### 3.420 $$\int x^3 \cosh ^2(x) \coth ^2(x) \, dx$$

Optimal. Leaf size=102 $3 x \text{PolyLog}\left (2,e^{2 x}\right )-\frac{3}{2} \text{PolyLog}\left (3,e^{2 x}\right )+\frac{3 x^4}{8}-x^3+\frac{3 x^2}{8}+3 x^2 \log \left (1-e^{2 x}\right )-\frac{3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+\frac{1}{2} x^3 \sinh (x) \cosh (x)-\frac{3 \cosh ^2(x)}{8}+\frac{3}{4} x \sinh (x) \cosh (x)$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.182629, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.833, Rules used = {5450, 3311, 30, 3310, 3720, 3716, 2190, 2531, 2282, 6589} $3 x \text{PolyLog}\left (2,e^{2 x}\right )-\frac{3}{2} \text{PolyLog}\left (3,e^{2 x}\right )+\frac{3 x^4}{8}-x^3+\frac{3 x^2}{8}+3 x^2 \log \left (1-e^{2 x}\right )-\frac{3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+\frac{1}{2} x^3 \sinh (x) \cosh (x)-\frac{3 \cosh ^2(x)}{8}+\frac{3}{4} x \sinh (x) \cosh (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3*Cosh[x]^2*Coth[x]^2,x]

[Out]

(3*x^2)/8 - x^3 + (3*x^4)/8 - (3*Cosh[x]^2)/8 - (3*x^2*Cosh[x]^2)/4 - x^3*Coth[x] + 3*x^2*Log[1 - E^(2*x)] + 3
*x*PolyLog[2, E^(2*x)] - (3*PolyLog[3, E^(2*x)])/2 + (3*x*Cosh[x]*Sinh[x])/4 + (x^3*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 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 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int x^3 \cosh ^2(x) \coth ^2(x) \, dx &=\int x^3 \cosh ^2(x) \, dx+\int x^3 \coth ^2(x) \, dx\\ &=-\frac{3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+\frac{1}{2} x^3 \cosh (x) \sinh (x)+\frac{\int x^3 \, dx}{2}+\frac{3}{2} \int x \cosh ^2(x) \, dx+3 \int x^2 \coth (x) \, dx+\int x^3 \, dx\\ &=-x^3+\frac{3 x^4}{8}-\frac{3 \cosh ^2(x)}{8}-\frac{3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+\frac{3}{4} x \cosh (x) \sinh (x)+\frac{1}{2} x^3 \cosh (x) \sinh (x)+\frac{3 \int x \, dx}{4}-6 \int \frac{e^{2 x} x^2}{1-e^{2 x}} \, dx\\ &=\frac{3 x^2}{8}-x^3+\frac{3 x^4}{8}-\frac{3 \cosh ^2(x)}{8}-\frac{3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+\frac{3}{4} x \cosh (x) \sinh (x)+\frac{1}{2} x^3 \cosh (x) \sinh (x)-6 \int x \log \left (1-e^{2 x}\right ) \, dx\\ &=\frac{3 x^2}{8}-x^3+\frac{3 x^4}{8}-\frac{3 \cosh ^2(x)}{8}-\frac{3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+3 x \text{Li}_2\left (e^{2 x}\right )+\frac{3}{4} x \cosh (x) \sinh (x)+\frac{1}{2} x^3 \cosh (x) \sinh (x)-3 \int \text{Li}_2\left (e^{2 x}\right ) \, dx\\ &=\frac{3 x^2}{8}-x^3+\frac{3 x^4}{8}-\frac{3 \cosh ^2(x)}{8}-\frac{3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+3 x \text{Li}_2\left (e^{2 x}\right )+\frac{3}{4} x \cosh (x) \sinh (x)+\frac{1}{2} x^3 \cosh (x) \sinh (x)-\frac{3}{2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 x}\right )\\ &=\frac{3 x^2}{8}-x^3+\frac{3 x^4}{8}-\frac{3 \cosh ^2(x)}{8}-\frac{3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+3 x \text{Li}_2\left (e^{2 x}\right )-\frac{3 \text{Li}_3\left (e^{2 x}\right )}{2}+\frac{3}{4} x \cosh (x) \sinh (x)+\frac{1}{2} x^3 \cosh (x) \sinh (x)\\ \end{align*}

Mathematica [C]  time = 0.146842, size = 94, normalized size = 0.92 $\frac{1}{16} \left (48 x \text{PolyLog}\left (2,e^{2 x}\right )-24 \text{PolyLog}\left (3,e^{2 x}\right )+6 x^4-16 x^3+48 x^2 \log \left (1-e^{2 x}\right )+4 x^3 \sinh (2 x)-6 x^2 \cosh (2 x)-16 x^3 \coth (x)+6 x \sinh (2 x)-3 \cosh (2 x)+2 i \pi ^3\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3*Cosh[x]^2*Coth[x]^2,x]

[Out]

