### 3.418 $$\int x \cosh ^2(x) \coth ^2(x) \, dx$$

Optimal. Leaf size=33 $\frac{3 x^2}{4}-\frac{\cosh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))+\frac{1}{2} x \sinh (x) \cosh (x)$

[Out]

(3*x^2)/4 - Cosh[x]^2/4 - x*Coth[x] + Log[Sinh[x]] + (x*Cosh[x]*Sinh[x])/2

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Rubi [A]  time = 0.0543393, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {5450, 3310, 30, 3720, 3475} $\frac{3 x^2}{4}-\frac{\cosh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))+\frac{1}{2} x \sinh (x) \cosh (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Cosh[x]^2*Coth[x]^2,x]

[Out]

(3*x^2)/4 - Cosh[x]^2/4 - x*Coth[x] + Log[Sinh[x]] + (x*Cosh[x]*Sinh[x])/2

Rule 5450

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x \cosh ^2(x) \coth ^2(x) \, dx &=\int x \cosh ^2(x) \, dx+\int x \coth ^2(x) \, dx\\ &=-\frac{1}{4} \cosh ^2(x)-x \coth (x)+\frac{1}{2} x \cosh (x) \sinh (x)+\frac{\int x \, dx}{2}+\int x \, dx+\int \coth (x) \, dx\\ &=\frac{3 x^2}{4}-\frac{\cosh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))+\frac{1}{2} x \cosh (x) \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0288381, size = 33, normalized size = 1. $\frac{3 x^2}{4}+\frac{1}{4} x \sinh (2 x)-\frac{1}{8} \cosh (2 x)-x \coth (x)+\log (\sinh (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Cosh[x]^2*Coth[x]^2,x]

[Out]

(3*x^2)/4 - Cosh[2*x]/8 - x*Coth[x] + Log[Sinh[x]] + (x*Sinh[2*x])/4

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Maple [A]  time = 0.04, size = 48, normalized size = 1.5 \begin{align*}{\frac{3\,{x}^{2}}{4}}+ \left ( -{\frac{1}{16}}+{\frac{x}{8}} \right ){{\rm e}^{2\,x}}+ \left ( -{\frac{1}{16}}-{\frac{x}{8}} \right ){{\rm e}^{-2\,x}}-2\,x-2\,{\frac{x}{{{\rm e}^{2\,x}}-1}}+\ln \left ({{\rm e}^{2\,x}}-1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(x)^2*coth(x)^2,x)

[Out]

3/4*x^2+(-1/16+1/8*x)*exp(2*x)+(-1/16-1/8*x)*exp(-2*x)-2*x-2*x/(exp(2*x)-1)+ln(exp(2*x)-1)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(x)^2*coth(x)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 2.15228, size = 1022, normalized size = 30.97 \begin{align*} \frac{{\left (2 \, x - 1\right )} \cosh \left (x\right )^{6} + 6 \,{\left (2 \, x - 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} +{\left (2 \, x - 1\right )} \sinh \left (x\right )^{6} +{\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{4} +{\left (15 \,{\left (2 \, x - 1\right )} \cosh \left (x\right )^{2} + 12 \, x^{2} - 34 \, x + 1\right )} \sinh \left (x\right )^{4} + 4 \,{\left (5 \,{\left (2 \, x - 1\right )} \cosh \left (x\right )^{3} +{\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} -{\left (12 \, x^{2} + 2 \, x + 1\right )} \cosh \left (x\right )^{2} +{\left (15 \,{\left (2 \, x - 1\right )} \cosh \left (x\right )^{4} + 6 \,{\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{2} - 12 \, x^{2} - 2 \, x - 1\right )} \sinh \left (x\right )^{2} + 16 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (3 \,{\left (2 \, x - 1\right )} \cosh \left (x\right )^{5} + 2 \,{\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{3} -{\left (12 \, x^{2} + 2 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 2 \, x + 1}{16 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(x)^2*coth(x)^2,x, algorithm="fricas")

[Out]

1/16*((2*x - 1)*cosh(x)^6 + 6*(2*x - 1)*cosh(x)*sinh(x)^5 + (2*x - 1)*sinh(x)^6 + (12*x^2 - 34*x + 1)*cosh(x)^
4 + (15*(2*x - 1)*cosh(x)^2 + 12*x^2 - 34*x + 1)*sinh(x)^4 + 4*(5*(2*x - 1)*cosh(x)^3 + (12*x^2 - 34*x + 1)*co
sh(x))*sinh(x)^3 - (12*x^2 + 2*x + 1)*cosh(x)^2 + (15*(2*x - 1)*cosh(x)^4 + 6*(12*x^2 - 34*x + 1)*cosh(x)^2 -
12*x^2 - 2*x - 1)*sinh(x)^2 + 16*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 -
cosh(x)^2 + 2*(2*cosh(x)^3 - cosh(x))*sinh(x))*log(2*sinh(x)/(cosh(x) - sinh(x))) + 2*(3*(2*x - 1)*cosh(x)^5 +
2*(12*x^2 - 34*x + 1)*cosh(x)^3 - (12*x^2 + 2*x + 1)*cosh(x))*sinh(x) + 2*x + 1)/(cosh(x)^4 + 4*cosh(x)*sinh(
x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cosh(x)^3 - cosh(x))*sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh ^{2}{\left (x \right )} \coth ^{2}{\left (x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(x)**2*coth(x)**2,x)

[Out]

Integral(x*cosh(x)**2*coth(x)**2, x)

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Giac [B]  time = 1.19143, size = 136, normalized size = 4.12 \begin{align*} \frac{12 \, x^{2} e^{\left (4 \, x\right )} - 12 \, x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{\left (6 \, x\right )} - 34 \, x e^{\left (4 \, x\right )} - 2 \, x e^{\left (2 \, x\right )} + 16 \, e^{\left (4 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 16 \, e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, x - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}{16 \,{\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(x)^2*coth(x)^2,x, algorithm="giac")

[Out]

1/16*(12*x^2*e^(4*x) - 12*x^2*e^(2*x) + 2*x*e^(6*x) - 34*x*e^(4*x) - 2*x*e^(2*x) + 16*e^(4*x)*log(e^(2*x) - 1)
- 16*e^(2*x)*log(e^(2*x) - 1) + 2*x - e^(6*x) + e^(4*x) - e^(2*x) + 1)/(e^(4*x) - e^(2*x))