 5602

Int[Cosh[(a_.) + (b_.)*(x_)]*Coth[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[1/(E^(a + b*x)*2) + E^(a + b*x)/2 - 1/
(E^(a + b*x)*(1 - E^(2*(c + d*x)))) - E^(a + b*x)/(1 - E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
^2 - d^2, 0]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

Rubi steps

\begin{align*} \int \cosh (x) \coth (n x) \, dx &=\int \left (\frac{e^{-x}}{2}+\frac{e^x}{2}-\frac{e^{-x}}{1-e^{2 n x}}-\frac{e^x}{1-e^{2 n x}}\right ) \, dx\\ &=\frac{1}{2} \int e^{-x} \, dx+\frac{\int e^x \, dx}{2}-\int \frac{e^{-x}}{1-e^{2 n x}} \, dx-\int \frac{e^x}{1-e^{2 n x}} \, dx\\ &=-\frac{e^{-x}}{2}+\frac{e^x}{2}+e^{-x} \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );e^{2 n x}\right )\\ \end{align*}

Mathematica [B]  time = 0.169179, size = 156, normalized size = 2.05 \[ \frac{1}{2} e^{-2 x} \left (-\frac{e^{2 n x+x} \, _2F_1\left (1,1-\frac{1}{2 n};2-\frac{1}{2 n};e^{2 n x}\right )}{2 n-1}-\frac{e^{(2 n+3) x} \, _2F_1\left (1,1+\frac{1}{2 n};2+\frac{1}{2 n};e^{2 n x}\right )}{2 n+1}+e^x \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};e^{2 n x}\right )-e^{3 x} \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};e^{2 n x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Coth[n*x],x]

[Out]

(-((E^(x + 2*n*x)*Hypergeometric2F1[1, 1 - 1/(2*n), 2 - 1/(2*n), E^(2*n*x)])/(-1 + 2*n)) - (E^((3 + 2*n)*x)*Hy
pergeometric2F1[1, 1 + 1/(2*n), 2 + 1/(2*n), E^(2*n*x)])/(1 + 2*n) + E^x*Hypergeometric2F1[1, -1/(2*n), 1 - 1/
(2*n), E^(2*n*x)] - E^(3*x)*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), E^(2*n*x)])/(2*E^(2*x))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int \cosh \left ( x \right ){\rm coth} \left (nx\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*coth(n*x),x)

[Out]

int(cosh(x)*coth(n*x),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} - \frac{1}{2} \, \int \frac{e^{\left (2 \, x\right )} + 1}{e^{\left (n x + x\right )} + e^{x}}\,{d x} + \frac{1}{2} \, \int \frac{e^{\left (2 \, x\right )} + 1}{e^{\left (n x + x\right )} - e^{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(n*x),x, algorithm="maxima")

[Out]

1/2*(e^(2*x) - 1)*e^(-x) - 1/2*integrate((e^(2*x) + 1)/(e^(n*x + x) + e^x), x) + 1/2*integrate((e^(2*x) + 1)/(
e^(n*x + x) - e^x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (x\right ) \coth \left (n x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(n*x),x, algorithm="fricas")

[Out]

integral(cosh(x)*coth(n*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \coth{\left (n x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(n*x),x)

[Out]

Integral(cosh(x)*coth(n*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (x\right ) \coth \left (n x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(n*x),x, algorithm="giac")

[Out]

integrate(cosh(x)*coth(n*x), x)