((2*I)*Pi^3 - 16*x^3 + 6*x^4 - 3*Cosh[2*x] - 6*x^2*Cosh[2*x] - 16*x^3*Coth[x] + 48*x^2*Log[1 - E^(2*x)] + 48*x
*PolyLog[2, E^(2*x)] - 24*PolyLog[3, E^(2*x)] + 6*x*Sinh[2*x] + 4*x^3*Sinh[2*x])/16

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 117, normalized size = 1.2 \begin{align*}{\frac{3\,{x}^{4}}{8}}+ \left ( -{\frac{3}{32}}+{\frac{3\,x}{16}}-{\frac{3\,{x}^{2}}{16}}+{\frac{{x}^{3}}{8}} \right ){{\rm e}^{2\,x}}+ \left ( -{\frac{3}{32}}-{\frac{3\,x}{16}}-{\frac{3\,{x}^{2}}{16}}-{\frac{{x}^{3}}{8}} \right ){{\rm e}^{-2\,x}}-2\,{\frac{{x}^{3}}{{{\rm e}^{2\,x}}-1}}-2\,{x}^{3}+3\,{x}^{2}\ln \left ({{\rm e}^{x}}+1 \right ) +6\,x{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) -6\,{\it polylog} \left ( 3,-{{\rm e}^{x}} \right ) +3\,{x}^{2}\ln \left ( 1-{{\rm e}^{x}} \right ) +6\,x{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) -6\,{\it polylog} \left ( 3,{{\rm e}^{x}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(x)^2*coth(x)^2,x)

[Out]

3/8*x^4+(-3/32+3/16*x-3/16*x^2+1/8*x^3)*exp(2*x)+(-3/32-3/16*x-3/16*x^2-1/8*x^3)*exp(-2*x)-2*x^3/(exp(2*x)-1)-
2*x^3+3*x^2*ln(exp(x)+1)+6*x*polylog(2,-exp(x))-6*polylog(3,-exp(x))+3*x^2*ln(1-exp(x))+6*x*polylog(2,exp(x))-
6*polylog(3,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^3*cosh(x)^2*coth(x)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [C]  time = 2.32171, size = 2541, normalized size = 24.91 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(x)^2*coth(x)^2,x, algorithm="fricas")

[Out]

1/32*((4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^6 + 6*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)*sinh(x)^5 + (4*x^3 - 6*x^2 + 6
*x - 3)*sinh(x)^6 + (12*x^4 - 68*x^3 + 6*x^2 - 6*x + 3)*cosh(x)^4 + (12*x^4 - 68*x^3 + 15*(4*x^3 - 6*x^2 + 6*x
- 3)*cosh(x)^2 + 6*x^2 - 6*x + 3)*sinh(x)^4 + 4*(5*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^3 + (12*x^4 - 68*x^3 + 6
*x^2 - 6*x + 3)*cosh(x))*sinh(x)^3 + 4*x^3 - (12*x^4 + 4*x^3 + 6*x^2 + 6*x + 3)*cosh(x)^2 + (15*(4*x^3 - 6*x^2
+ 6*x - 3)*cosh(x)^4 - 12*x^4 - 4*x^3 + 6*(12*x^4 - 68*x^3 + 6*x^2 - 6*x + 3)*cosh(x)^2 - 6*x^2 - 6*x - 3)*si
nh(x)^2 + 6*x^2 + 192*(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)*s
inh(x)^2 + 2*(2*x*cosh(x)^3 - x*cosh(x))*sinh(x))*dilog(cosh(x) + sinh(x)) + 192*(x*cosh(x)^4 + 4*x*cosh(x)*si
nh(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))*d
ilog(-cosh(x) - sinh(x)) + 96*(x^2*cosh(x)^4 + 4*x^2*cosh(x)*sinh(x)^3 + x^2*sinh(x)^4 - x^2*cosh(x)^2 + (6*x^
2*cosh(x)^2 - x^2)*sinh(x)^2 + 2*(2*x^2*cosh(x)^3 - x^2*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 96*(x^2
*cosh(x)^4 + 4*x^2*cosh(x)*sinh(x)^3 + x^2*sinh(x)^4 - x^2*cosh(x)^2 + (6*x^2*cosh(x)^2 - x^2)*sinh(x)^2 + 2*(
2*x^2*cosh(x)^3 - x^2*cosh(x))*sinh(x))*log(-cosh(x) - sinh(x) + 1) - 192*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + s
inh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cosh(x)^3 - cosh(x))*sinh(x))*polylog(3, cosh(x) + s
inh(x)) - 192*(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*co
sh(x)^3 - cosh(x))*sinh(x))*polylog(3, -cosh(x) - sinh(x)) + 2*(3*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^5 + 2*(12*
x^4 - 68*x^3 + 6*x^2 - 6*x + 3)*cosh(x)^3 - (12*x^4 + 4*x^3 + 6*x^2 + 6*x + 3)*cosh(x))*sinh(x) + 6*x + 3)/(co
sh(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**3*cosh(x)**2*coth(x)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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^3*cosh(x)^2*coth(x)^2,x, algorithm="giac")

[Out]

integrate(x^3*cosh(x)^2*coth(x)^2, x)