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]

3/4*x^2-1/4*cosh(x)^2-x*coth(x)+ln(sinh(x))+1/2*x*cosh(x)*sinh(x)

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Rubi [A]  time = 0.05, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 30

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

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 3475

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

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 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]

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*}

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Mathematica [A]  time = 0.03, size = 33, normalized size = 1.00 \[ \frac {3 x^2}{4}+\frac {1}{4} x \sinh (2 x)-\frac {1}{8} \cosh (2 x)-x \coth (x)+\log (\sinh (x)) \]

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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fricas [B]  time = 0.58, size = 336, normalized size = 10.18 \[ \frac {{\left (2 \, x - 1\right )} \cosh \relax (x)^{6} + 6 \, {\left (2 \, x - 1\right )} \cosh \relax (x) \sinh \relax (x)^{5} + {\left (2 \, x - 1\right )} \sinh \relax (x)^{6} + {\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \relax (x)^{4} + {\left (15 \, {\left (2 \, x - 1\right )} \cosh \relax (x)^{2} + 12 \, x^{2} - 34 \, x + 1\right )} \sinh \relax (x)^{4} + 4 \, {\left (5 \, {\left (2 \, x - 1\right )} \cosh \relax (x)^{3} + {\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \relax (x)\right )} \sinh \relax (x)^{3} - {\left (12 \, x^{2} + 2 \, x + 1\right )} \cosh \relax (x)^{2} + {\left (15 \, {\left (2 \, x - 1\right )} \cosh \relax (x)^{4} + 6 \, {\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \relax (x)^{2} - 12 \, x^{2} - 2 \, x - 1\right )} \sinh \relax (x)^{2} + 16 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - \cosh \relax (x)^{2} + 2 \, {\left (2 \, \cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (3 \, {\left (2 \, x - 1\right )} \cosh \relax (x)^{5} + 2 \, {\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \relax (x)^{3} - {\left (12 \, x^{2} + 2 \, x + 1\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 2 \, x + 1}{16 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - \cosh \relax (x)^{2} + 2 \, {\left (2 \, \cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x)\right )}} \]

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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giac [B]  time = 0.12, size = 101, normalized size = 3.06 \[ \frac {12 \, x^{2} e^{\left (4 \, x\right )} - 12 \, x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{\left (6 \, x\right )} - 34 \, x e^{\left (4 \, x\right )} - 2 \, x e^{\left (2 \, x\right )} + 16 \, e^{\left (4 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 16 \, e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, x - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}{16 \, {\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}\right )}} \]

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))

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maple [A]  time = 0.21, size = 48, normalized size = 1.45 \[ \frac {3 x^{2}}{4}+\left (-\frac {1}{16}+\frac {x}{8}\right ) {\mathrm e}^{2 x}+\left (-\frac {1}{16}-\frac {x}{8}\right ) {\mathrm e}^{-2 x}-2 x -\frac {2 x}{{\mathrm e}^{2 x}-1}+\ln \left ({\mathrm e}^{2 x}-1\right ) \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 0.12, size = 48, normalized size = 1.45 \[ \ln \left ({\mathrm {e}}^{2\,x}-1\right )-2\,x-{\mathrm {e}}^{-2\,x}\,\left (\frac {x}{8}+\frac {1}{16}\right )+{\mathrm {e}}^{2\,x}\,\left (\frac {x}{8}-\frac {1}{16}\right )-\frac {2\,x}{{\mathrm {e}}^{2\,x}-1}+\frac {3\,x^2}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cosh ^{2}{\relax (x )} \coth ^{2}{\relax (x )}\, dx \]

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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