### 3.419 $$\int x^2 \cosh ^2(x) \coth ^2(x) \, dx$$

Optimal. Leaf size=73 $\text{PolyLog}\left (2,e^{2 x}\right )+\frac{x^3}{2}-x^2-x^2 \coth (x)+\frac{1}{2} x^2 \sinh (x) \cosh (x)+\frac{x}{4}+2 x \log \left (1-e^{2 x}\right )-\frac{1}{2} x \cosh ^2(x)+\frac{1}{4} \sinh (x) \cosh (x)$

[Out]

x/4 - x^2 + x^3/2 - (x*Cosh[x]^2)/2 - x^2*Coth[x] + 2*x*Log[1 - E^(2*x)] + PolyLog[2, E^(2*x)] + (Cosh[x]*Sinh
[x])/4 + (x^2*Cosh[x]*Sinh[x])/2

________________________________________________________________________________________

Rubi [A]  time = 0.157967, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.833, Rules used = {5450, 3311, 30, 2635, 8, 3720, 3716, 2190, 2279, 2391} $\text{PolyLog}\left (2,e^{2 x}\right )+\frac{x^3}{2}-x^2-x^2 \coth (x)+\frac{1}{2} x^2 \sinh (x) \cosh (x)+\frac{x}{4}+2 x \log \left (1-e^{2 x}\right )-\frac{1}{2} x \cosh ^2(x)+\frac{1}{4} \sinh (x) \cosh (x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

x/4 - x^2 + x^3/2 - (x*Cosh[x]^2)/2 - x^2*Coth[x] + 2*x*Log[1 - E^(2*x)] + PolyLog[2, E^(2*x)] + (Cosh[x]*Sinh
[x])/4 + (x^2*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 3311

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

Rule 30

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

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int x^2 \cosh ^2(x) \coth ^2(x) \, dx &=\int x^2 \cosh ^2(x) \, dx+\int x^2 \coth ^2(x) \, dx\\ &=-\frac{1}{2} x \cosh ^2(x)-x^2 \coth (x)+\frac{1}{2} x^2 \cosh (x) \sinh (x)+\frac{\int x^2 \, dx}{2}+\frac{1}{2} \int \cosh ^2(x) \, dx+2 \int x \coth (x) \, dx+\int x^2 \, dx\\ &=-x^2+\frac{x^3}{2}-\frac{1}{2} x \cosh ^2(x)-x^2 \coth (x)+\frac{1}{4} \cosh (x) \sinh (x)+\frac{1}{2} x^2 \cosh (x) \sinh (x)+\frac{\int 1 \, dx}{4}-4 \int \frac{e^{2 x} x}{1-e^{2 x}} \, dx\\ &=\frac{x}{4}-x^2+\frac{x^3}{2}-\frac{1}{2} x \cosh ^2(x)-x^2 \coth (x)+2 x \log \left (1-e^{2 x}\right )+\frac{1}{4} \cosh (x) \sinh (x)+\frac{1}{2} x^2 \cosh (x) \sinh (x)-2 \int \log \left (1-e^{2 x}\right ) \, dx\\ &=\frac{x}{4}-x^2+\frac{x^3}{2}-\frac{1}{2} x \cosh ^2(x)-x^2 \coth (x)+2 x \log \left (1-e^{2 x}\right )+\frac{1}{4} \cosh (x) \sinh (x)+\frac{1}{2} x^2 \cosh (x) \sinh (x)-\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 x}\right )\\ &=\frac{x}{4}-x^2+\frac{x^3}{2}-\frac{1}{2} x \cosh ^2(x)-x^2 \coth (x)+2 x \log \left (1-e^{2 x}\right )+\text{Li}_2\left (e^{2 x}\right )+\frac{1}{4} \cosh (x) \sinh (x)+\frac{1}{2} x^2 \cosh (x) \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0928496, size = 64, normalized size = 0.88 $\frac{1}{8} \left (-8 \text{PolyLog}\left (2,e^{-2 x}\right )+4 x^3+8 x^2+2 x^2 \sinh (2 x)-8 x^2 \coth (x)+16 x \log \left (1-e^{-2 x}\right )+\sinh (2 x)-2 x \cosh (2 x)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(8*x^2 + 4*x^3 - 2*x*Cosh[2*x] - 8*x^2*Coth[x] + 16*x*Log[1 - E^(-2*x)] - 8*PolyLog[2, E^(-2*x)] + Sinh[2*x] +
2*x^2*Sinh[2*x])/8

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 87, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}}{2}}+ \left ({\frac{1}{16}}-{\frac{x}{8}}+{\frac{{x}^{2}}{8}} \right ){{\rm e}^{2\,x}}+ \left ( -{\frac{1}{16}}-{\frac{x}{8}}-{\frac{{x}^{2}}{8}} \right ){{\rm e}^{-2\,x}}-2\,{\frac{{x}^{2}}{{{\rm e}^{2\,x}}-1}}-2\,{x}^{2}+2\,x\ln \left ({{\rm e}^{x}}+1 \right ) +2\,{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) +2\,x\ln \left ( 1-{{\rm e}^{x}} \right ) +2\,{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [B]  time = 2.17395, size = 1840, normalized size = 25.21 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((2*x^2 - 2*x + 1)*cosh(x)^6 + 6*(2*x^2 - 2*x + 1)*cosh(x)*sinh(x)^5 + (2*x^2 - 2*x + 1)*sinh(x)^6 + (8*x
^3 - 34*x^2 + 2*x - 1)*cosh(x)^4 + (8*x^3 + 15*(2*x^2 - 2*x + 1)*cosh(x)^2 - 34*x^2 + 2*x - 1)*sinh(x)^4 + 4*(
5*(2*x^2 - 2*x + 1)*cosh(x)^3 + (8*x^3 - 34*x^2 + 2*x - 1)*cosh(x))*sinh(x)^3 - (8*x^3 + 2*x^2 + 2*x + 1)*cosh
(x)^2 + (15*(2*x^2 - 2*x + 1)*cosh(x)^4 - 8*x^3 + 6*(8*x^3 - 34*x^2 + 2*x - 1)*cosh(x)^2 - 2*x^2 - 2*x - 1)*si
nh(x)^2 + 2*x^2 + 32*(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))*dilog(cosh(x) + sinh(x)) + 32*(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))*dilog(-cosh(x) - sinh(x)) + 32*
(x*cosh(x)^4 + 4*x*cosh(x)*sinh(x)^3 + x*sinh(x)^4 - x*cosh(x)^2 + (6*x*cosh(x)^2 - x)*sinh(x)^2 + 2*(2*x*cosh
(x)^3 - x*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 32*(x*cosh(x)^4 + 4*x*cosh(x)*sinh(x)^3 + x*sinh(x)^4
- x*cosh(x)^2 + (6*x*cosh(x)^2 - x)*sinh(x)^2 + 2*(2*x*cosh(x)^3 - x*cosh(x))*sinh(x))*log(-cosh(x) - sinh(x)
+ 1) + 2*(3*(2*x^2 - 2*x + 1)*cosh(x)^5 + 2*(8*x^3 - 34*x^2 + 2*x - 1)*cosh(x)^3 - (8*x^3 + 2*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))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*cosh(x)^2*coth(x)^2, x)